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[Перевод] Самопаркующийся авто за 500 строк кода

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TLDR


В этой статье мы научим авто самостоятельно парковаться с помощью генетического алгоритма.


Мы создадим первое поколение авто с произвольными геномами, которое будет вести себя примерно так:


8gbsullqgo6q2vikath_8qrhuim.gif



Примерно на сороковом поколении авто начнут понимать, что такое авто-парковка, и начнут приближаться к парковочному месту:


k96gh0vysbjgtgrea-2-6wc-9kg.gif



Другой пример с более сложной отправной точкой:


uskc64_n8y2t_l5rsr2rzigwjhi.gif



Да-да, авто врезаются в другие авто и паркуются не идеально, но это лишь 40 поколение с момента создания их мира, поэтому будьте милостивы и позвольте авто расти :D

Вы можете запустить симулятор, чтобы увидеть процесс эволюции прямо в браузере. Симулятор предоставляет следующие возможности:



Генетический алгоритм для этого проекта реализован на TypeScript. Полный исходный код проекта можно найти в этом репозитории.


Мы будем использовать генетический алгоритм для эволюционирования геномов авто. Однако статья затрагивает только основы алгоритма и не является исчерпывающим руководством по генетическим алгоритмам.

Готовы? Тогда поехали.


План


Шаг за шагом мы перейдем от высокоуровневой задачи создания самопаркующегося авто к простой низкоуровневой задаче поиска оптимальной комбинации 180 битов (поиска оптимального генома авто).


Мы собираемся сделать следующее:


  1. 💪🏻 Дать авто мышцы (мускулы) (двигатель, рулевое колесо), чтобы оно могло двигаться к парковочному месту.
  2. 👀 Дать авто глаза (сенсоры или датчики), чтобы оно могло видеть препятствия вокруг.
  3. 🧠 Дать авто мозг, чтобы оно могло управлять мышцами (движениями) на основе того, что оно видит (препятствия через сенсоры). Мозг будет простой чистой функцией движения = f(сенсоры)
  4. 🧬 Эволюционировать мозг, чтобы совершать правильные движения на основе данных сенсоров. Здесь мы будем применять генетический алгоритм. Поколение за поколением наша функция мозга движения = f(сенсоры) будет учиться парковаться.

Даем авто мышцы


Для того, чтобы двигаться, авто нужны "мышцы". Дадим авто 2 типа мышц:


  1. Мышцы двигателя — позволяют авто двигаться ↓ назад, ↑ вперед или ◎ продолжать движение (нейтральная передача)
  2. Мышцы руля — позволяют авто поворачивать ← влево, → вправо или ◎ двигаться прямо

С этими двумя мышцами авто сможет выполнять следующие движения:


me0fr_70mnb0amrrqomvygdx858.gif



В нашем случае мышцы — это приемники сигналов, поступающих от мозга каждые 100 мс. Мышцы ведут себя по-разному в зависимости от сигнала мозга. Мы поговорим о "мозге" ниже, а сейчас допустим, что мозг может посылать 3 сигнала для каждой мышцы: -1, 0 или +1.


type MuscleSignal = -1 | 0 | 1;

Например, мозг может послать сигнал со значением +1 в мышцу двигателя и авто начнет двигаться вперед. Сигнал -1 в мышцу двигателя двигает авто назад. Если мозг посылает сигнал -1 в мышцу руля, авто поворачивает налево и т.д.


Вот как значения сигнала мозга сопоставляются с действиями мышц в нашем случае:


Мышца -1 0 1
Двигатель ↓ Назад ◎ Нейтраль ↑ Вперед
Руль ← Налево ◎ Прямо → Направо

В случае ручной парковки в симуляторе при нажатии одной из клавиш WASD (или касании сенсорного экрана) в мышцы посылаются сигналы -1, 0 или +1.

Даем авто глаза


Перед тем, как наше авто научится парковаться с помощью мышц, оно должно иметь возможность "видеть" окружение. Дадим ему 8 глаз в форме сенсоров расстояния:


  • каждый сенсор обнаруживает препятствие на расстоянии от 0 до 4 м
  • каждый сенсор отправляет информацию о видимом препятствии в мозг каждые 100 мс
  • если сенсор не видит препятствия, он отправляет в мозг значение 0. Если значение сенсора чуть больше нуля (например, 0.01), значит, препятствие находится очень близко

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С помощью симулятора можно увидеть, как меняется цвет каждого сенсора в зависимости от близости препятствия.

type Sensors = number[];

Даем авто мозг


На данный момент наше авто может видеть и двигаться, но отсутствует "координатор", который будет трансформировать сигналы от глаз в соответствующие движения мышц. Нам нужен мозг.


