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[Перевод] KAN 2.0: Kolmogorov-Arnold Networks Meet Science

Предлагаю полный перевод второй статьи на тему нейронных сетей на основе алгоритма Колмогорова-Арнольда (KAN), опубликованной в августе 2024 года. В этой работе исследователи продолжают развивать тему KAN, раскрывают ее связь с наукой, а также приводят некоторые практические советы по использованию библиотеки pykan, написанной на python, в которой реализован алгоритм KAN.
Перевод первой статьи размещен по адресу https://habr.com/ru/articles/856776/.

Аннотация

Основная проблема ИИ + Наука (AI + Science) заключается в их изначальной несовместимости: сегодняшний ИИ в первую очередь основан на коннекционизме, в то время как наука опирается на символизм. Чтобы соединить два мира, мы предлагаем структуру для бесшовной синергии сетей Колмогорова-Арнольда (KAN) и науки. Структура подчеркивает использование KAN для трех аспектов научного открытия: определение соответствующих признаков, выявление модульных структур и открытие символических формул. Синергия двунаправленная: наука к KAN (включение научных знаний в KAN) и KAN к науке (извлечение научных идей из KAN). Мы выделяем основные новые функции в pykan: (1) MultKAN: KAN с узлами умножения. (2) kanpiler: компилятор KAN, который компилирует символические формулы в KAN. (3) преобразователь деревьев: преобразование KAN (или любых нейронных сетей) в древовидные графы. На основе этих инструментов мы демонстрируем способность KAN открывать различные типы физических законов, включая сохраняющиеся величины, лагранжианы, симметрии и конститутивные законы.

Рисунок 1: Синергия науки и сети Колмогорова-Арнольда (KAN)

1 Введение

В последние годы ИИ + Наука стали многообещающей новой областью, что привело к значительным научным достижениям, включая прогнозирование сворачивания белков [37], автоматическое доказательство теорем [95, 83], прогноз погоды [41] и другие. Общей чертой этих задач является то, что все они могут быть хорошо сформулированы в виде проблем с четкими целями, оптимизируемыми системами ИИ-черного ящика. Хотя эта парадигма работает исключительно хорошо для прикладной науки, существует другой вид науки: наука, движимая любопытством. В исследованиях, движимых любопытством, процедура является более исследовательской, часто не имеющей четких целей, кроме «получения большего понимания». Для ясности, наука, движимая любопытством, далеко не бесполезна; совсем наоборот. Научные знания и понимание, полученные посредством любопытства, часто закладывают прочную основу для завтрашних технологий и способствуют широкому спектру приложений.

Хотя и прикладная, и любознательная наука бесценны и незаменимы, они задают разные вопросы. Когда астрономы наблюдают движение небесных тел, прикладные исследователи сосредотачиваются на прогнозировании их будущих состояний, в то время как любознательные исследователи изучают физику, лежащую в основе движения. Другим примером является AlphaFold, которая, несмотря на свой огромный успех в прогнозировании белковых структур, остается в сфере прикладной науки, поскольку она не предоставляет новых знаний на более фундаментальном уровне (например, атомных сил). Гипотетически AlphaFold должен был открыть важную неизвестную физику, чтобы достичь своих высокоточных предсказаний. Однако эта информация остается скрытой от нас, оставляя AlphaFold в значительной степени черным ящиком. Поэтому мы выступаем за новые парадигмы ИИ для поддержки любознательной науки. Эта новая парадигма ИИ + Наука требует более высокой степени интерпретируемости и интерактивности в инструментах ИИ, чтобы их можно было легко интегрировать в научные исследования.

Недавно новый тип нейронной сети, называемый сетью Колмогорова-Арнольда (KAN) [57], показал себя многообещающим для задач, связанных с наукой. В отличие от многослойных персептронов (MLP), которые имеют фиксированные функции активации на узлах, KAN имеют обучаемые функции активации на ребрах. Поскольку KAN могут раскладывать многомерные функции на одномерные функции, интерпретируемость может быть получена путем символической регрессии этих 1D-функций. Однако их определение интерпретируемости несколько узкое, поскольку оно приравнивается почти исключительно к способности извлекать символические формулы. Это ограниченное определение сужает сферу их применения, поскольку символические формулы не всегда необходимы или возможны в науке. Например, если в физике символьные уравнения являются мощным инструментом и широко распространены, то в химии и биологии системы часто слишком сложны, чтобы их можно было представить с помощью таких уравнений. В этих областях модульные структуры и ключевые особенности могут быть достаточными для характеристики интересных аспектов этих систем. Еще одним упущенным из виду аспектом является обратная задача - внедрение знаний в KAN: как мы можем внедрить предварительные знания в KAN в духе обучения на основе физики?

Мы совершенствуем и расширяем KAN, чтобы их было легко использовать в науке, движимой любопытством. Цель этой статьи можно кратко сформулировать следующим образом:

Чтобы быть более конкретным, научные объяснения могут иметь разные уровни, начиная от самых грубых / простых / корреляционных и заканчивая самыми тонкими / сложными / причинно-следственными:
• Важные характеристики: Например, " полностью определяется x_1 и x_2 , в то время как другие факторы не имеют значения". Другими словами, существует функция , такая, что y = f(x_1, x_2) .
• Модульные структуры: например, " x_1 и x_2 независимо друг от друга аддитивно влияют на ". Это означает, что существуют такие функции и , что y = g(x_1) + h(x_2) .
• Символические формулы: Например, " зависит от x_1 как синусоидальная функция и от x_2 как экспоненциальная функция". Другими словами, y = sin(x_1) + exp(x_2) .

В статье рассказывается о том, как включить и извлечь эти свойства из KAN. Структура статьи выглядит следующим образом (показано на рисунке 1): В разделе 2 мы дополняем исходную CAN узлами умножения, представляя новую модель под названием MultKAN. В разделе 3 мы исследуем способы внедрения научных индуктивных методов в KAN, уделяя особое внимание важным функциям (раздел 3.1), модульным структурам (раздел 3.2) и символическим формулам (раздел 3.3).

Рисунок 2: Вверху: сравнение диаграмм KAN и MultKAN. MultKAN имеет дополнительные слои умножения M. Внизу: После обучения на KAN реализует алгоритм, требующий двух узлов сложения, тогда как MultKAN требует только один узел умножения.

Рисунок 2: Вверху: сравнение диаграмм KAN и MultKAN. MultKAN имеет дополнительные слои умножения M. Внизу: После обучения на f(x, y) = xy

KAN реализует алгоритм, требующий двух узлов сложения, тогда как MultKAN требует только один узел умножения.

В разделе 4 мы предлагаем методы извлечения научных знаний из KAN, снова охватывая важные функции (раздел 4.1), модульные структуры (раздел 4.2) и символьные формулы (раздел 4.3). В разделе 5 мы применяем KAN к различным задачам научного поиска, используя инструменты, разработанные в предыдущих разделах. Эти задачи включают в себя обнаружение сохраняющихся величин, симметрий, лагранжианов и определяющих законов. Коды доступны по адресу https://github.com/KindXiaoming/pykan и также могут быть установлены с помощью pip install pykan. Хотя статья называется “KAN 2.0”, выпущенная версия pykan - 0.2.x.

2 MultKAN: Увеличение количества KAN за счет умножения

Теорема о представлении Колмогорова-Арнольда (Kolmogorov-Arnold representation theorem KART) утверждает, что любая непрерывная функция большой размерности может быть разложена на конечную композицию одномерных непрерывных функций и переменной:

Это означает, что сложение является единственной истинной многомерной операцией, в то время как другие многомерные операции (включая умножение) могут быть выражены как операции сложения в сочетании с одномерными функциями. Например, чтобы умножить два положительных числа x и y, мы можем выразить это как xy = exp(logx + logy) *, правая часть которого состоит только из сложения и одномерных функций (log и exp). (*Значения и могут быть отрицательными, поэтому можно использовать большое значение c > 0 и выразить xy = exp(log(x + c) + log(y +c)) − c(x + y) − c^2 . Другие конструкции включают квадратичные функции, такие как xy = ((x + y)^2 − (x − y)^2)/4 или xy = ((x + y)^2 − x^2 − y^2)/2) .
Однако, учитывая распространенность умножений как в науке, так и в повседневной жизни, желательно явно включать умножения в KAN, что потенциально может повысить как интерпретируемость, так и потенциал модели.

Kolmogorov-Arnold Network (KAN) В то время как уравнение KART (1) соответствует двухслойной сети, Лю и др. [57] удалось расширить его на произвольную глубину, признав, что кажущиеся различными внешние функции Φ_q и внутренние функции $ϕ_{q,p}$ могут быть объединены с помощью предложенных ими слоев KAN. Глубину L KANN можно построить, просто наложив L слоев KAN. Форма глубины L KAN представлена целочисленным массивом [n_0, n_1, ··· , n_L] , где n_l обозначает количество нейронов в -м нейронном слое. Слой KAN с входными размерами n_l и выходными размерами $n_{l+1}$ преобразует входной вектор $x_l \in \mathbb{R}^{n_l}$ в $x_{l+1} \in \mathbb{R}^{n_{l+1}}$

и вся сеть представляет собой композицию из L слоев KAN, т.е.,

На диаграммах KAN можно интуитивно представить сеть, состоящую из узлов (суммирование) и ребер (обучаемые активации), как показано на рисунке 2 вверху слева. При обучении на наборе данных, сгенерированном из f(x, y) = xy , KAN (рисунок 2 внизу слева) использует два дополнительных узла, что делает неясным, действия сети. Однако после некоторого размышления мы понимаем, что она использует равенство xy = ((x + y)^2 − (x − y)^2)/4 , но это далеко не очевидно.

Multiplicative Kolmogorov-Arnold Networks (MultKAN) Для явного введения операций умножения мы предлагаем MultKAN, который может более четко выявлять мультипликативные структуры в данных. MultKAN (показан на рисунке 2 вверху справа) похож на KAN, причем оба имеют стандартные слои KAN. Мы называем входные узлы слоя KAN узлами, а выходные узлы слоя KAN подузлами. Разница между KAN и MultKAN заключается в преобразованиях из подузлов текущего слоя в узлы следующего слоя. В KAN узлы напрямую копируются из подузлов предыдущего слоя. В MultKAN некоторые узлы (узлы сложения) копируются из соответствующих подузлов, в то время как другие узлы (узлы умножения) выполняют умножение на k подузлах из предыдущего слоя. Для простоты далее мы установим k = 2 *.
*(Для простоты мы задали k = 2 , но пакет pykan позволяет использовать любое целое число k ≥ 2 . Пользователи даже могут задавать разные значения для разных узлов умножения. Однако если в одном слое используются разные значения , распараллеливание этих умножений может быть затруднено.)

На основе диаграммы MultKAN (рисунок 2 вверху справа) можно интуитивно понять, что MultKAN — это обычный KAN с необязательными вставленными умножениями. Для математической точности мы определяем следующие обозначения: Количество операций сложения (умножения) в слое обозначается как n_l^a ( n_l^m) соответственно. Они собираются в массивы: ширина сложения n^a ≡[n_0^a,n_1^a,⋅⋅⋅n_L ^a] и ширина умножения n^m ≡[n_0^m,n_1^m,⋅⋅⋅n_L ^m] . Когда n_0^m = n_1^m =⋅⋅⋅n_L^m=0 , MultKAN сводится к KAN. Например, на рисунке 2 (вверху справа) показан MultKAN с $\mathbf{n}^a = [2, 2, 1]$ и $\mathbf{n}^m = [0, 2, 0]$ .

Слой MultKAN состоит из стандартного KANLayer $\mathbf{Φ}_l$ и слоя умножения M_l . $\mathbf{Φ}_l$ принимает входной вектор $x_l ∈ \mathbb{R}^{n_l^a+n_l^m}$ и выводит $\mathbf{z}_l=\mathbf{\Phi}_l(x) \in \mathbb{R}^{n_{l+1}^a + 2n_{l+1}^m}$ . Слой умножения состоит из двух частей: часть умножения выполняет умножения пар подузлов, в то время как другая часть выполняет тождественное преобразование. Написанный на Python, M_l преобразует $\mathbf{z}_l$ следующим образом:

где — поэлементное умножение. MultKANLayer можно кратко представить как Ψ_l≡M_l∘ Φ_l . Таким образом весь MultKAN:

Рисунок 3: Добавление вспомогательных переменных к входным данным улучшает интерпретируемость. Для уравнения релятивистской массы , (a) двухслойная KAN необходима, если в качестве входных данных используются только . (b) Если мы добавим и в качестве вспомогательных переменных к KAN, то будет достаточно однослойной KAN (seed 0). (c) seed 1 находит другое решение, которое не является оптимальным и которого можно избежать с помощью проверки гипотез (раздел 4.3).

Рисунок 3: Добавление вспомогательных переменных к входным данным улучшает интерпретируемость. Для уравнения релятивистской массы ${m_0}/{\sqrt{{1-v^2}/c^2}}$ , (a) двухслойная KAN необходима, если в качестве входных данных используются только (m0, v, c)

. (b) Если мы добавим β≡v/c

и $γ ≡1/\sqrt{1-β^2}$ в качестве вспомогательных переменных к KAN, то будет достаточно однослойной KAN (seed 0). (c) seed 1 находит другое решение, которое не является оптимальным и которого можно избежать с помощью проверки гипотез (раздел 4.3).

Поскольку в слоях умножения нет обучаемых параметров, все методы разреженной регуляризации (например, ℓ_1 и энтропийная регуляризация) для KAN [57] могут быть напрямую применены к MultKAN. Для задачи умножения f(x, y) = xy MultKAN действительно учится использовать один узел умножения, что позволяет ему выполнять простое умножение, поскольку все изученные функции активации являются линейными (рисунок 2 внизу справа).
Хотя KAN ранее рассматривались как особый случай MultKAN, мы расширяем определение и рассматриваем «KAN» и «MultKAN» как синонимы. По умолчанию, когда мы ссылаемся на KAN, умножение разрешено. Если мы конкретно ссылаемся на KAN без умножения, мы явно это укажем.

3 Наука в KAN

В науке знание предметной области имеет решающее значение, позволяя нам эффективно работать даже с небольшими или нулевыми данными. Поэтому полезно принять подход, основанный на физике, для KAN: мы должны включить доступные индуктивные предубеждения в KAN, сохраняя их гибкость для открытия новой физики из данных.
Мы исследуем три типа индуктивных предубеждений, которые могут быть интегрированы в KAN. От самых грубых/легких/корреляционных до самых тонких/сложных/причинных – это: важные признаки (раздел 3.1), модульные структуры (раздел 3.2) и символические формулы (раздел 3.3).

3.1 Добавление важных признаков в KAN

В задаче регрессии цель состоит в том, чтобы найти функцию , такую, что y = f(x_1, x_2, · · · , x_n) . Предположим, мы хотим ввести вспомогательную входную переменную a = a(x_1, x_2,···, x_n) , преобразуя функцию в y = f(x_1,···,x_n, a) . Хотя вспомогательная переменная a не добавляет новой информации, она может увеличить выразительную силу нейронной сети. Это связано с тем, что сети не нужно тратить ресурсы на вычисление вспомогательной переменной. Кроме того, вычисления могут стать проще, что приведет к улучшению интерпретируемости. Пользователи могут добавлять вспомогательные функции к входам с помощью метода augment_input:

В качестве примера рассмотрим формулу для релятивистской массы $m(m_0,v,c) = m_0/\sqrt{1 -(v/c)^2}$ , где — масса покоя, — скорость точечной массы, а — скорость света. Поскольку физики часто работают с безразмерными числами β ≡ v/c и $γ ≡ 1/\sqrt{1 - β^2} ≡ 1/\sqrt{1 - (v/c)^2}$ , они могут ввести и наряду с и в качестве входных данных. Рисунок 3 показывает KAN с этими вспомогательными переменными и без них: (a) иллюстрирует KAN, составленный из символической формулы (см. раздел 3.3 для компилятора KAN), которая требует 5 ребер; (b)(c) показывает KAN со вспомогательными переменными, требующими только 2 или 3 ребра и достигающими потерь 10⁻⁶ и 10⁻⁴ соответственно. Обратите внимание, что (b) и (c) отличаются только величиной seed. Seed равный 1 представляет собой неоптимальное решение, поскольку оно также определяет β = v/c как ключевой признак (feature). Это неудивительно, так как в классическом пределе v ≪ c , $γ ≡ 1/\sqrt{1 - (v/c)^2} ≈ 1 + (v/c)^2/2 = 1 + β^2/2$ . Разнообразие, вызванное различными seed, можно рассматривать либо как особенность, либо как ошибку (bug): Как особенность, это разнообразие может помочь найти неоптимальные решения, которые, тем не менее, могут предложить интересные идеи; как ошибка, оно может быть устранено с помощью метода проверки гипотез, предложенного в разделе 4.3.

Рисунок 4: Построение модульных структур для KAN: (a) мультипликативная разделимость; (b) симметрии

3.2 Строительство модульных конструкций для KAN

Модульность распространена в природе: например, кора головного мозга человека разделена на несколько функционально различных модулей, каждый из которых отвечает за определенные задачи, такие как восприятие или принятие решений. Эта модульность упрощает понимание нейронных сетей, поскольку позволяет нам интерпретировать кластеры нейронов совместно, а не анализировать каждый нейрон по отдельности.
Структурная модульность характеризуется кластерами связей, где внутрикластерные связи намного сильнее межкластерных. Чтобы обеспечить модульность, мы вводим модульный метод, который сохраняет внутрикластерные связи, удаляя межкластерные связи. Модули указываются пользователями. Синтаксис:

Например, если пользователь хочет назначить модулю определенные узлы/подузлы - скажем, 0-й узел в слой 1, 1-й и 3-й подузлы в слой 1, 1-й и 3-й узлы в слой 2 - он может использовать module(1, '[0] -> [1,3] -> [1,3]'). Если говорить конкретно, то существует два типа модульности: разделимость и симметрия.

Разделимость Мы говорим, что функция считается разделимой, если ее можно выразить как сумму или произведение функций непересекающихся групп переменных. Например, функция четырех переменных f(x_1, x_2, x_3, x_4) является максимально мультипликативно разделимой, если она имеет вид f_1(x_1)f_2(x_2)f_3(x_3)f_4(x_4) , создавая четыре различные группы (1), (2), (3), (4). Пользователи могут создавать эти модули, вызывая метод module четыре раза: module(0, '[ i ]->[ i ]'), = 0, 1, 2, 3, как показано на рисунке 4 (а). Последний вызов можно пропустить, так как первых трех достаточно для определения групп. Более слабые формы мультипликативной разделимости могут быть f_1 (x_1, x_2)f_2 (x_3,x_4) (вызов module(0, '[ 0,1 ]->[ 0,1 ]' )) или f_1 (x_1)f_2 (x_2,x_3,x_4) (вызов module(0, '[ 0 ]->[ 0 ]' )).

Обобщенная симметрия Мы говорим, что функция симметрична относительно переменных (x_1,x_2) , если f(x_1, x_2,x_3,· · · ) = g(h(x_1, x_2),x_3,· · · ) . Это свойство называется симметрией, поскольку значение остается неизменным до тех пор, пока h(x_1,x_2) остается постоянным, даже если x_1 и x_2 изменяются. Например, функция является вращательно-инвариантной в 2D, если f(x_1,x_2) = g(r) где $r ≡ \sqrt{x_1^2 + x_2^2}$ . Когда симметрия затрагивает только подмножество переменных, ее можно считать иерархической, поскольку x_1 и x_2 сначала взаимодействуют через (2-слойный KAN), а затем взаимодействует с другими переменными через (2-слойный KAN). Предположим, что функция с четырьмя переменными имеет иерархическую форму f(x_1,x_2,x_3,x_4) = h(f(x_1,x_2),g(x_3,x_4)) ,

Рисунок 5: Компилятор KAN (kanpiler) преобразует символьные выражения в KAN. (a) Как работает kanpiler: символьная формула сначала разбирается на дерево выражений, которое затем преобразуется в KAN. (b) Применение KAN к 10 уравнениям (выбранным из набора данных Фейнмана). (c) Расширение скомпилированного KAN для увеличения его выразительной мощности.

как показано на рисунке 4 (b). Мы можем использовать метод module для создания этой структуры, вызвав module(0, '[ 0,1 ] -> [ 0,1 ] -> [ 0,1 ] -> [ 0 ]'), гарантируя, что группы переменных (x_1,x_2) и (x_3,x_4) не будут взаимодействовать в первых двух слоях.

3.3 Составление символических формул в KAN

Ученые часто находят удовлетворение в представлении сложных явлений посредством символических уравнений. Однако, хотя эти уравнения и лаконичны, им может не хватать выразительной силы, необходимой для охвата всех нюансов из-за их специфических функциональных форм. Напротив, нейронные сети очень выразительны, но могут неэффективно тратить время обучения и данные на изучение знаний предметной области, уже известных ученым. Чтобы использовать сильные стороны обоих подходов, мы предлагаем двухэтапную процедуру:
(1) компилировать символические уравнения в KAN и
(2) настраивать эти KAN с использованием данных.
Первый шаг направлен на внедрение известных знаний предметной области в KAN, в то время как второй шаг фокусируется на изучении новой «физики» из данных.

kanpiler (KAN compiler) Цель kanpiler — преобразовать символическую формулу в KAN. Процесс, показанный на рисунке 5 (a), включает три основных шага: (1) Символическая формула разбирается на древовидную структуру, где узлы представляют выражения, а ребра обозначают операции/функции. (2) Затем это дерево модифицируется для приведения в соответствие со структурой графа KAN. Изменения включают перемещение всех листовых узлов на входной слой через фиктивные ребра и добавление фиктивных подузлов/узлов для соответствия архитектуре KAN. Эти фиктивные ребра/узлы/подузлы выполняют только тождественное преобразование. (3) Переменные объединяются в первом слое, эффективно преобразуя дерево в граф. Для наглядности на ребрах размещены одномерные кривые для представления функций. Мы протестировали kanpiler на наборе данных Фейнмана, и он успешно обрабатывает все 120 уравнений. Примеры показаны на рисунке 5 (b). Kanpiler принимает входные переменные (как символы sympy) и выходное выражение (как выражение sympy) и возвращает модель KAN.

Обратите внимание, что возвращаемая модель KAN находится в символическом режиме, т.е. символические функции точно закодированы. Если вместо этого мы используем кубические сплайны для аппроксимации этих символических функций, мы получим потери MSE $ℓ ∝ N^{−8}$ [57], где N — число интервалов сетки (пропорционально числу параметров модели).
Расширение ширины/глубины для увеличения выразительной мощности Сеть KAN, сгенерированная kanpiler, компактна, без избыточных ребер, которые могут ограничить ее выразительную мощность и помешать дальнейшей тонкой настройке. Для решения этой проблемы мы предлагаем методы expand_width и expand_depth для расширения сети, чтобы она стала шире и глубже, как показано на рисунке 5 (c). Методы расширения изначально добавляют нулевые функции активации, которые страдают от нулевых градиентов во время обучения. Поэтому метод возмущения следует использовать для возмущения этих нулевых функций до ненулевых значений, что делает их обучаемыми с ненулевыми градиентами.

4 KAN в науку

Сегодняшние черные ящики глубоких нейронных сетей являются мощными, но интерпретация этих моделей остается сложной задачей. Ученые ищут не только высокопроизводительные модели, но и возможность извлекать значимые знания из моделей. В этом разделе мы сосредоточимся на повышении интерпретируемости научных целей KAN. Мы рассмотрим три уровня извлечения знаний из KAN, от самых базовых до самых сложных: важные признаки (раздел 4.1), модульные структуры (раздел 4.2) и символические формулы (раздел 4.3).

4.1 Определение важности переменных KAN

Определение важности переменных имеет решающее значение для многих задач. Учитывая регрессионную модель , где y ≈ f(x_1 ,x_2 ,...,x_n) , мы стремимся назначить баллы (scores) входным переменным для оценки их важности. Лю и др. [57] использовали функцию L1 norm для указания важности ребер, но эта метрика может быть проблематичной, поскольку она учитывает только локальную информацию.

Чтобы решить эту проблему, мы вводим более эффективную оценку атрибуции, которая лучше отражает важность переменных, чем норма L1. Для простоты предположим, что есть узлы умножения, поэтому нам не нужно различать узлы и подузлы* (*для подузлов, принадлежащих узлу умножения, подузлы наследуют свои оценки от узла умножения). Предположим, что у нас есть L-слой KAN с шириной [n_0, n_1, · · · , n_L] . Мы определяем $E_{l,i,j}$ как стандартное отклонение активаций на ребре (l, i, j) , а $N_{l,i}$ как стандартное отклонение активаций на узле (l, i) . Затем мы определяем оценку узла (атрибуции) $A_{l,i}$ и оценку ребра (атрибуции) $B_{l,i,j}$ . В [57] мы просто определили $B_{l,i,j} = E_{l,i,j}$ и $A_{l,i} = N_{l,i}$ . Однако это определение не учитывает более поздние сегменты сети; даже если узел или ребро имеет бо́льшую норму, они могут не вносить вклад в выход, если остальная часть сети фактически является нулевой функцией. Поэтому теперь мы вычисляем оценки узлов и ребер итеративно от выходного слоя до входного слоя. Мы устанавливаем все выходные измерения так, чтобы они имели единичные оценки, т. е. $A_{L,i} = 1$ , i = 0, 1, · · · , n_L – 1 * , и вычисляем оценки следующим образом:
(*другие варианты могут быть сделаны на основе воспринимаемой важности каждого выходного измерения, хотя это менее критично, когда выходные данные, как правило, одномерны)

Рисунок 6: Определение важных признаков в KAN. (a) сравнение оценки атрибуции с нормой L1, используемой в работе Лю и др. [57]. В двух синтетических задачах оценка атрибуции дает больше информации, чем норма L1. (b) Оценки атрибуции можно вычислить для входных данных и использовать для обрезки входных данных.

Сравнение $E_{l,i,j}$ и $B_{l,i,j}$ Мы обнаружили, что $B_{l,i,j}$ обеспечивает более точное отражение важности ребер. На рисунке 6 мы сравниваем KAN, обученные на двух уравнениях y = exp(sin(πx_1) + x_2^2) и y= (x_1^2 + x_2^2)^2 + (x_3^2+ x_4^2)^2 , и визуализируем KAN с оценками важности (L1 norm) или (оценка атрибуции). Для первого уравнения оценки атрибуции показывают более чистый график, чем нормы L1, так как многие активные ребра в первом слое не вносят вклад в конечный результат из-за неактивных последующих ребер. Оценка атрибуции учитывает это, что приводит к более осмысленному графику. Для второго уравнения y= (x_1^2 + x_2^2)^2 + (x_3^2+ x_4^2)^2 , мы можем сказать исходя из её символического вида, что все четыре переменные одинаково важны. Оценки атрибуции правильно отражают равную важность всех четырех переменных, тогда как норма L1 неверно предполагает, что x_3 и x_4 важнее, чем x_1 и x_2 .
Отсечение входных данных на основе оценок атрибуции В реальных наборах данных размерность входных данных может быть большой, но только несколько переменных могут быть релевантными. Чтобы решить эту проблему, мы предлагаем отсечение нерелевантных признаков на основе оценок атрибуции, чтобы мы могли сосредоточиться на наиболее релевантных. Пользователи могут применить prune_input, чтобы сохранить только наиболее релевантные переменные. Например, если есть 100 входных признаков, упорядоченных по убыванию релевантности в функции $y =∑_{i=0}^{99} {x_i^2/2^i}$ , x_i ∈ [-1,1] , и после обучения только первые пять признаков показывают значительно более высокие оценки атрибуции, метод prune_input сохранит только эти пять признаков. Отсеченная сеть становится компактной и интерпретируемой, тогда как исходная KAN со 100 входными данными слишком плотна для прямой интерпретации.