Входные данные мозга


В качестве входных данных от сенсоров каждые 100 мс мозг будет получать 8 чисел с плавающей запятой, каждое в диапазоне [0...4]. Входные данные могут выглядеть так:


const sensors: Sensors = [s0, s1, s2, s3, s4, s5, s6, s7];
// например, 🧠 ← [0, 0.5, 4, 0.002, 0, 3.76, 0, 1.245]

Выходные данные


Каждые 100 мс мозг должен генерировать 2 целых числа:


  1. Сигнал для двигателя — engineSignal.
  2. Сигнал для руля — wheelSignal.

Каждое число должно иметь тип MuscleSignal и принимать одно из 3 значений: -1, 0 или +1.


Формулы/функции мозга


Принимая во внимание только что написанное, можно сказать, что мозг — это просто функция:


const { engineSignal, wheelSignal } = brainToMuscleSignal(
  brainFunction(sensors)
);
// например, { engineSignal: 0, wheelSignal: -1 } ← 🧠 ← [0, 0.5, 4, 0.002, 0, 3.76, 0, 1.245]

Где brainToMuscleSignal — это функция, конвертирующая сырые сигналы мозга (числа с плавающей запятой) в сигналы мышц (числа -1, 0 или +1), которые мышцы способны понять. Мы реализуем эту функцию преобразования позже.


Сейчас главный вопрос — что представляет собой brainFunction?


Чтобы сделать автомобиль умнее, а его движения более сложными, можно использовать многослойный перцептрон. Название немного пугающее, но это просто нейронная сеть с базовой архитектурой (думайте об этом как о формуле с большим количеством параметров/коэффициентов).


Я рассказывал о многослойном перцептроне в проектах homemade-machine-learning, machine-learning-experiments и nano-neuron.

Однако, вместо нейронных сетей, мы выберем гораздо более простой подход и будем использовать два линейных полинома с несколькими переменными (если быть точнее, то в каждом полиноме будет ровно 8 переменных, по 1 на сенсор), которые будут выглядеть примерно так:


engineSignal = brainToMuscleSignal(
  (e0 * s0) + (e1 * s1) + ... + (e7 * s7) + e8 // <- brainFunction
)

wheelSignal = brainToMuscleSignal(
  (w0 * s0) + (w1 * s1) + ... + (w7 * s7) + w8 // <- brainFunction
)

Где:


  • [s0, s1, ..., s7] — 8 переменных: значения сенсоров, являются динамическими
  • [e0, e1, ..., e8] — 9 коэффициентов для полинома двигателя. Авто должен будет этому научиться, значения будут статическими
  • [w0, w1, ..., w8] — 9 коэффициентов для полинома руля. Авто должен будет этому научиться, значения будут статическими

Ценой использования простой функции будет то, что мозг авто не сможет учиться сложным движениям, будет плохо обобщать и адаптироваться к неизвестному окружению. Но для парковки и в целях демонстрации работы генетического алгоритма этого должно быть достаточно.


Общую полиномиальную функцию можно реализовать следующим образом:


type Coefficients = number[];

// Функция для вычисления линейного полинома на основе коэффициентов и переменных
const linearPolynomial = (coefficients: Coefficients, variables: number[]): number => {
  if (coefficients.length !== (variables.length + 1)) {
    throw new Error('Количество коэффициентов и переменных не совпадает');
  }
  let result = 0;
  coefficients.forEach((coefficient: number, coefficientIndex: number) => {
    if (coefficientIndex < variables.length) {
      result += coefficient * variables[coefficientIndex];
    } else {
      // Последний коэффициент добавляется без умножения
      result += coefficient
    }
  });
  return result;
};

В этом случае мозг авто будет состоять из 2 полиномов и будет выглядеть так:


const engineSignal: MuscleSignal = brainToMuscleSignal(
  linearPolynomial(engineCoefficients, sensors)
);

const wheelSignal: MuscleSignal = brainToMuscleSignal(
  linearPolynomial(wheelCoefficients, sensors)
);