4.2 Определение модульных структур из KAN

Хотя оценка атрибуции дает ценную информацию о том, какие ребра или узлы важны, она не раскрывает модульные структуры, т. е. как связаны важные ребра и узлы. В этой части мы стремимся раскрыть модульные структуры из обученных KAN и MLP, исследуя два типа модульности: анатомическую модульность и функциональную модульность.

Рисунок 7: Индуцирование анатомической модульности в нейронных сетях посредством обмена нейронами. Подход включает назначение пространственных координат нейронам и их перестановку для минимизации общей стоимости соединения. Для двух задач (слева: многозадачный паритет, справа: иерархическое голосование большинства) обмен нейронами работает для KAN (вверху) в обоих случаях и работает для MLP (внизу) для многозадачного паритета.

4.2.1 Анатомическая модульность

Анатомическая модульность относится к тенденции нейронов, расположенных близко друг к другу в пространстве, иметь более сильные связи, чем те, которые находятся дальше друг от друга. Хотя искусственные нейронные сети не имеют физических пространственных координат, было показано, что введение концепции физического пространства улучшает интерпретируемость [51, 52]. Мы принимаем метод обмена нейронами из [51, 52], который сокращает связи, сохраняя при этом функциональность сети. Мы называем этот метод auto_swap. Анатомическая модульная структура, выявленная путем обмена нейронами, облегчает идентификацию модулей, даже визуально, для двух задач, показанных на рисунке 7: (1) многозадачный разреженный паритет; и (2) иерархическое голосование большинства. Для многозадачной разреженной четности у нас есть 10 входных бит $x_i ∈ \{0, 1\}, i = 1, 2, ···, 10$ , и выход $y_j = x_{2j−1} ⊕ x_{2j}, j = 1,···, 5$ , где обозначает сложение по модулю 2. Задача демонстрирует модульность, поскольку каждый выход зависит только от подмножества входов. Функция auto_swap успешно идентифицирует модули как для KAN, так и для MLP, при этом KAN обнаруживает более простые модули. Для иерархического голосования большинством с 9 входными битами $x_i ∈ \{0, 1\}$ , i=1,···,9 и выходом y = maj(maj(x_1,x_2,x_3),maj(x_4,x_5,x_6),maj(x_7,x_8,x_9)) , где maj обозначает голосование большинством (выход 1, если два или три входа равны 1, в противном случае 0). KAN раскрывает модульную структуру даже до auto_swap, и диаграмма становится более организованной после auto_swap. MLP показывает некоторую модульную структуру из шаблона весов первого слоя, указывая на взаимодействия между переменными, но глобальная модульная структура остается неясной независимо от auto_swap.

4.2.2 Функциональная модульность

Функциональная модульность относится к общей функции, представленной нейронной сетью. Учитывая сеть Oracle, где внутренние детали, такие как веса и активации скрытых слоев, недоступны (слишком сложны для анализа), мы все равно можем собирать информацию о функциональной модульности посредством прямых и обратных проходов на входах и выходах. Мы определяем три типа функциональной модульности (см. Рисунок 8 (a)), основываясь в основном на [84].

Разделимость: Функция f аддитивно разделима, если

Обратите внимание, что $\frac{∂^2 f}{∂x_i ∂x_j }=0$ , когда 1≤i≤k , k+ 1 ≤ j≤ n .

Чтобы обнаружить разделимость, мы можем вычислить матрицу Гессиан

$\mathbf{H} ≡ ∇^T ∇f \left(\mathbf{H}_{ij} = \frac{∂^2 f}{∂x_i ∂x_j}\right)$ и проверить блочную структуру.

Рисунок 8: Обнаружение функциональной модульности в KAN. (a) Мы изучаем три типа функциональной модульности: разделимость (аддитивная или мультипликативная), общая разделимость и симметрия. (b) Применение этих тестов рекурсивно преобразует функцию в дерево. Здесь функция может быть символической функцией (вверху), KAN (в середине) или MLP (внизу). Как KAN, так и MLP создают правильные графы деревьев в конце обучения, но показывают различную динамику обучения.

Если $\mathbf{H}_{ij}=0$ для всех 1≤i≤k и k+1≤j≤n , то мы знаем, что аддитивно разделима. Для мультипликативной разделимости мы можем преобразовать ее в аддитивную разделимость, взяв логарифм:

Для обнаружения мультипликативной разделимости мы определяем $\mathbf{H}_{ij} ≡ \frac{∂^2 log|f|}{∂x_i ∂x_j}$ и

проверяем блочную структуру. Пользователи могут вызвать test_separability для проверки общей разделимости.

Обобщенная разделимость: Функция f имеет обобщенную разделимость, если

Для обнаружения обобщенной отделимости мы вычисляем

где мы использовали $\frac{∂F}{∂g} = \frac{∂F}{∂h}$ . Обратите внимание, что $\frac{{∂f}/{∂x_i}}{∂f/∂x_j}$ является

мультипликативно разделимым, это может быть обнаружено с помощью теста на разделимость, предложенного выше. Пользователи могут вызвать test_general_separability для проверки на аддитивную или мультипликативную разделимость.

Обобщенная симметрия: Функция имеет обобщенную симметрию (по первым k переменным), если

Обозначим $\mathbf{y} = (x_1,· · · ,x_k)$ и $\mathbf{z} = (x_{k+1},· · · ,x_n)$ . Это свойство называется обобщенной симметрией, поскольку сохраняет одно и то же значение до тех пор, пока остается постоянным, независимо от индивидуальных значений x_1,· · · ,x_k . Вычислим градиент f относительно : $∇_y f = \frac{∂g}{∂h} ∇_y h$ . Поскольку $\frac{∂g}{∂h}$ является скалярной функцией, она не

меняет направление ∇_y h . Таким образом, направление $\widehat{{∇_y f}} ≡ \frac{∇_y f}{|∇_y f|}$ не зависит от $\mathbf{z}$ , т. е.

что является условием симметрии. Пользователи могут вызвать метод test_symmetry для проверки симметрии.

Древовидный преобразователь Три типа функциональной модульности образуют иерархию: симметрия является наиболее общей, общая разделимость является промежуточной, а разделимость является наиболее специфической. Математически,

Чтобы получить максимальную иерархию модульных структур, мы применяем обобщенное обнаружение симметрии рекурсивно, формируя группы всего с k = 2 переменными и расширяя их до всех k = n переменных. Например, рассмотрим функцию с 8 переменными

которая имеет четыре обобщенные симметрии k = 2 , включающие группы (x_1,x_2),(x_3,x_4),(x_5,x_6),(x_7,x_8) ; две обобщенные симметрии k = 2 , включающие группы (x_1,x_2,x_3,x_4) и (x_5,x_6,x_7,x_8) . Таким образом, каждая группа k = 4 содержит две группы k = 2 , демонстрируя иерархию. Для каждой обобщенной симметрии мы также можем проверить, является ли обобщенная симметрия далее обобщенно разделимой или разделимой. Пользователи могут использовать метод plot_tree для получения графа дерева для функции (функцией могут быть любые выражения Python, нейронные сети и т. д.). Для модели нейронной сети пользователи могут просто вызвать model.tree(). Граф дерева может иметь стиль «дерево» (по умолчанию) или «ящик» («box»).

Примеры На рисунке 8 (b) приведены два примера. Когда точные символьные функции вводятся в plot_tree, получаются графы истинного дерева. Нас особенно интересует, работает ли преобразователь деревьев для нейронных сетей. Для этих простых случаев и KAN, и MLP могут найти правильный граф, если они достаточно обучены. На рисунке 8 (b) (внизу) показана эволюция графов деревьев во время обучения KAN и MLP. Особенно интересно наблюдать, как нейронные сети постепенно усваивают правильную модульную структуру. В первом случае f(x_1,x_2,x_3,x_4) = (x_1^2 + x_2^2)^2 + (x_3^2 + x_4^2)^2 , и KAN, и MLP постепенно приобретают больше индуктивных смещений (их промежуточные состояния различны), пока не достигнут правильной структуры.

Рисунок 9: Три приема для упрощения символической регрессии. Прием A (верхний ряд): обнаружение и использование модульных структур. Прием B (средний ряд): инициализация разреженных соединений. Прием C (нижний ряд): проверка гипотез.

Во втором случае, $f(x_1,x_2,x_3) = sin(x_1)/\sqrt{x_2^2 + x_3^2}$ , обе модели изначально обнаруживают мультипликативную разделимость для всех трех переменных, показывая даже более высокую симметрию, чем правильная структура. После обучения обе модели «понимают», что: для лучшего соответствия данным (потери становятся ниже), такая высокосимметричная структура больше не может быть выполнена и должна быть ослаблена до менее строгой структуры. Дополнительное наблюдение заключается в том, что KAN имеет промежуточную структуру, не найденную в MLP. Есть два предостережения, которые мы хотели бы упомянуть: (1) результаты могут зависеть от начального значения и/или порога. (2) все тесты полагаются на производные второго порядка, которые могут быть ненадежными из-за того, что модель обучается только на информации нулевого порядка. Состязательные конструкции, такие как $f_ϵ (x) = f(x)+ϵsin \left( \frac xϵ\right)$ могут привести к проблемам, потому что хотя |f_ϵ (x)-f(x)| → 0 при ϵ → 0, |f_ϵ^" (x) - f^"(x)| → ∞ при . Хотя такие крайние случаи маловероятны на практике, гладкость необходима для обеспечения успешности наших методов.

4.3 Определение символических формул из KAN

Символические формулы являются наиболее информативными, поскольку они четко раскрывают как важные характеристики, так и модульные структуры, как только они становятся известны. В работе Лю и др. [57] авторы показали ряд примеров, из которых они могут извлекать символические формулы, имея при необходимости некоторые предварительные знания. С помощью новых инструментов, предложенных выше (важность характеристик, модульные структуры и символические формулы), пользователи могут использовать эти новые инструменты для легкого взаимодействия и совместной работы с KAN, что упрощает символическую регрессию. Ниже мы представляем три приема, проиллюстрированные на рисунке 9.

Прием A: обнаружить и использовать модульные структуры Сначала мы можем обучить общую сеть и исследовать ее модульность. После того, как модульная структура идентифицирована, мы инициализируем новую модель с этой модульной структурой в качестве индуктивных смещений. Например, рассмотрим функцию f(q,v,B,m) = qvB/m Сначала мы инициализируем большую KAN (предположительно достаточно выразительную), чтобы подогнать набор данных с разумной точностью. После обучения из обученной KAN извлекается древовидный граф (ссылка на раздел 4.2), который показывает мультипликативную разделимость. Затем мы можем построить модульную структуру во второй KAN (ссылка на раздел 3.2), обучить ее, а затем символизировать все одномерные функции, чтобы вывести формулу.

Прием B: разреженная инициализация Символические формулы обычно соответствуют KAN с разреженными связями (см. Рисунок 5 (b)), поэтому разреженная инициализация KAN лучше согласует их с индуктивными смещениями символических формул. В противном случае, плотно инициализированные KAN требуют тщательной регуляризации для обеспечения разреженности. Разреженная инициализация может быть достигнута путем передачи аргумента «sparse_init=True» инициализатору KAN. Например, для функции f(q,E,v,B,θ) = q(E + vBsinθ) разреженно инициализированная KAN очень похожа на окончательно обученную KAN, требуя лишь незначительных корректировок в обучении. В отличие от этого, плотная инициализация потребует длительного обучения для удаления ненужных ребер.

Прием C: Проверка гипотезы Когда мы сталкиваемся с несколькими разумными гипотезами, мы можем попробовать их все (ветвясь в «параллельные вселенные»), чтобы проверить, какая гипотеза является наиболее точной и/или самой простой. Чтобы облегчить проверку гипотез, мы создаем систему контрольных точек, которая автоматически сохраняет версии модели всякий раз, когда вносятся изменения (например, обучение, обрезка). Например, рассмотрим функцию $f(m_0,v,c) = m_0/ \sqrt{1 - (v/c)^2}$ . Начнем со случайно инициализированного KAN, имеющего версию 0.0. После обучения он эволюционирует до версии 0.1, где активируется как при β = v/c , так и при $γ = 1/ \sqrt{1 - (v/c)^2}$ . Выдвигаем гипотезу, что могут понадобиться только или . Сначала мы устанавливаем ребро на ноль и обучаем модель, получая $6,5×10^{−4}$ тестового RMSE (версия 0.2). Чтобы проверить альтернативную гипотезу, мы хотим вернуться к точке ветвления (версия 0.1) для чего вызываем model.rewind( '0.1' ), которая возвращает модель обратно к версии 0.1. Чтобы указать, что вызвана перемотка, версия 0.1 переименовывается в версию 1.1. Теперь мы устанавливаем ребро равным нулю, обучаем модель, получая $2.0×10^{−6}$ тестового RMSE (версия становится 1.2). Сравнение версий 0.2 и 1.2 показывает, что вторая гипотеза лучше из-за меньших потерь при одинаковой сложности (обе гипотезы имеют два ненулевых ребра).

5 Применение

Предыдущие разделы в основном были сосредоточены на проблемах регрессии в педагогических целях. В этом разделе мы применяем KAN для обнаружения физических концепций, таких как сохраняющиеся величины, лагранжианы, скрытые симметрии и конститутивные законы. Эти примеры иллюстрируют, как инструменты, предложенные в этой статье, могут быть эффективно интегрированы в реальные научные исследования для решения этих сложных задач.

5.1 Открытие сохраняющихся величин

Рисунок 10: Использование KAN для обнаружения сохраняющихся величин для двумерного гармонического осциллятора.

Сохраняющиеся величины — это физические величины, которые остаются постоянными с течением времени. Например, свободно падающий мяч преобразует свою гравитационную потенциальную энергию в кинетическую энергию, в то время как полная энергия (сумма обеих форм энергии) остается постоянной (при условии пренебрежимо малого сопротивления воздуха). Сохраняющиеся величины имеют решающее значение, поскольку они часто соответствуют симметриям в физических системах и могут упростить вычисления за счет уменьшения размерности системы. Традиционно вывод сохраняющихся величин с помощью бумаги и карандаша может быть трудоемким и требует обширных знаний предметной области. Недавно были исследованы методы машинного обучения для обнаружения сохраняющихся величин [55, 53, 54, 58, 32, 89]. Мы следуем подходу Лю и др. [53], определивших дифференциальное уравнение, которому должны удовлетворять сохраняющиеся величины, тем самым преобразуя проблему поиска сохраняющихся величин в решение дифференциальных уравнений. Они использовали многослойные персептроны (MLP) для параметризации сохраняющихся величин. Мы в основном следуем их процедуре, но заменяем MLP на KAN. Конкретнее, они рассматривают динамическую систему с переменной состояния $\mathbf{z} ∈ \mathbb{R}^d$ , управляемую уравнением $\frac{d \mathbf{z}}{dt} = \mathbf{f}(\mathbf{z})$ . Необходимым и достаточным условием для того, чтобы функция $H(\mathbf{z})$ была сохраняющейся величиной, является выполнение условия $\mathbf{f}(\mathbf{z}) · ∇H(\mathbf{z}) = 0$ для всех $\mathbf{z}$ . Например, в одномерном гармоническом осцилляторе фазовое пространство характеризуется положением и импульсом, $\mathbf{z} = (x,p)$ , а уравнение эволюции имеет вид d(x,p)/dt = (p,-x) . Энергия $H= \frac12 (x^2 + p^2)$ является сохраняющейся величиной, поскольку

$\mathbf{f}(\mathbf{z})·∇H(\mathbf{z}) = (p,-x)·(x,p)=0$ . Мы параметризуем с помощью KAN и обучаем его, используя функции потерь $\ell = ∑_{i=1}^N|\mathbf{f}(\mathbf{z}^{(i)} )· \hat{∇}H{(\mathbf{z}^{\left(i\right)})}|^2$ где $\hat{∇}$ — нормализованный градиент, а $\mathbf{z}^{(i)}$ — i-я точка данных, равномерно взятая из гиперкуба [-1,1]^d .
Мы выбираем 2D гармонический осциллятор для тестирования KAN, характеризующийся (x,y,p_x,p_y) . Он имеет три сохраняющиеся величины: (1) энергия

вдоль направления : $H_1 = \frac12 (x^2 + p_x^2)$ ; (2) энергия вдоль направления :

$H_2 =\frac12 (y^2+p_y^2)$ ; (3) угловой момент H_3 = xp_y-yp_x . Мы обучаем [4, [0, 2], 1] KAN с

тремя различными случайными начальными числами, как показано на рисунке 10, которые соответствуют H1, H2 и H3 соответственно.

5.2 Открытие лагранжианов

Рисунок 11: Использование KAN для изучения лагранжианов для одиночного маятника (вверху) и релятивистской массы в однородном поле (внизу).

В физике механика Лагранжа — это формулировка классической механики, основанная на принципе стационарного действия. Она описывает механическую систему с использованием фазового пространства и гладкой функции L, известной как лагранжиан. Для многих систем $\mathcal{L}= T - V$ , где и представляют собой кинетическую и потенциальную энергию системы соответственно. Фазовое пространство обычно описывается как $(\mathbf{\text{q}},\mathbf{\dot{\text{q}}})$ , где $\mathbf{\text{q}}$ и $\mathbf{\dot{\text{q}}}$ обозначают координаты и скорости соответственно. Уравнение движения может быть

выведено из лагранжиана через уравнение Эйлера-Лагранжа: $\frac d{dt}\left(\frac{∂\mathcal{L}}{∂\mathbf{\dot{q}}}\right) = \frac{∂\mathcal{L}}{∂\mathbf{q}}$ , или эквивалентно

Учитывая фундаментальную роль лагранжиана, интересным вопросом является то, можем ли мы вывести лагранжиан из данных. Следуя [19], мы обучаем лагранжеву нейронную сеть предсказывать $\mathbf{\ddot{q}}$ из $(\mathbf{\text{q}},\mathbf{\dot{\text{q}}})$ . LNN использует MLP для параметризации $\mathcal{L} (\mathbf{\text{q}},\mathbf{\dot{\text{q}}})$ и вычисляет уравнение (18) для предсказания мгновенных ускорений $\mathbf{\ddot{q}}$ . Однако LNN сталкиваются с двумя основными проблемами: (1) Обучение LNN может быть нестабильным из-за производных второго порядка и инверсии матрицы в уравнении (18). (2) LNN не обладают интерпретируемостью, поскольку сами MLP нелегко интерпретировать. Мы решаем эти проблемы с помощью KAN.
Чтобы решить первую задачу, отметим, что матричное обращение гессиана $(\nabla_{\dot{q}} \nabla_{\dot{q}}^T \mathcal{L})^{-1}$ становится проблематичным, когда гессиан имеет собственные значения, близкие к нулю. Чтобы смягчить это, мы инициализируем $(\nabla_{\dot{q}} \nabla_{\dot{q}}^T \mathcal{L})$ как положительно определенную матрицу (или положительное число в 1D). Поскольку $(\nabla_{\dot{q}} \nabla_{\dot{q}}^T \mathcal{L})$ — это масса в классической механике, а кинетическая энергия обычно равна $T=\frac12m\mathbf{\dot{q}}^2$ , кодирование этих априорных знаний в KAN проще, чем в MLP (с использованием kanpiler, представленного в разделе 3.3). Kanpiler может преобразовать символическую формулу T в KAN (как показано на рисунке 11). Мы используем эту преобразованную KAN для инициализации и продолжаем обучение, что приводит к гораздо большей стабильности по сравнению со случайной инициализацией. После обучения символическая регрессия может быть применена к каждому ребру для извлечения символических формул, решая вторую задачу.
Мы показываем два 1D примера на рисунке 11, одиночный маятник и релятивистская масса в однородном поле. Скомпилированные KAN показаны слева, с ребрами на $\dot{q}$ , отображающими квадратичные функции, и ребрами на как нулевые функции.

Одиночный маятник Часть $\dot{q}$ остается квадратичной функцией $T(\dot{q}) = \frac12\dot{q}^2$ , в то время

как часть учится быть функцией косинуса, так как V(q)=1-cos(q) . На рисунке 11 вверху результаты suggest_symbolic отображают пять функций, которые лучше всего соответствуют сплайнам, учитывая как пригодность, так и простоту. Как и ожидалось, косинус и квадратичная функция появляются в верхней части списков.

Релятивистская масса в однородном поле После обучения кинетическая энергия отклоняется от $T = ½ \dot{q}^2$ , поскольку для релятивистской частицы $T_r=(1 - \dot{q}^2)^{-1/2}-1$ . На рисунке 11 (внизу) символическая регрессия успешно находит V(q) = q , но не может идентифицировать T_r из-за своей композиционной природы, поскольку наша символическая регрессия ищет только простые функции. Предполагая, что первая композиция функций является квадратичной, мы создаем еще один [1, 1, 1] KAN для соответствия T_r и устанавливаем первую функцию как квадратичную функцию с помощью fix_symbolic и обучаем только вторую функцию. После обучения мы видим, что истина $x^{−1/2}$ появляется среди пяти лучших кандидатов. Однако $x^{1/2}$ немного лучше подходит для сплайна, на что указывает более высокое значение R-квадрата. Это говорит о том, что символическая регрессия чувствительна к шуму (из-за несовершенного обучения), и априорные знания имеют решающее значение для правильного суждения. Например, знание того, что кинетическая энергия должна стремиться к бесконечности по мере приближения скорости к скорости света, помогает подтвердить, что $x^{−1/2}$ является правильным членом, поскольку $x^{1/2}$ не демонстрирует ожидаемого стремления к бесконечности.

5.3 Обнаружение скрытой симметрии

Рисунок 12: Повторное открытие скрытой симметрии черной дыры Шварцшильда с помощью MLP и KAN. (a) ∆t(r), изученный MLP, является глобально гладким решением; (b) ∆t(r), изученный KAN, является решением "доменной стенки"; (c) KAN показывает всплеск потерь на "доменной стенке"; (d) KAN можно использовать для точной настройки решения MLP, близкой к машинной точности.

Филип Андерсон утверждал, что «было бы лишь небольшим преувеличением сказать, что физика — это изучение симметрии», подчеркивая, насколько открытие симметрии было бесценным как для углубления нашего понимания, так и для более эффективного решения проблем.
Однако симметрии иногда не проявляются, а скрываются, только проявляясь путем применения некоторого преобразования координат. Например, после того, как Шварцшильд открыл свою одноименную метрику черной дыры, Painlevé, Gullstrand и Lemaître потребовалось 17 лет, чтобы раскрыть ее скрытую трансляционную симметрию. Они продемонстрировали, что пространственные сечения можно сделать трансляционно инвариантными с помощью умного преобразования координат, тем самым углубив наше понимание черных дыр [65]. Лю и Тегмарк [56] показали, что преобразование Gullstrand-Painlevé можно обнаружить, обучив MLP за считанные минуты. Однако они не получили чрезвычайно высокой точности (т.е. машинной точности) для решения. Мы пытаемся вернуться к этой проблеме с помощью KAN.
Предположим, что в пространстве-времени (t,x,y,z) имеется черная дыра Шварцшильда с массой 2M = 1 , центром в точке x = y = z = 0 и радиусом r_s=2M=1 . Метрика Шварцшильда описывает, как искажаются пространство и время вокруг нее:

Применяя преобразование Gullstrand-Painlevé $t'=t+2M\left(2u+ln⁡\left(\frac{u-1}{u+1}\right) \right)$ ,

$u ≡ \sqrt{\frac{r}{2M}}$ , x'=x, y'=y, z'=z , метрика в новых координатах становится:

которая демонстрирует трансляционную инвариантность в пространственном сечении (нижний правый блок 3×3 представляет собой евклидову метрику). Лю и Тегмарк [56] использовали многослойный перцептрон для обучения отображению из (t,x,y,z) в

(t',x',y',z') . Определяя матрицу Якоби $\mathbf{J}≡\frac{∂(t', x', y', z')}{∂(t,x,y,z)}$ , $\mathbf{\text{g}}$ преобразуется в

$\mathbf{g}'= \mathbf{J}^{-T}\mathbf{g}\mathbf{J}^{-1}$ . Мы берем нижний правый блок 3×3 $\mathbf{\text{g}}'$ ′ и берем его разность с евклидовой метрикой, чтобы получить потерю MSE. Потери минимизируются путем выполнения градиентных спусков на MLP. Чтобы упростить задачу, они предполагают знание x'=x, y'=y, z'=z и используют только MLP (1 вход и 1 выход) для прогнозирования

временной разницы $∆t(r)=t'-t=2M\left(2u+ln⁡\left(\frac{u-1}{u+1}\right) \right)$ , $u≡\sqrt{\frac{r}{2M}}$ из радиуса .
MLP и KAN находят разные решения Мы обучили как MLP, так и KAN минимизировать эту функцию потерь, и результаты показаны на рисунке 12. Поскольку задача имеет 1 входное измерение и 1 выходное измерение, KAN эффективно сводится к сплайну. Изначально мы ожидали, что KAN превзойдут MLP, поскольку сплайны, как известно, превосходят MLP в низкоразмерных условиях [63]. Однако, в то время как MLP может достичь $10^{−8}$ потерь, KAN застревает на $10^{−3}$ потерях, несмотря на уточнения сетки. Оказалось, что KAN и MLP узнали два разных решения: в то время как MLP нашел глобально гладкое решение (рисунок 12 (a)), KAN узнал решение доменной стенки (рисунок 12 (b)). Решение доменной стенки имеет особую точку, которая разделяет всю кривую на два сегмента. Левый сегмент правильно изучает ∆t(r) , в то время как правый сегмент изучает −∆t(r) , что также является допустимым решением, но отличается от левого сегмента знаком минус. В особой точке появляется пик потерь (рисунок 12 (c)). Можно считать это особенностью KAN, поскольку решения с доменными стенками распространены в природе. Однако, если считать это недостатком, KAN все еще могут получать глобально гладкие решения, добавляя регуляризации (для уменьшения колебаний сплайна) или экспериментируя с различными случайными начальными числами (примерно 1 из 3 случайных начальных чисел находит глобально гладкие решения).
KAN могут достигать чрезвычайной точности Хотя MLP находит глобально гладкое решение и достигает потерь $10^{−8}$ , потери все еще далеки от машинной точности. Мы обнаружили, что ни более длительное обучение, ни увеличение размера MLP не привели к значительному уменьшению потерь. Поэтому мы обратились к KAN, которые, как сплайны в 1D, могут достигать произвольной точности за счет уточнения сетки (при бесконечных данных). Сначала мы использовали MLP в качестве учителя, генерируя контролируемые пары (x,y) для обучения KAN для соответствия контролируемым данным. Таким образом, KAN инициализируется для глобально гладкого решения. Затем мы итеративно уточняли KAN, увеличивая количество интервалов сетки до 1000. В конечном итоге точно настроенные KAN достигают потерь $10^{−15}$ , близких к машинной точности (рисунок 12 (d)).