Результатом функции linearPolynomial является число с плавающей запятой. Функция brainToMuscleSignal должна конвертировать широкий диапазон чисел с плавающей запятой в 3 целых числа. Она будет это делать в 2 этапа:


  1. Преобразование любого числа с плавающей запятой (например, 0.456, 3673.45 или -280) в число с плавающей запятой в диапазоне (0...1) (например, 0.05 или 0.86).
  2. Преобразование числа с плавающей запятой в диапазоне (0...1) в целые числа -1, 0 или +1. Например, число, близкое к 0, будет преобразовано в -1, число, близкое к 0.5, будет преобразовано в 0, а число, близкое к 1, будет преобразовано в 1.

Для выполнения первого преобразования нам потребуется сигмоида, реализующая следующую формулу:


3ugloonpejunico7hnqwdrrdopa.png



Она преобразует любое число с плавающей запятой (ось x) в число в диапазоне (0...1) (ось y). Это именно то, что нам нужно.


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Вот как выглядят шаги преобразования на сигмовидном графике:


lxo-5lpjeaie9yosx-cpydvoiuq.png



Возможная реализация двух упомянутых выше преобразований:


// Вычисляет сигмоидное значение для переданного числа
const sigmoid = (x: number): number => {
  return 1 / (1 + Math.E ** -x);
};

// Преобразует сигмоидное значение (0...1) в сигналы мышц (-1, 0, +1).
// Параметр `margin` - это значение между 0 и 0.5:
// [0 ... (0.5 - margin) ... 0.5 ... (0.5 + margin) ... 1]
const sigmoidToMuscleSignal = (sigmoidValue: number, margin: number = 0.4): MuscleSignal => {
  if (sigmoidValue < (0.5 - margin)) {
    return -1;
  }
  if (sigmoidValue > (0.5 + margin)) {
    return 1;
  }
  return 0;
};

// Преобразует сигнал мозга в сигнал мышц
const brainToMuscleSignal = (rawBrainSignal: number): MuscleSignal => {
  const normalizedBrainSignal = sigmoid(rawBrainSignal);
  return sigmoidToMuscleSignal(normalizedBrainSignal);
}

Геном авто (ДНК)


Основной вывод предыдущего раздела: коэффициенты [e0, e1, ..., e8] и [w0, w1, ..., w8] определяют поведение авто. Эти 18 чисел вместе формируют уникальный геном авто (его ДНК, если угодно).

Геном авто в десятичной форме


Объединим коэффициенты [e0, e1, ..., e8] и [w0, w1, ..., w8] вместе для формирования генома авто в десятичной форме:


// Геном авто - это список десятичных чисел (коэффициентов)
const carGenomeBase10 = [e0, e1, ..., e8, w0, w1, ..., w8];
// например, carGenomeBase10 = [17.5, 0.059, -46, 25, 156, -0.085, -0.207, -0.546, 0.071, -58, 41, 0.011, 252, -3.5, -0.017, 1.532, -360, 0.157]

Геном авто в двоичной форме


Сделаем шаг вперед (на уровень генов) и преобразуем десятичные числа генома авто в двоичный формат (в 1 и 0).


Я подробно описал процесс преобразования чисел с плавающей запятой в двоичные числа в статье "Двоичное представление чисел с плавающей запятой". Взгляните на нее, если возникнут вопросы.

Пример преобразования числа с плавающей запятой в 16-битное двоичное число:


q_9whgngpzhj9yfbcp-zz7ufyx8.png



В нашем случае для уменьшения длины генома мы конвертируем каждое число в нестандартное 10-битное число (1 бит знака, 4 бита экспоненты и 5 дробных битов).


Всего у нас 18 коэффициентов, каждый будет конвертирован в 10-битное число. Это означает, что геном авто будет представлять собой массив из 0 и 1 длиной 18 * 10 = 180 бит.