5.4 Изучение конститутивных законов

Конститутивный закон определяет поведение и свойства материала, моделируя его реакцию на внешние силы или деформации. Одной из простейших форм конститутивного закона является закон Гука [34], который линейно связывает деформацию и напряжение эластичных материалов. Конститутивные законы охватывают широкий спектр материалов, включая эластичные материалы [80, 68], пластичные материалы [64] и жидкости [8]. Традиционно эти законы были выведены из первых принципов, основанных на теоретических и экспериментальных исследованиях [79, 81, 6, 29]. Однако недавние достижения представили подходы, основанные на данных, которые используют машинное обучение для обнаружения и уточнения этих законов из выделенных наборов данных [73, 91, 59, 60].

Рисунок 13: Открытие конститутивных законов (соотношений между тензором давления P и тензором деформации F) с помощью KAN, взаимодействуя с ними. Вверху: предсказание диагонального элемента P₁₁; внизу: предсказание внедиагонального элемента P₁₂.

Мы следуем стандартным обозначениям и экспериментальным установкам в упругой части NCLaw [59] и определяем материальный закон как параметризованную функцию $\mathcal{E}_\theta(\mathbf{F})\rightarrow \mathbf{P}$ , где $\mathbf{F}$ обозначает тензор деформации, $\mathbf{P}$ — первый тензор напряжений Piola–Kirchhoff , а — параметры в материнском законе.
Многие изотропные материалы имеют линейные материальные законы при малой деформации:

Однако, когда деформация становится больше, начинают проявляться нелинейные эффекты. Например, материал NeoHookean имеет следующий конститутивный закон:

где и — так называемые параметры Lamé, определяемые так называемым модулем

Юнга и коэффициентом Пуассона как $µ =\frac Y{2(1+v)}$ , $λ=\frac{Yv}{(1+ν )(1-2ν)}$ .

Для простоты выберем Y = 1 и ν = 0,2 , отсюда $µ = \frac {5}{12}≈0,42$ и $λ = \frac{5}{18}≈0,28$ .

Предположим, что мы работаем с нео-гуковскими материалами, и наша цель — использовать KAN для предсказания тензора $\mathbf{P}$ из тензора $\mathbf{F}$ . Предположим, мы не знаем, что это нео-гуковские материалы, но у нас есть предварительное знание того, что линейный конститутивный закон приблизительно действителен для малой деформации. Из-за симметрии достаточно продемонстрировать, что мы можем точно предсказать $P_{11}$ и $P_{12}$ из 9 матричных элементов $\mathbf{F}$ . Мы хотим скомпилировать линейные конститутивные законы в KAN, которые представляют собой $P_{11}= 2µ(F_{11}– 1)+ λ(F_{11}+ F_{22}+ F_{33}– 3)$ , и $P_{12}= µ(F_{12}+ F_{21} )$ . Мы хотим извлечь неогуковские законы из обученных KAN, представляющих собой $P_{11}= µ(F_{11}^2+ F_{12}^2+ F_{13}^2– 1)+λ log⁡(det⁡(\mathbf{F}))$ , и $P_{12}= µ(F_{11} F_{21}+ F_{12} F_{22}+ F_{13} F_{23} )$ . Мы генерируем синтетический набор данных, выбирая $F_{ij}$ из $U[δ_{ij}– w,δ_{ij}+ w](w = 0,2)$ и используя неогуковский конститутивный закон для вычисления P. Наше взаимодействие с KAN проиллюстрировано на рисунке 13. В обоих случаях мы успешно вычислили истинные символические формулы с помощью некоторых индуктивных предубеждений. Однако главный вывод заключается не в том, что мы можем заново найти точные символьные формулы - учитывая, что предварительные знания искажают процесс, - а в том, что в реальных сценариях, где ответы неизвестны и пользователи могут делать предположения на основе предварительных знаний, пакет pykan позволяет легко проверить или учесть предварительные знания.

Предсказание $P_{11}$ На шаге 1 мы компилируем линейный конститутивный закон $P_{11}= 2µ(F_{11}– 1)+ λ(F_{11}+F_{22}+ F_{33}– 3)$ в KAN с помощью kanpiler, что приводит к потере $10^{−2}$ . На шаге 2 мы привносим возмущение в KAN так, чтобы она стала обучаемой (на что указывает изменение цвета с красного на фиолетовый; красный обозначает чисто символическую часть, в то время как фиолетовый указывает на то, что активны как символическая, так и сплайновая части). На шаге 3 мы обучаем возмущенную модель до сходимости, что дает потерю $6×10^{−3}$ . На шаге 4, предполагая, что детерминант является ключевой вспомогательной переменной, мы используем expand_width (для KAN) и augment_input (для набора данных), чтобы включить детерминант |F |. На шаге 5 мы обучаем KAN до сходимости, что дает потерю $2×10^{−4}$ . На шаге 6 мы символизируем KAN, чтобы получить символическую формулу $P_{11} = 0,42(F_{11}^2+ F_{12}^2+F_{13}^2 – 1) + 0,28log⁡(|F|)$ , что дает потерю $3×10^{−11}$ .

Прогнозирование $P_{12}$ Мы экспериментировали с кодированием линейного конститутивного закона в качестве априорного знания и без него. С предварительными знаниями: на шаге 1 мы компилируем линейный конститутивный закон в KAN, что приводит к потере $10^{−2}$ . Затем мы выполняем ряд операций, включая расширение (шаг 2), возмущение (шаг 3), обучение (шаг 4), сокращение (шаг 5) и, наконец, символизацию (шаг 6). Влияние предварительных знаний очевидно, поскольку окончательный KAN определяет только незначительные корректирующие члены линейного конститутивного закона. Окончательный KAN символизируется как $P_{12}=0,42(F_{12}+F_{21})+0,44F_{13} F_{23}-0,03F_{21}^2+0,02 F_{12}^2$ , что дает потерю $7×10^{−3}$ , лишь немного лучше, чем линейный конститутивный закон. Без предварительных знаний: на шаге 1 мы случайным образом инициализируем модель KAN. На шаге 2 мы обучаем KAN с регуляризацией. На шаге 3 мы обрезаем KAN, чтобы сделать ее более компактной. На шаге 4 мы символизируем KAN, получая $P_{12}=0,42(F_{11} F_{21}+F_{12} F_{22}+F_{13} F_{23} )$ , что близко соответствует точной формуле, достигая потери $6×10^{−9}$ . Сравнение двух сценариев — одного с предварительными знаниями и одного без них — показывает неожиданный результат: в этом примере предварительные знания кажутся вредными, возможно, потому, что линейный конститутивный закон, вероятно, находится вблизи (плохого) локального минимума, из которого модели трудно выйти. Однако мы, вероятно, не должны случайным образом экстраполировать этот вывод на более сложные задачи и более крупные сети. Для более сложных задач нахождение локального минимума с помощью градиентного спуска может быть достаточно сложным, что делает желательным приблизительное начальное решение. Кроме того, более крупные сети могут быть достаточно перепараметризованы, чтобы исключить плохие локальные минимумы, гарантируя, что все локальные минимумы будут глобальными и взаимосвязанными.

Рисунок 14: KAN интерполирует между программным обеспечением 1.0 и 2.0. (a) KAN находят баланс между интерпретируемостью (программное обеспечение 1.0) и обучаемостью (программное обеспечение 2.0). (b) Граница Парето KAN на плоскости шкалы интерпретируемости. Количество интерпретации, которое мы можем получить от KAN, зависит от масштабов проблемы и методов интерпретируемости.

6 Связанные работы

Сети Колмогорова-Арнольда (KAN), вдохновленные теоремой Колмогорова-Арнольда о представлении (KART), были недавно предложены Лю и др. [57]. Хотя связь между KART и сетями долгое время считалась несущественной [30], Лю и др. обобщили исходную двухслойную сеть до произвольной глубины и продемонстрировали их перспективность для научно-ориентированных задач с учетом их точности и интерпретируемости. Последующие исследования изучали применение KAN в различных областях, включая графы [12, 22, 38, 99], уравнения в частных производных [87, 78] и обучение операторов [1, 78, 67], табличные данные [70], временные ряды [85, 28, 93, 27], распознавание человеческой активности [49, 50], нейронауку [96, 33], квантовую науку [40, 46, 4], компьютерное зрение [17, 7, 44, 16, 76, 10], обучение ядра [101], ядерную физику [48], электротехнику [69], биологию [71]. Лю и др. использовали B-сплайны для параметризации 1D-функций, а другие исследования изучали различные функции активации, включая wavelet [11, 76], радиальную базисную функцию [47], ряды Фурье [92]), конечный базис [35, 82], базисные функции Якоби [2], полиномиальные базисные функции [75], рациональные функции [3]. Также были предложены другие методы для KAN, включая регуляризацию [5], Kansformer (объединяющий трансформатор и KAN) [15], адаптивное обновление сетки [72], federated learning [98], сверточные KAN [10]. Продолжаются дебаты относительно того, действительно ли KAN превосходят другие нейронные сети (особенно MLP) в различных областях [7, 16, 42, 77, 97], что говорит о том, что, хотя KAN показывают перспективы для задач машинного обучения, необходимы дальнейшие разработки, чтобы превзойти самые современные модели.

Машинное обучение для физических законов Основная цель KAN — помочь в открытии новых физических законов из данных. Предыдущие исследования показали, что машинное обучение можно использовать для изучения различных типов физических законов, включая уравнения движения [90, 13, 43, 20], законы сохранения [55, 53, 54, 58, 32, 89], симметрии [39, 56, 94], фазовые переходы [88, 14], лагранжианы и гамильтонианы [19, 31] и символическую регрессию [18, 61, 23, 74] и т. д. Однако для того, чтобы сделать нейронные сети интерпретируемыми, часто требуются знания, специфичные для предметной области, что ограничивает их общность. Мы надеемся, что KAN превратятся в универсальные базовые модели для физических открытий.

Механистическая интерпретируемость стремится понять, как нейронные сети работают на фундаментальном уровне [21, 62, 86, 25, 66, 100, 51, 24, 45, 26]. Некоторые исследования в этой области сосредоточены на разработке моделей, которые по своей сути являются интерпретируемыми [24] или на предложении методов обучения, которые явно способствуют интерпретируемости [51]. KAN попадают в эту категорию, поскольку теорема Колмогорова-Арнольда раскладывает многомерную функцию на набор одномерных функций, которые значительно легче интерпретировать, чем многомерные функции.

7 Обсуждение

KAN интерполирует между программным обеспечением 1.0 и 2.0 Ключевое различие между сетями Колмогорова-Арнольда (KAN) и другими нейронными сетями (программное обеспечение 2.0, термин, введенный Andrej Karpathy) заключается в их большей интерпретируемости, что позволяет пользователям манипулировать ими, аналогично традиционному программному обеспечению (программное обеспечение 1.0). Однако KAN не являются полностью традиционным программным обеспечением, поскольку они (1) обучаемы (хорошо), что позволяет им изучать новые вещи из данных, и (2) имеют пониженную интерпретируемость (плохо), поскольку они становятся менее интерпретируемыми и контролируемыми по мере увеличения масштабов сети. На рисунке 14 (a) визуализируется положение программного обеспечения 1.0, программного обеспечения 2.0 и KAN на плоскости интерпретируемости-обучаемости, иллюстрируя, как KAN могут сбалансировать компромиссы между этими двумя парадигмами. Целью данной статьи является предложение различных инструментов, которые делают KAN более похожими на программное обеспечение 1.0, используя при этом обучаемость программного обеспечения 2.0.

Повышение эффективности. Первоначальный пакет pykan [57] был плох в эффективности. Мы включили несколько методов для повышения его эффективности.
1. Эффективные оценки сплайнов. Вдохновленные Efficient KAN [9], мы оптимизировали оценки сплайнов, избегая ненужных расширений входных данных. Для KAN с L слоями, N нейронами на слой и размером сетки G использование памяти было сокращено с O(LN^2 G) до O(LNG) .
2. Включение символической ветви (symbolic branch) только при необходимости. Слой KAN содержит как сплайновую, так и символическую ветвь. Символическая ветвь требует гораздо больше времени, чем сплайновая, поскольку ее нельзя распараллелить (нужны катастрофические двойные циклы). Однако во многих приложениях символическая ветвь не нужна, поэтому мы можем пропустить ее, когда это возможно, что значительно сокращает время выполнения, особенно когда сеть большая.
3. Сохранение промежуточных активаций только при необходимости. Для построения диаграмм KAN необходимо сохранять промежуточные активации. Изначально активации сохранялись по умолчанию, что приводило к замедлению времени выполнения и чрезмерному использованию памяти. Теперь мы сохраняем промежуточные активации только при необходимости (например, для построения графиков или применения регуляризаций при обучении). Пользователи могут включить эти улучшения эффективности с помощью одной строки: model.speed().
4. Ускорение на GPU. Изначально все модели запускались на CPU из-за мелкомасштабного характера проблем. Теперь мы сделали модель совместимой с GPU* (*Модели можно обучать на графических процессорах, но не все функции уже поддерживаются графическими процессорами) . Например, обучение [4,100,100,100,1] с Adam для 100 шагов раньше занимало целый день на CPU (до внедрения 1, 2, 3), но теперь занимает 20 секунд на CPU и менее одной секунды на GPU. Однако KAN все еще отстают от MLP по эффективности, особенно в больших масштабах. Сообщество работает над бенчмаркингом и повышением эффективности KAN, и разрыв в эффективности был значительно сокращен [36].

Поскольку цель этой статьи — сделать KAN более похожими на программное обеспечение 1.0, при выборе между 1.0 (интерактивность и универсальность) и 2.0 (эффективность и конкретика) мы отдаем приоритет интерактивности и универсальности, а не эффективности. Например, мы храним афишированные данные в моделях (что потребляет дополнительную память), поэтому пользователи могут просто вызвать model.plot() для генерации диаграммы KAN без ручного выполнения прямого прохода для сбора данных.

Интерпретируемость Хотя обучаемые одномерные функции в KAN более интерпретируемы, чем весовые матрицы в MLP, масштабируемость остается проблемой. По мере масштабирования моделей KAN, даже если все сплайн-функции интерпретируются по отдельности, становится все труднее управлять объединенным выводом этих 1D-функций. Следовательно, KAN может оставаться интерпретируемой только тогда, когда масштаб сети относительно мал (рисунок 14 (b), толстая красная линия). Важно отметить, что интерпретируемость зависит как от внутренних факторов (связанных с самой моделью), так и от внешних факторов (связанных с методами интерпретируемости). Расширенные методы интерпретируемости должны быть способны обрабатывать интерпретируемость на различных уровнях. Например, интерпретируя KAN с помощью символической регрессии, обнаружения модульности и атрибуции признаков (рисунок 14 (b), тонкие красные линии), граница Парето интерпретируемости по сравнению с масштабом выходит за рамки того, чего может достичь только KAN. Перспективным направлением будущих исследований является разработка более продвинутых методов интерпретируемости, которые могут еще больше раздвинуть текущие границы Парето.

Будущая работа В этой статье представлена структура, которая интегрирует KAN с научными знаниями, фокусируясь в первую очередь на мелкомасштабных примерах, связанных с физикой. В дальнейшем два перспективных направления включают применение этой структуры к более масштабным проблемам и ее распространение на другие научные дисциплины за пределами физики.

Признание

Мы хотели бы поблагодарить Yizhou Liu, Di Luo, Akash Kundu и многих пользователей GitHub за плодотворное обсуждение и конструктивные предложения. Мы выражаем особую благодарность пользователю GitHub Blealtan за публикацию их потрясающей работы по повышению эффективности KAN.
Z.L. и M.T. поддерживаются IAIFI через грант NSF PHY-2019786.

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26 нояб. 2024 г.

15.08.25 03:10 KEVIN3

HOW I RECOVER MY LOST FUNDS AFTER INVESTING Exercise extreme caution when dealing with this company. They scammed me $800,000 out of my money and even gained unauthorized access to my banking information, which severely impacted my financial well-being. I had initially approached them with the hope of improving my life through investment, but instead, I was deceived. This company should be thoroughly investigated and shut down to prevent further harm. Fortunately, I was able to recover my funds thanks to Mrs. Rose. Although I was initially hesitant and unsure about contacting her, I decided to take the risk and I’m so glad I did. You can reach her via telegram (@RoseCyberHives) and email rosecryptorecovery@gmail. com With the help of her skilled team of cybersecurity professionals and cryptocurrency experts, I was able to recover my lost funds quickly and with minimal stress, a sharp contrast to my previous experience. If you’ve been scammed or are in a similar situation, I highly recommend reaching out to her for assistance. via telegram (@RoseCyberHives) rosecryptorecovery@gmail. com
15.08.25 04:59 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.08.25 04:59 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.08.25 04:59 marcushenderson624

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15.08.25 04:59 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.08.25 09:13 eva117

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15.08.25 13:27 wendytaylor015

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15.08.25 13:27 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.08.25 20:26 caitlinfedex

After my wife passed, I found myself lonely and searching for connection. I joined a senior living forum just to talk with others my age to share stories, maybe feel a little less isolated. That’s where I met a man who claimed to be helping retirees like me manage their savings with cryptocurrency. He spoke warmly and convincingly about a retirement care fund that guaranteed lifetime monthly payouts. He said it was specifically designed for people in my stage of life safe secure and backed by health organizations. He gained my trust over weeks of conversation. We spoke daily. He called me by name asked about my late wife, and remembered little details about my life. I believed he genuinely cared. Eventually, I transferred $200000 in USDT the majority of my retirement savings, into the wallet address he provided. That was the last time I heard from him. The forum profile vanished. His phone number stopped working. No response to emails. I was stunned. I couldn’t believe I had been tricked. I didn’t know who to turn to. I felt ashamed and foolish. At my age recovering from a financial loss like this is nearly impossible.My relative stepped in. They searched online for solutions and found TRUST GEEKS HACK EXPERT. I was skeptical at first but they insisted we give them a chance. From the moment we contacted them they treated me with respect patience and genuine concern. Their team didn’t speak in technical riddles. They explained every step. They used blockchain analysis tools to trace the movement of my funds through several addresses.It turned out the wallet was linked to a scam platform. TRUST GEEKS HACK EXPERT worked directly with the USDT issuer and coordinated with relevant platforms to freeze the funds before they were fully laundered.In just two weeks they managed to recover $155000 in USDT.The money helped yes but it was more than that. I no longer felt powerless. I had been taken advantage of but I wasn’t alone in the fight. Thanks to TRUST GEEKS HACK EXPERT I got back not only a large part of my savings but also peace of mind a sense of dignity and hope for the future. If you ever find yourself in such situation I encourage you to contact TRUST GEEKS HACK EXPERT Email.trustgeekshackexpert{At}fastservice{Dot}com — Telegram:: Trustgeekshackexpert & W h a t's A p p +1 7 1 9 4 9 2 2 6 9 3
15.08.25 20:27 caitlinfedex

After my wife passed, I found myself lonely and searching for connection. I joined a senior living forum just to talk with others my age to share stories, maybe feel a little less isolated. That’s where I met a man who claimed to be helping retirees like me manage their savings with cryptocurrency. He spoke warmly and convincingly about a retirement care fund that guaranteed lifetime monthly payouts. He said it was specifically designed for people in my stage of life safe secure and backed by health organizations. He gained my trust over weeks of conversation. We spoke daily. He called me by name asked about my late wife, and remembered little details about my life. I believed he genuinely cared. Eventually, I transferred $200000 in USDT the majority of my retirement savings, into the wallet address he provided. That was the last time I heard from him. The forum profile vanished. His phone number stopped working. No response to emails. I was stunned. I couldn’t believe I had been tricked. I didn’t know who to turn to. I felt ashamed and foolish. At my age recovering from a financial loss like this is nearly impossible.My relative stepped in. They searched online for solutions and found TRUST GEEKS HACK EXPERT. I was skeptical at first but they insisted we give them a chance. From the moment we contacted them they treated me with respect patience and genuine concern. Their team didn’t speak in technical riddles. They explained every step. They used blockchain analysis tools to trace the movement of my funds through several addresses.It turned out the wallet was linked to a scam platform. TRUST GEEKS HACK EXPERT worked directly with the USDT issuer and coordinated with relevant platforms to freeze the funds before they were fully laundered.In just two weeks they managed to recover $155000 in USDT.The money helped yes but it was more than that. I no longer felt powerless. I had been taken advantage of but I wasn’t alone in the fight. Thanks to TRUST GEEKS HACK EXPERT I got back not only a large part of my savings but also peace of mind a sense of dignity and hope for the future. If you ever find yourself in such situation I encourage you to contact TRUST GEEKS HACK EXPERT Email.trustgeekshackexpert{At}fastservice{Dot}com — Telegram:: Trustgeekshackexpert & W h a t's A p p +1 7 1 9 4 9 2 2 6 9 3
15.08.25 21:04 vallatjosette

My Wild Ride: How a Wizard Hilton Cyber Tech Helped Me Recover Lost Bitcoin of 103,000 It all started when I had the misfortune of misplacing the digital wallet containing my substantial cryptocurrency holdings - a staggering 103,000 bitcoins. I had amassed this sizable fortune over the years through strategic investments and careful trading, but in a moment of carelessness, the private keys to my digital wallet had vanished. Panic set in as I grappled with the gravity of my situation - those bitcoins, worth millions at the time, were effectively lost to the ether, inaccessible and seemingly unrecoverable. Just when I had resigned myself to the devastating financial blow, a chance encounter with a self-proclaimed "digital wizard" Wizard Hilton Cyber Tech presented a glimmer of hope. This enigmatic individual, who operated in the shadowy corners of the crypto underworld, claimed to possess the arcane knowledge and skills necessary to retrieve my lost digital wealth. Intrigued and desperate, I put my skepticism aside and entrusted this mysterious wizard with the task. What followed was a wild, suspenseful journey into the depths of blockchain technology and cryptography, as the wizard navigated complex algorithms, exploited vulnerabilities, and pieced together the puzzle of my missing private keys. After days of intense effort, Wizard Hilton Cyber Tech emerged triumphant, having successfully recovered the entirety of my 103,000 bitcoin stash. The sheer relief and elation I felt in that moment was indescribable, as I regained access to my small digital fortune and was able to secure my financial future. This harrowing experience taught me the invaluable lesson of never taking the security of my cryptocurrency holdings for granted, and the importance of always maintaining the utmost diligence when it comes to protecting one's digital wealth. Email: wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 Thanks.
16.08.25 06:53 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.08.25 06:53 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.08.25 06:53 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.08.25 07:01 lisared

Losing nearly 500,000 USDT to hackers was one of the most devastating experience of my life. I felt powerless as my hard-earned assets vanished into anonymous blockchain transactions until I found Agent Jasmine Lopez. Her deep expertise, tireless effort, and unwavering focus gave me hope and delivered results. She traced complex transactions, pursued every lead, and recovered a large portion of my stolen funds something I never thought possible. If you’ve lost digital assets, don’t give up. Contact Recoveryfundprovider@gmail. com WhatsApp at +{44 736-644-5035}. She turned my nightmare into a recovery story, and she could do the same for you.
16.08.25 17:05 mark

I see a lot of recommendations online and it’s already obvious there are bad eggs online who will only add to your mystery. I can only recommend one and you can reach them via mail on (zattechrecovery @ gmail c0 m ) if you need help on recovering what you lost to scammers.
17.08.25 03:33 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.08.25 03:33 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.08.25 05:22 aveasley4

I’m a California resident who was contacted by someone named “Lin” on WhatsApp. Our conversation eventually turned to cryptocurrency investments. Lin introduced me to a trading platform called CBOTBIT (https://cbotbit.com), walked me through setting up an account, and guided me on transferring funds. Under Lin’s direction, I began trading and was led to believe that my account was growing significantly. However, when I attempted to withdraw my funds, I was unable to do so — and shortly after, the website went offline entirely. I had invested over $340,000 in Bitcoin, planning to use it for my retirement. The experience was devastating, and I was at a loss for what to do next. Eventually, I came across [email protected] and decided to reach out. I followed their instructions, and within a few days, I was able to recover my original investment. I'm incredibly relieved and grateful for their assistance, and I recommend contacting them if you’re in a similar situation and need help recovering lost funds.
17.08.25 17:49 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.08.25 17:49 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.08.25 17:49 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.08.25 21:10 Connorhelms

I can honestly say losing my bitcoin savings was one of the darkest moments of my life. I had put everything I could into that investment, believing it would secure my future, only to wake up one day and realize the platform had vanished with all my funds. The feeling was indescribable, fear, shame, and helplessness all at once. I thought I’d never see that money again. Then I came across Alpha Spy Nest. At first, I was hesitant, I had already been deceived once, and the thought of trusting anyone else terrified me. But something about their professionalism and the way they patiently explained things gave me a little hope. For the first time in weeks, I felt like I wasn’t completely alone in the fight. They listened to my story without judgment, broke down the technical side in a way I could understand, and reassured me that there was still a chance. Watching them work was incredible. They traced every transaction, uncovered where my bitcoin had been moved, and kept me updated through every stage. It felt like they cared as much about my recovery as I did. The day they told me they had successfully recovered my lost savings, I actually cried. Seeing those funds back in my wallet felt unreal, it was like a huge weight had been lifted off my shoulders. Beyond just recovering my bitcoin, Alpha Spy Nest gave me back my peace of mind and restored a trust I thought I had lost forever. I’m forever grateful to them. If I hadn’t found Alpha Spy Nest, I’d still be stuck in regret and despair. Instead, I now share my story as proof that recovery is possible. You can also reach out to them via: Email: [email protected], WhatsApp: +15132924878‬, Telegram: https://t.me/Alphaspynest, Website: www.alphaspynest.org
18.08.25 22:17 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.08.25 22:17 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
19.08.25 02:26 Jacksonbash

You are not alone if you have had your bitcoins taken from your wallet or if you invested in an initial coin offering (ICO) that was a scam. I experienced the same thing. I first lost $173,000 to a fraudulent investment scam company in less than five months. I was able to get all of my money in five days thanks to a hacker my friend recommended. In order to raise awareness about these cryptocurrency criminals and do my part to minimize the number of victims, I'm speaking out. To report a victim, simply send an email to [email protected].
19.08.25 18:15 keith

RECOVER FROM CRYPTO SCAMS WITH THE HELP OF BONES RECOVERY BITCOIN EXPERT If someone needs to recover cryptocurrency that has been lost to wallet hackers, internet scammers, or BTC transferred to the wrong address, I suggest BONES RECOVERY BITCOIN, a very trustworthy recovery organization . Simply presenting my case to the expert and supplying the necessary data allowed BONES RECOVERY BITCOIN to assist me in recovering my Bitcoin. It's a relief that I recovered this much after losing considerably more to the fraudulent broker I initially put my trust in. Because we will inevitably make mistakes, we can never be diligent enough. If you have lost money to cryptocurrency scams, I strongly advise getting in touch with BONES RECOVERY BITCOIN. Contact address WhatsApp....‪+1-60-621-024-64‬ Email [email protected]
19.08.25 18:54 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
19.08.25 18:54 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
19.08.25 18:54 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected] You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1
20.08.25 12:01 Official