Например, геном в десятичном формате, упомянутый выше, в двоичном формате будет выглядеть так:


type Gene = 0 | 1;

type Genome = Gene[];

const genome: Genome = [
  // Коэффициенты двигателя
  0, 1, 0, 1, 1, 0, 0, 0, 1, 1, // <- 17.5
  0, 0, 0, 1, 0, 1, 1, 1, 0, 0, // <- 0.059
  1, 1, 1, 0, 0, 0, 1, 1, 1, 0, // <- -46
  0, 1, 0, 1, 1, 1, 0, 0, 1, 0, // <- 25
  0, 1, 1, 1, 0, 0, 0, 1, 1, 1, // <- 156
  1, 0, 0, 1, 1, 0, 1, 1, 0, 0, // <- -0.085
  1, 0, 1, 0, 0, 1, 0, 1, 0, 1, // <- -0.207
  1, 0, 1, 1, 0, 0, 0, 0, 1, 1, // <- -0.546
  0, 0, 0, 1, 1, 0, 0, 1, 0, 0, // <- 0.071

  // Коэффициенты руля
  1, 1, 1, 0, 0, 1, 1, 0, 1, 0, // <- -58
  0, 1, 1, 0, 0, 0, 1, 0, 0, 1, // <- 41
  0, 0, 0, 0, 0, 0, 1, 0, 1, 0, // <- 0.011
  0, 1, 1, 1, 0, 1, 1, 1, 1, 1, // <- 252
  1, 1, 0, 0, 0, 1, 1, 0, 0, 0, // <- -3.5
  1, 0, 0, 0, 1, 0, 0, 1, 0, 0, // <- -0.017
  0, 0, 1, 1, 1, 1, 0, 0, 0, 1, // <- 1.532
  1, 1, 1, 1, 1, 0, 1, 1, 0, 1, // <- -360
  0, 0, 1, 0, 0, 0, 1, 0, 0, 0, // <- 0.157
];

О боже! Бинарный геном выглядит так загадочно. Но эти 180 нулей и единиц определяют поведение авто на парковке! Это все равно, что хакнуть чужую ДНК и точно определить, что делает каждый ген. Потрясающе!


Кстати, точные значения геномов и коэффициентов лучшего авто можно видеть на панели управления симулятора:


mydackjh-pnfcjtrnywoe0xxnow.png



Вот исходный код, который выполняет преобразование чисел с плавающей запятой из двоичного в десятичный формат (это понадобится мозгу для декодирования генома и создания мышечных сигналов на основе данных генома):


type Bit = 0 | 1;

type Bits = Bit[];

type PrecisionConfig = {
  signBitsCount: number,
  exponentBitsCount: number,
  fractionBitsCount: number,
  totalBitsCount: number,
};

type PrecisionConfigs = {
  custom: PrecisionConfig,
};

const precisionConfigs: PrecisionConfigs = {
  //  Специальный 10-битный объект для более быстрой эволюции
  custom: {
    signBitsCount: 1,
    exponentBitsCount: 4,
    fractionBitsCount: 5,
    totalBitsCount: 10,
  },
};

// Преобразует числа с плавающей запятой из бинарного в десятичный формат
function bitsToFloat(bits: Bits, precisionConfig: PrecisionConfig): number {
  const { signBitsCount, exponentBitsCount } = precisionConfig;

  // Определяем знак
  const sign = (-1) ** bits[0]; // -1^1 = -1, -1^0 = 1

  // Вычисляем значение экспоненты
  const exponentBias = 2 ** (exponentBitsCount - 1) - 1;
  const exponentBits = bits.slice(signBitsCount, signBitsCount + exponentBitsCount);
  const exponentUnbiased = exponentBits.reduce(
    (exponentSoFar: number, currentBit: Bit, bitIndex: number) => {
      const bitPowerOfTwo = 2 ** (exponentBitsCount - bitIndex - 1);
      return exponentSoFar + currentBit * bitPowerOfTwo;
    },
    0,
  );
  const exponent = exponentUnbiased - exponentBias;

  // Вычисляем значение дроби
  const fractionBits = bits.slice(signBitsCount + exponentBitsCount);
  const fraction = fractionBits.reduce(
    (fractionSoFar: number, currentBit: Bit, bitIndex: number) => {
      const bitPowerOfTwo = 2 ** -(bitIndex + 1);
      return fractionSoFar + currentBit * bitPowerOfTwo;
    },
    0,
  );

  // Объединяем все вместе
  return sign * (2 ** exponent) * (1 + fraction);
}

// Преобразует 8-битное двоичное число с плавающей запятой в десятичное
function bitsToFloat10(bits: Bits): number {
  return bitsToFloat(bits, precisionConfigs.custom);
}