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20.08.25 17:00 lisared

Few months ago I experienced one of the most devastating moments of my life €300k was stolen by hackers . When my money was stolen I felt hopeless, until I was introduced to Agent Jasmine Lopez her expertise and dedication gave me peace of mind. She skillfully traced complex transactions and tracked down the scammers, which would have been impossible for me to do on my own. Thanks to her tireless efforts, I was able to recover a significant portion of my lost funds. If you've fallen victim of hackers, I highly recommend reaching out to her. You can contact her via email at Recoveryfundprovider@gmail. com WhatsApp at +{44 736-644-5035}. Her attention to detail and commitment to her work are truly impressive. Don't hesitate to reach out her expertise could be the key to recovering your lost funds.
20.08.25 20:42 vallatjosette

My Wild Ride: How a Wizard Hilton Cyber Tech Helped Me Recover Lost Bitcoin of 103,000 It all started when I had the misfortune of misplacing the digital wallet containing my substantial cryptocurrency holdings - a staggering 103,000 bitcoins. I had amassed this sizable fortune over the years through strategic investments and careful trading, but in a moment of carelessness, the private keys to my digital wallet had vanished. Panic set in as I grappled with the gravity of my situation - those bitcoins, worth millions at the time, were effectively lost to the ether, inaccessible and seemingly unrecoverable. Just when I had resigned myself to the devastating financial blow, a chance encounter with a self-proclaimed "digital wizard" Wizard Hilton Cyber Tech presented a glimmer of hope. This enigmatic individual, who operated in the shadowy corners of the crypto underworld, claimed to possess the arcane knowledge and skills necessary to retrieve my lost digital wealth. Intrigued and desperate, I put my skepticism aside and entrusted this mysterious wizard with the task. What followed was a wild, suspenseful journey into the depths of blockchain technology and cryptography, as the wizard navigated complex algorithms, exploited vulnerabilities, and pieced together the puzzle of my missing private keys. After days of intense effort, Wizard Hilton Cyber Tech emerged triumphant, having successfully recovered the entirety of my 103,000 bitcoin stash. The sheer relief and elation I felt in that moment was indescribable, as I regained access to my small digital fortune and was able to secure my financial future. This harrowing experience taught me the invaluable lesson of never taking the security of my cryptocurrency holdings for granted, and the importance of always maintaining the utmost diligence when it comes to protecting one's digital wealth. Email: wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 Thanks.
21.08.25 01:54 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.08.25 01:54 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.08.25 01:54 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.08.25 14:56 elainemurray

E.M.A.I.L. [email protected] W.h.a.t.s.A.p.p. +1(561)(726)(3697) It all began with an Instagram advertisement. A man presented himself as a cryptocurrency investment coach, showcasing a luxurious lifestyle and screenshots of substantial profits. He claimed his trading algorithm could double any investment in a week. Curious, I decided to reach out to him. He said the minimum investment was $50,000 and sent me a link to a platform that looked legitimate. At first, everything seemed promising. My balance increased steadily, charts showed growth, and I felt excited. When I tried to withdraw the money, the system blocked the transaction and demanded a release fee of $15,000. I paid it, but my account was soon marked under review, and the coach stopped responding. I realised I had been scammed. Desperate, I turned to Techy Force Cyber Retrieval. From the first consultation, they carefully reviewed my case and assured me the funds could be traced. They explained how they would track the wallet addresses and follow the path of the stolen money. Their clear guidance made me feel confident. Over the next few weeks, Techy Force Cyber Retrieval kept me updated on every step. They traced the transactions. Their persistence and expertise were obvious, and I could see real progress. Even when the process felt slow, Techy Force Cyber Retrieval maintained clear communication. They explained each action and why it mattered. Their careful attention made the ordeal manageable and reassuring. A month later, thanks to Techy Force Cyber Retrieval, I was able to recover 80% of my money and have it safely returned to my bank account. Relief and gratitude overwhelmed me. What started as a tempting opportunity had turned into a stressful ordeal, but Techy Force Cyber Retrieval transformed it into a successful resolution.
21.08.25 21:17 monte5757

I went through one of the most painful experiences of my life I lost over $850,000 to scammers. They had no mercy,things got so bad that I even sold some of my properties just to keep up with their demands. At one point, I felt completely hopeless and trapped. But everything changed when I came across a legit recovery agent who actually helped me recover my funds and return them safely to my wallet. It was like getting my life back. If you’ve ever been scammed, please don’t give up. Reach out to Recoveryfundprovider [at] gmail [dot] com WhatsApp at +{44 736-644-5035}.she’s professional, reliable, and truly knows how to get results. I’m genuinely grateful I found her, and I believe you’ll thank me later too.
22.08.25 08:00 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.08.25 08:00 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.08.25 08:00 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.08.25 05:11 ramiroalcazar

A hacker skillfully impersonated one of our clients’ trusted suppliers and manipulated their finance department into transferring $180,000 worth of Bitcoin. The request appeared authentic, reinforced by falsified documents and meticulously crafted communication that mirrored the supplier’s established style. By the time the company uncovered the deception, the funds had already been funneled through numerous digital wallets. When they appealed to their bank for assistance, they were told recovery was impossible since cryptocurrency transactions are irreversible. The company suddenly faced the prospect of financial ruin. Confronted with this devastating loss, the client sought a specialized solution and turned to Techy Force Cyber Retrieval. From the initial consultation, our team demonstrated that hope remained. Unlike the bank, which dismissed their plea, Techy Force Cyber Retrieval meticulously analyzed every aspect of the fraud and assured the client that our blockchain specialists had the tools to trace and potentially reclaim the stolen assets. Our investigation began with a comprehensive audit of the fraudulent transactions. Using advanced blockchain forensics, Techy Force Cyber Retrieval’s analysts traced the stolen Bitcoin across a labyrinth of wallets and exchanges. Every development was documented with precision, and regular updates were delivered to the client, ensuring transparency and trust. This combination of technical mastery and clear communication provided reassurance during an otherwise overwhelming crisis. After persistent tracking and strategic intervention, Techy Force Cyber Retrieval achieved a remarkable outcome. From the original $180,000 that had been siphoned away, our team successfully recovered the equivalent of $160,000 in USDT. This restitution enabled the company to meet payroll obligations, stabilize its operations, and avert what had seemed like an inevitable collapse. What distinguished Techy Force Cyber Retrieval in the eyes of the client was not only our technical expertise but also our dedication and professionalism. We approached the case with urgency, maintained open communication at every stage, and treated the recovery effort with the utmost seriousness. At a moment when the client felt powerless, Techy Force Cyber Retrieval restored both their assets and their confidence. Now, victims of cryptocurrency fraud do not have to accept total loss. With the right expertise, recovery is achievable. Techy Force Cyber Retrieval stands as proof that even in situations deemed hopeless, we can deliver tangible results and provide businesses and individuals with a genuine second chance. CONSULT TECHY FORCE CYBER RETRIEVAL FOR A GREAT SOLUTION RECOVERY......... WHATSAPP.......... +15617263697 WEBSITE......... https://techyforcecyberretrieval.com
24.08.25 01:05 samshaffer

Have been on the quest for a way to recover my $195,000 from a forex broker. I was fortunate to meet Darek at ([email protected]) through a friend who was recommended as the best crypto recovery specialist he was the one who assisted me and restore my money back to my atomic wallet. I did reach him at [email protected] and it was an amazing experience. Never knew there was an expert who could trace back crypto until I met this great team. Don’t think twice, get connected with the right personnel now to avoid been fooled again..
24.08.25 06:32 amy

I have lost all hope to recover my investment in pragmatic trade which I lost $125,000 in the process of investment. I'm here to say thank you to anthony davies for being a man of your word got my funds recovered back plus a bonus within a week of working with [email protected] I hope people make use of this opportunity as well and don't give up on getting your funds back. is also available on Telegram: @anthonydavies01
24.08.25 17:13 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 17:13 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 17:13 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 18:39 nelly young

Getting your USDC stolen feels like the ground has been pulled out from under you. It’s not just about the money you’ve been violated, and your security is at risk. And since crypto lives on the blockchain, there’s no bank to call and no quick refund. That’s why you need someone who knows exactly what to do. Agent Jasmine Lopez has helped countless victims trace stolen USDC and lock down their accounts before more damage happens. Time matters reach her today at [email protected] WhatsApp at +44 736-644-5035 and take the first step toward getting your funds and peace of mind back.
24.08.25 23:56 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: Recoverycapital @fastservice. com Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 23:56 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: Recoverycapital @fastservice. com Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.08.25 23:56 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: Recoverycapital @fastservice. com Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
25.08.25 00:55 vallatjosette

My Wild Ride: How a Wizard Hilton Cyber Tech Helped Me Recover Lost Bitcoin of 103,000 It all started when I had the misfortune of misplacing the digital wallet containing my substantial cryptocurrency holdings - a staggering 103,000 bitcoins. I had amassed this sizable fortune over the years through strategic investments and careful trading, but in a moment of carelessness, the private keys to my digital wallet had vanished. Panic set in as I grappled with the gravity of my situation - those bitcoins, worth millions at the time, were effectively lost to the ether, inaccessible and seemingly unrecoverable. Just when I had resigned myself to the devastating financial blow, a chance encounter with a self-proclaimed "digital wizard" Wizard Hilton Cyber Tech presented a glimmer of hope. This enigmatic individual, who operated in the shadowy corners of the crypto underworld, claimed to possess the arcane knowledge and skills necessary to retrieve my lost digital wealth. Intrigued and desperate, I put my skepticism aside and entrusted this mysterious wizard with the task. What followed was a wild, suspenseful journey into the depths of blockchain technology and cryptography, as the wizard navigated complex algorithms, exploited vulnerabilities, and pieced together the puzzle of my missing private keys. After days of intense effort, Wizard Hilton Cyber Tech emerged triumphant, having successfully recovered the entirety of my 103,000 bitcoin stash. The sheer relief and elation I felt in that moment was indescribable, as I regained access to my small digital fortune and was able to secure my financial future. This harrowing experience taught me the invaluable lesson of never taking the security of my cryptocurrency holdings for granted, and the importance of always maintaining the utmost diligence when it comes to protecting one's digital wealth. Email: wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 Thanks.
25.08.25 05:14 samshaffer

Have been on the quest for a way to recover my $195,000 from a forex broker. I was fortunate to meet Darek at ([email protected]) through a friend who was recommended as the best crypto recovery specialist he was the one who assisted me and restore my money back to my atomic wallet. I did reach him at [email protected] and it was an amazing experience. Never knew there was an expert who could trace back crypto until I met this great team. Don’t think twice, get connected with the right personnel now to avoid been fooled again..
26.08.25 10:14 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.08.25 10:14 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.08.25 10:14 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.08.25 23:59 floridaberna

It began as an entirely ordinary afternoon when an unexpected message appeared in my Facebook inbox. The sender was a woman I had never encountered. She introduced herself with warmth and courtesy, explaining that her late husband had been deeply invested in Bitcoin before his sudden passing. According to her narrative, he had left behind a vast fortune in cryptocurrency. However, due to intricate international banking regulations and legal entanglements, she was unable to access the funds without external assistance. Her words carried an air of urgency yet also sincerity. She claimed she had come across my profile in a cryptocurrency discussion forum and believed I might be the right person to help. The obstacle she faced was a set of legal fees amounting to $25,000. Once those costs were paid, she assured me the Bitcoin would be unlocked and a generous portion would be mine in gratitude. Against my better instincts, I allowed myself to be persuaded. She provided what appeared to be legitimate documents and even screenshots of supposed legal correspondence. The level of detail made her story feel authentic. After several exchanges, I transferred the $25,000.Almost immediately, her demeanor shifted. Responses became sporadic and vague. Within days, communication ceased entirely. My optimism dissolved into dread as I recognized the truth. I had been deceived. Refusing to surrender, I searched for solutions and discovered TECHY FORCE CYBER RETRIEVAL, a firm specializing in tracing and reclaiming stolen cryptocurrency. Their team outlined a meticulous plan to track the blockchain trail, identify the sequence of wallets involved, and coordinate with local authorities to freeze the assets before they disappeared into the labyrinth of the crypto underworld. Time was critical. In the volatile realm of digital currency, stolen funds can be laundered through countless transactions within hours. Yet TECHY FORCE CYBER RETRIEVAL pursued each lead with relentless precision. They provided regular progress updates and explained how every crypto transaction leaves a digital footprint, no matter how deeply it is buried. Weeks later, the breakthrough came. The fraudulent account was located, the illicit funds were flagged, and to my immense relief, I recovered 100% of my stolen money. This left me with an invaluable lesson. The internet is full of opportunists who prey on trust and desperation. But should you ever fall victim, know that there are experts like TECHY FORCE CYBER RETRIEVAL who can help you reclaim what is rightfully yours. MAil support (@) techyforcecyberretrieval (.) com WhatsApp +1 561 726 369 7 Website ht tp s : / / techy force cyber retrieval . com
27.08.25 17:24 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.08.25 17:24 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.08.25 17:24 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.08.25 20:27 maximeden68

Top Crypto Bitcoin Wallet Recovery Services Contact iFORCE HACKER RECOVERY I wish to express my sincere gratitude to iForce Hacker Recovery for their exceptional service in recovering my lost bitcoin. Their expertise and professionalism were evident throughout the process, providing me with reassurance during a challenging time. The team's dedication to client satisfaction and their effective strategies were instrumental in resolving my issue. I highly recommend their services to anyone facing similar challenges. Thank you, iForce Hacker Recovery. Contact iForce Hacker Recovery: Email: iforcehk @ consultant. com WhatsApp: +1 (240) 803-3706 Website: iforcehackers. com/
27.08.25 21:21 dbeverly

One morning I received an email that looked like it came from my cryptocurrency exchange, OKX. The subject warned: Suspicious activity detected and urged me to log in via a link. The message seemed authentic — perfect logos, formatting, even the sender’s address. My pulse spiked and I clicked. The site that opened was indistinguishable from OKX’s homepage. Convinced it was real, I entered my username and password. Moments later, dread set in: my account was emptied. $12,500 in Bitcoin was gone. I called OKX, hoping for a reversal, but the representative explained transfers were irreversible. Desperate, I turned to GRAYWARE TECH SERVICE. From the first call, their professionalism was clear. They carefully examined my case, tracing the stolen Bitcoin across the blockchain. They identified the destination wallets and mapped every move the hackers made. Their regular updates turned fear into cautious optimism. Acting swiftly, GRAYWARE coordinated with exchanges along the path of the stolen funds, freezing assets before they could vanish. Their vigilance and expertise kept me informed and reassured. After two tense weeks, the breakthrough came: my full $12,500 was recovered and returned to my OKX account. What began as a nightmare became a success thanks to GRAYWARE TECH SERVICE’s relentless determination. They didn’t just recover my money — they restored my confidence in digital security. Contact: WhatsApp: +1 858 275 9508 Website: https://graywaretechservice.com Email: [email protected]
28.08.25 10:08 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 10:08 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 10:08 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 13:50 Maureensels

STEPS TO RECOVERING MONEY FROM A CRYPTO INVESTMENT SCAM CONTACT WIZARD LARRY RECOVERY EXPERTS Are you trying to find a means to get your other cryptocurrency, including Bitcoin, back from scammers? Stop searching! Call or text +1 (616) - (292) - (4789) or send a mail to (WIZARDLARRY(@)MAIL. C OM) is the greatest Scammed Crypto Recovery Agency I've seen on Google Earth. I'm delighted I made the brave decision to get in touch with them. In a matter of hours, all of the cryptocurrency that I had stolen was returned to my Coin Base wallet. For assistance with their strategies, I will advise you all to get in touch with their customer service. You will thank me later. Website... Wizardlarryrecovery. com Email [email protected] o m
28.08.25 13:50 Maureensels

STEPS TO RECOVERING MONEY FROM A CRYPTO INVESTMENT SCAM CONTACT WIZARD LARRY RECOVERY EXPERTS Are you trying to find a means to get your other cryptocurrency, including Bitcoin, back from scammers? Stop searching! Call or text +1 (616) - (292) - (4789) or send a mail to (WIZARDLARRY(@)MAIL. C OM) is the greatest Scammed Crypto Recovery Agency I've seen on Google Earth. I'm delighted I made the brave decision to get in touch with them. In a matter of hours, all of the cryptocurrency that I had stolen was returned to my Coin Base wallet. For assistance with their strategies, I will advise you all to get in touch with their customer service. You will thank me later. Website... Wizardlarryrecovery. com Email [email protected] o m
28.08.25 16:31 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 16:31 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.08.25 19:17 leticiazavala

Earlier this year, I found what looked like an unbeatable offer on first-class airline tickets through an online travel agency. They advertised heavily discounted fares for international travel for only 0.11 BTC, which was worth around 7300 USD at the time. The only condition was that payment had to be made in Bitcoin. The website looked professional, had convincing reviews, and even offered live chat support. After some hesitation, I decided to go ahead with the booking. Two days later, I received an email from the airline saying my reservation had been canceled. I went back to the agency’s website only to find it had disappeared. Their customer support was gone, and their social media accounts had vanished. That was when I realized I had been scammed. I thought my Bitcoin was lost for good. Unlike a credit card transaction, there was no way to reverse it. A friend recommended a company called Techy Force Cyber Retrieval that specializes in tracking and recovering stolen crypto. I reached out not expecting much. To my surprise, their team responded quickly. I sent them the transaction details, and they immediately began tracing the movement of the 0.11 BTC. They followed the funds through multiple wallets and discovered the Bitcoin had ended up at a large exchange known for cooperating in fraud investigations. Techy Force Cyber Retrieval worked fast. They submitted the evidence to the exchange and requested that the wallet be frozen. Over the next few weeks, they handled all communication and submitted the necessary documentation. Just under three weeks later, the exchange approved the release of my funds. I was stunned when I saw the 0.11 BTC back in my wallet. I had lost hope, but Techy Force Cyber Retrieval delivered. They were professional, honest, and transparent throughout the entire process. If you have been scammed in a crypto deal, do not assume the money is gone forever. With the right help, it is possible to trace and recover stolen assets. Techy Force Cyber Retrieval gave me a second chance, and I would not hesitate to recommend them to anyone who finds themselves in a similar situation. FOR SERVICE HELP Email: [email protected] [email protected] CALL +15617263697
29.08.25 09:20 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.08.25 09:20 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.08.25 09:20 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
30.08.25 01:40 Moises Velez

In a single transaction, my Coinbase wallet was destroyed. I was at a loss as to what to do and was unable to contact Coinbase support Teams. This transaction resulted in the complete depletion of my Ethereum and Bitcoin. I was not the individual in question, and I required assistance after losing approximately $821,000. I visited the police station to submit a report, but they were also unable to provide any assistance. A gentleman approached me at the stiction and presented me with a note that included an email address (recoveryhacker101[at]gmail[dot]com). He informed me that this recovery company could be of assistance. Upon contacting them and submitting all necessary documentation within 72 hours, I received a letter stating that she had successfully recovered all of my funds and identified the individuals responsible for the criminal activity. I would like to inform everyone that it is feasible to retrieve their funds provided that they correspond with the appropriate individual and adhere to the prescribed procedures.
31.08.25 14:19 jack118

In 2018, Marie assisted in the recovery of stolen bitcoin. The service was excellent. The coins were stolen by hackers, creating a nightmare. Marie was suggested by a friend. Contact her at [email protected] and @MARIE_CONSULTANCY on Telegram or WhatsApp at +1 7127594675. She was responsible for recovering Bitcoin. The trading goes on.
31.08.25 23:14 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
31.08.25 23:14 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
31.08.25 23:14 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.09.25 04:55 Maurice Cherry

Through Hackrontech recovery, Bitcoin scam victims can retrieve their money. I recommed hackrontech to anyone who has fallen victim to a scam and has been looking for methods and techniques to recover their lost cryptocurrency or wallets. Hackrontech Recovery is a reliable cryptocurrency recovery firm that assists victims in recovering their stolen cryptocurrency and offers secure solutions to protect your wallets from online scammers. I must admit that I was deeply melancholy and had given up on life until these experts could restore my 4.1 BTC to my wallet. If you’ve lost your cryptocurrency and you are helpless about it, contact hackrontech Recovery to get your money back. Email : hackrontech @gmail.com One key aspect that makes hackrontech Recovery stand out is its focus on providing secure solutions to protect wallets from online scammers. It’s not just about recovering lost funds; it’s also about preventing future incidents and ensuring that clients’ digital assets are safeguarded against potential threats. This proactive approach demonstrates their commitment to the long-term financial security of their clients. Furthermore, for individuals who have lost their cryptocurrency and are feeling helpless, reaching out to hackrontech Recovery could be a turning point in their situation. The reassurance that they are legitimate for seeking help and recovering lost funds can provide much-needed relief and a sense of empowerment. hackrontech Recovery as a reliable cryptocurrency recovery firm is certainly well-founded. Their ability to assist scam victims in recovering stolen cryptocurrency, their focus on providing secure solutions, and their commitment to supporting clients through challenging situations make them a valuable resource for individuals navigating the complex world of digital currencies. If you or someone you know has fallen victim to a cryptocurrency scam, contacting hackrontech Recovery could be the first step towards reclaiming lost funds and regaining peace of mind. Contact info; email: hackrontech @gmail.com
01.09.25 11:06 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.09.25 11:07 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text: +1 (336) 390-6684 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.09.25 16:41 Anitastar

Wizard Hilton Cyber Tech's strategic approach to Bitcoin recovery represents a mutually beneficial collaboration that offers valuable advantages for all involved. At the core of this innovative partnership is a shared commitment to leveraging cutting-edge technology and expertise to tackle the increasingly complex challenges of cryptocurrency theft and loss. By combining Wizard Hilton's world-class cybersecurity capabilities with the deep industry insights and recovery methodologies of Cyber Tech, this alliance is poised to revolutionize the way individuals and organizations safeguard their digital assets. Through a meticulous, multi-layered process, the team meticulously analyzes each case, employing advanced forensic techniques to trace the flow of stolen funds and identify potential recovery avenues. This rigorous approach not only maximizes the chances of successful Bitcoin retrieval, but also provides invaluable intelligence to enhance future security measures and prevention strategies. Importantly, the collaboration is built on a foundation of trust, transparency, and a genuine concern for the well-being of clients, ensuring that the recovery process is handled with the utmost care and discretion. As the cryptocurrency landscape continues to evolve, this strategic alliance between Wizard Hilton Cyber Tech stands as a shining example of how industry leaders can come together to safeguard digital assets, protect victims of cybercrime, and pave the way for a more secure and resilient cryptocurrency ecosystem. Email : wizardhiltoncybertech ( @ ) gmail (. ) com OR support ( @ ) wizardhiltoncybertech (.) com WhatsApp number +18737715701 Thanks.
01.09.25 17:03 WILLER GOODY

Don’t be misled by the countless stories you come across online many of them are unreliable and can easily steer you in the wrong direction. I personally tried several so-called recovery services, only to end up frustrated and disappointed every single time. But everything changed when I finally connected with a true tech professional who genuinely knows what they’re doing. Instead of wasting time and energy with unskilled hackers who make empty promises, it’s far better to work with a proven expert who can actually recover your stolen or lost cryptocurrency, including Bitcoins. [email protected] is, without a doubt, one of the most trustworthy and skilled blockchain specialists I’ve ever come across. If you’ve lost money to scammers, don’t give up. Reach out to their email today and take the first step toward getting your coins back. You can also contact them at +44 736-644-5035.
02.09.25 23:12 [email protected]

Sylvester Bryant has recovered over £151,000 in lost cryptocurrency. You can reach him for assistance via email at Yt7cracker@gmail. com. For direct communication, Sylvester Bryant is available on WhatsApp at +1 512 577 7957. This service proved invaluable when other recovery methods had failed. The money was recovered quickly and effectively. My financial situation was dire prior to this. This recovery has restored my financial stability. This reliable support was crucial during a difficult time. Mr. Bryant's skill in retrieving lost funds is remarkable.
03.09.25 07:17 Fabian3

Scammers have ruined the forex trading market, they have deceived many people with their fake promises on high returns. I learnt my lesson the hard way. I only pity people who still consider investing with them. Please try and make your researches, you will definitely come across better and reliable forex trading agencies that would help you yield profit and not ripping off your money. Also, if your money has been ripped-off by these scammers, you can report to a regulated crypto investigative unit who make use of software to get money back. They helped me recover my scammed funds back to me If you encounter with some issue make sure you contact ([email protected]) they're recovery expert and a very effective and reliable .
04.09.25 02:11 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.09.25 02:11 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.09.25 02:45 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.09.25 16:24 Melbourne

Sylvester Bryant has recovered over £151,000 in lost cryptocurrency. You can reach him for assistance via email at Yt7cracker@gmail. com. For direct communication, Sylvester Bryant is available on WhatsApp at +1 512 577 7957. This service proved invaluable when other recovery methods had failed. The money was recovered quickly and effectively. My financial situation was dire prior to this. This recovery has restored my financial stability. This reliable support was crucial during a difficult time. Mr. Bryant's skill in retrieving lost funds is remarkable.
04.09.25 18:11 WILLER GOODY

Don't fall for fake recovery scams online. Many people promise to help you get back lost crypto. They often just make excuses or don't deliver. This leads to more disappointment and lost hope. After many bad experiences, I finally found a real expert. They were professional and knew what they were doing. This person actually recovered my stolen coins. They gave me peace of mind back. If you lost crypto, contact them at [email protected] WhatsApp at +44 736-644-5035. Don't waste your time with others. They can help you get your money back.
05.09.25 08:14 Amostampari

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05.09.25 15:39 micheal1219

Losing USDT can be upsetting. This stablecoin usually stays close to the dollar's value. However, mistakes happen. Bugs in code, scams, or sending funds to the wrong place can cause losses. It might feel like the money is gone forever. But often, recovery is possible. Contact Marie at [email protected]. You can also reach her on Telegram at @marie_consultancy or WhatsApp at +17127594675.
06.09.25 09:21 Young Felix

[email protected] will Help you Recovery your Scammed Funds.
06.09.25 09:22 Young Felix

[email protected] will Help you Recovery your Scammed Funds. I just had to share my experience with Intel Fox Recovery Services. Initially, I was unsure if it would be possible to recover my BTC, however, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. Here is another caution; not all recovery services are legit. I personally lost 3 BTC from my Binance account due to a deceptive platform. If you happen to have suffered a similar loss, consider INTEL FOX RECOVERY SERVICES. Their services not only showcase a highly mannerism of professionalism but also effectiveness of how they handle and assist their clients. Therefore, I highly recommend their services to anyone seeking assistance in recovering their stolen BTC.
06.09.25 18:52 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
06.09.25 18:52 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
14:26 bajo171

Recovery experts and lawyers work together to get your stolen money back. They find lost assets. Lawyers guide you through legal steps. Watch out for scam signs. Be careful of offers promising big profits with no risk. Don't let anyone rush your investment decisions. Marie can help you get back lost Bitcoin. She also helps stop future scams. Contact her on WhatsApp at +1 7127594675. You can also email her at [email protected] or reach her on Telegram at @Marie_consultancy.
18:04 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18:04 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
19:21 DAVEortega1010

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