Функция мозга для работы с двоичным геномом


Ранее функция мозга работала с полиномиальными коэффициентами engineCoefficients и wheelCoefficients напрямую. Однако сейчас эти коэффициенты кодируются в бинарную форму генома. Добавим функцию decodeGenome для извлечения коэффициентов из генома и перепишем функции мозга:


// Авто имеет 8 сенсоров расстояния
const CAR_SENSORS_NUM = 8;

// Дополнительный коэффициент, не связанный с сенсорами
const BIAS_UNITS = 1;

// Количество генов для кодирования каждого числового параметра в формулах
const GENES_PER_NUMBER = precisionConfigs.custom.totalBitsCount;

// Нам требуется 2 формулы для определения поведения авто:
// 1. Формула двигателя (входные данные: 8 сенсоров; выходные данные: -1 (назад), 0 (тоже направление), +1 (вперед))
// 2. Формула руля (входные данные: 8 сенсоров; выходные данные: -1 (влево), 0 (прямо), +1 (вправо))
const ENGINE_FORMULA_GENES_NUM = (CAR_SENSORS_NUM + BIAS_UNITS) * GENES_PER_NUMBER;
const WHEELS_FORMULA_GENES_NUM = (CAR_SENSORS_NUM + BIAS_UNITS) * GENES_PER_NUMBER;

// Длина бинарного генома авто
const GENOME_LENGTH = ENGINE_FORMULA_GENES_NUM + WHEELS_FORMULA_GENES_NUM;

type DecodedGenome = {
  engineFormulaCoefficients: Coefficients,
  wheelsFormulaCoefficients: Coefficients,
}

// Преобразует геном из бинарной формы в десятичную
const genomeToNumbers = (genome: Genome, genesPerNumber: number): number[] => {
  if (genome.length % genesPerNumber !== 0) {
    throw new Error('Неверное количество генов');
  }
  const numbers: number[] = [];
  for (let numberIndex = 0; numberIndex < genome.length; numberIndex += genesPerNumber) {
    const number: number = bitsToFloat10(genome.slice(numberIndex, numberIndex + genesPerNumber));
    numbers.push(number);
  }
  return numbers;
};

// Преобразует геном из бинарной формы в десятичную и
// делит геном на 2 набора коэффициентов (по одному на каждую мышцу)
const decodeGenome = (genome: Genome): DecodedGenome => {
  const engineGenes: Gene[] = genome.slice(0, ENGINE_FORMULA_GENES_NUM);
  const wheelsGenes: Gene[] = genome.slice(
    ENGINE_FORMULA_GENES_NUM,
    ENGINE_FORMULA_GENES_NUM + WHEELS_FORMULA_GENES_NUM,
  );

  const engineFormulaCoefficients: Coefficients = genomeToNumbers(engineGenes, GENES_PER_NUMBER);
  const wheelsFormulaCoefficients: Coefficients = genomeToNumbers(wheelsGenes, GENES_PER_NUMBER);

  return {
    engineFormulaCoefficients,
    wheelsFormulaCoefficients,
  };
};

// Обновляет функцию мозга для мышцы двигателя
export const getEngineMuscleSignal = (genome: Genome, sensors: Sensors): MuscleSignal => {
  const { engineFormulaCoefficients: coefficients } = decodeGenome(genome);
  const rawBrainSignal = linearPolynomial(coefficients, sensors);
  return brainToMuscleSignal(rawBrainSignal);
};

// Обновляет функцию мозга для мышцы руля
export const getWheelsMuscleSignal = (genome: Genome, sensors: Sensors): MuscleSignal => {
  const { wheelsFormulaCoefficients: coefficients } = decodeGenome(genome);
  const rawBrainSignal = linearPolynomial(coefficients, sensors);
  return brainToMuscleSignal(rawBrainSignal);
};

Постановка проблемы самопаркующегося автомобиля


Итак, мы свели высокоуровневую задачу самопаркующегося авто к простой задаче оптимизации, заключающейся в поиске оптимальной комбинации 180 единиц и нулей (поиске "достаточно хорошего" генома авто). Звучит просто, не так ли?

Наивный подход


Наивный подход поиска "достаточно хорошего" генома заключается в переборе всех комбинаций генов:


  1. [0, ..., 0, 0]
  2. [0, ..., 0, 1]
  3. [0, ..., 1, 0]
  4. [0, ..., 1, 1]
  5. и т.д.

Но обратимся к математике. 180 нулей или единиц дают нам 2^180 (или 1.53 * 10^54) возможных комбинаций. Допустим, проверка успешности парковки каждого авто занимает 15 сек. Допустим, мы можем запускать симуляцию для 10 авто одновременно. Тогда нам потребуется 15 * (1.53 * 10^54) / 10 = 2.29 * 10^54 [сек] или 7.36 * 10^46 [лет]. Придется ждать довольно долго. К слову, с рождения Христа прошло всего 2.021 * 10^3 [лет].


Генетический подход


Нам нужен более быстрый алгоритм поиска оптимального значения генома. Здесь в игру вступает генетический алгоритм. Мы можем не найти лучшего значения генома, но существует вероятность нахождения оптимального значения. И, что еще важнее, нам не нужно долго ждать. С помощью симулятора я нашел довольно хороший геном за 24 [часа].


Основы генетического алгоритма


Генетические алгоритмы (ГА), вдохновленные процессом естественного отбора, обычно используются для нахождения высококачественных решений задач оптимизации, полагаясь на биологически обусловленные операторы, такие как скрещивание (crossover), мутация (mutation) и отбор (selection).


Задача нахождения "достаточно хорошей" комбинации генов для каждого авто выглядит как задача оптимизации, поэтому велика вероятность того, что ГА хорошо с ней справится.


Мы не будем рассматривать ГА во всех подробностях, но на высоком уровне нам необходимо выполнить следующие шаги:


  1. СОЗДАНИЕ — самое первое поколение авто не может возникнуть из ниоткуда, поэтому мы сгенерируем набор произвольных геномов (набор бинарных массивов длиной 180). Например, мы можем создать ~1000 авто. Чем больше популяция, тем выше шансы быстро найти оптимальное решение (но тем медленнее будет выполняться программа).
  2. ОТБОР — необходимо отобрать лучших (наиболее приспособленных — fittest) особей для дальнейшего скрещивания (см. следующий шаг). Приспособленность каждой особи будет определяться функцией приспособленности (fitness function), которая в нашем случае будет показывать, насколько близко авто подобралось к парковочному месту. Чем ближе, тем лучше.
  3. СКРЕЩИВАНИЕ — простыми словами, мы позволяем "♂ авто отцам" "скрещиваться" с "♀ авто матерями" для смешения их геномов в пропорции ~50/50 и производства геномов "♂♀ авто детей". Идея в том, что дети могут быть лучше (или хуже) в парковке, получая лучшие (или худшие) биты от родителей.
  4. МУТАЦИЯ — в процессе скрещивания некоторые гены могут произвольно мутировать (1 и 0 в геноме потомка могут меняться местами). Это может привести к более широкому разнообразию геномов потомков и, как следствие, к более вариативному поведению авто. Представьте, что первый бит всех ~1000 авто был случайно установлен в 0. Единственный способ попробовать авто с первым битом, установленным в 1 — произвольные мутации. В тоже время частые мутации могут испортить хорошие геномы.
  5. Возврат к шагу 2 до тех пор, пока количество поколений не достигнет лимита (например, 100), или пока лучшие особи не достигнут ожидаемого показателя приспособленности (например, лучшее авто приблизится к парковочному месту ближе чем на 1 метр).

jip3pubxavgmjxee4akdh08ls2y.png



Развитие мозга авто с помощью генетического алгоритма


Создадим функции для каждого шага генетического алгоритма.


Функции шага создания


Функция createGeneration будет создавать массив произвольных геномов (популяцию или поколение) и будет принимать 2 параметра:


  • generationSize — размер популяции. Он будет сохраняться между поколениями
  • genomeLength — длина генома каждой особи. В нашем случае длина генома будет составлять 180

Каждый ген может быть 1 или 0 с вероятностью 50/50.


type Generation = Genome[];

type GenerationParams = {
  generationSize: number,
  genomeLength: number,
};

function createGenome(length: number): Genome {
  return new Array(length)
    .fill(null)
    .map(() => (Math.random() < 0.5 ? 0 : 1));
}

function createGeneration(params: GenerationParams): Generation {
  const { generationSize, genomeLength } = params;
  return new Array(generationSize)
    .fill(null)
    .map(() => createGenome(genomeLength));
}

Функции шага мутации


Функция mutate будет произвольно мутировать некоторые гены на основе значения mutationProbability (вероятность мутации).


Например, mutationProbability = 0.1 означает 10% вероятность, что ген будет мутирован. Допустим, у нас есть геном длиной 10, который выглядит как [0, 0, 0, 0, 0, 0 ,0 ,0 ,0 ,0]. Тогда после мутации есть шанс мутирования одного гена и получения генома, который выглядит как [0, 0, 0, 1, 0, 0 ,0 ,0 ,0 ,0].


// Число между 0 и 1
type Probability = number;

// @see: https://en.wikipedia.org/wiki/Mutation_(genetic_algorithm)
function mutate(genome: Genome, mutationProbability: Probability): Genome {
  for (let geneIndex = 0; geneIndex < genome.length; geneIndex += 1) {
    const gene: Gene = genome[geneIndex];
    const mutatedGene: Gene = gene === 0 ? 1 : 0;
    genome[geneIndex] = Math.random() < mutationProbability ? mutatedGene : gene;
  }
  return genome;
}

Функции шага скрещивания


Функция mate принимает геномы отца и матери и производит 2 потомков. В процессе скрещивания также возможна мутация (как в реальной жизни).


Каждый бит потомка будет определяться на основе значений соответствующих битов генома отца или матери. Вероятность наследования бита отца или матери составляет 50%. Допустим, у нас есть геном длиной 4 (для простоты):


Геном отца:           [0, 0, 1, 1]
Геном матери:         [0, 1, 0, 1]
                       ↓  ↓  ↓  ↓
Возможный потомок #1: [0, 1, 1, 1]
Возможный потомок #2: [0, 0, 1, 1]

В приведенном примере не учитывается мутация.


Реализация функции:


// Выполняет равномерное скрещивание: каждый бит выбирается от каждого предка с равной вероятностью
// @see: https://en.wikipedia.org/wiki/Crossover_(genetic_algorithm)
function mate(
  father: Genome,
  mother: Genome,
  mutationProbability: Probability,
): [Genome, Genome] {
  if (father.length !== mother.length) {
    throw new Error('Невозможно скрестить разные виды');
  }

  const firstChild: Genome = [];
  const secondChild: Genome = [];

  for (let geneIndex = 0; geneIndex < father.length; geneIndex += 1) {
    firstChild.push(
      Math.random() < 0.5 ? father[geneIndex] : mother[geneIndex]
    );
    secondChild.push(
      Math.random() < 0.5 ? father[geneIndex] : mother[geneIndex]
    );
  }

  return [
    mutate(firstChild, mutationProbability),
    mutate(secondChild, mutationProbability),
  ];
}

Функции шага отбора


Для выбора лучших особей нам нужен способ определения приспособленности каждого генома. Для этого мы будем использовать так называемую функцию приспособленности (фитнес-функцию).


Фитнес-функция всегда связана с конкретной решаемой задачей, она не является универсальной. В нашем случае фитнес-функция будет измерять расстояние между авто и парковочным местом. Чем ближе авто к месту, тем лучше его приспособленность. Мы реализуем фитнес-функцию немного позже, сейчас же представим для нее интерфейс:


type FitnessFunction = (genome: Genome) => number;

Допустим, у нас есть показатели приспособленности для каждой особи в популяции. Допустим также, что мы отсортировали особей по этому показателю, поэтому первая особь является самой приспособленной. Как выбирать предков из этого массива? Отбор должен выполняться таким образом, чтобы особи с более высоким показателем приспособленности чаще выбирались для скрещивания. С этим нам поможет функция weightedRandom:


// Извлекает произвольный элемент на основе его веса.
// Элементы с большим весом извлекаются чаще
const weightedRandom = <T>(items: T[], weights: number[]): { item: T, index: number } => {
  if (items.length !== weights.length) {
    throw new Error('Разное количество элементов и весов');
  }

  // Готовим массив совокупных весов.
  // Например:
  // - weights = [1, 4, 3]
  // - cumulativeWeights = [1, 5, 8]
  const cumulativeWeights: number[] = [];
  for (let i = 0; i < weights.length; i += 1) {
    cumulativeWeights[i] = weights[i] + (cumulativeWeights[i - 1] || 0);
  }

  // Получаем произвольное число в диапазоне [0...sum(weights)]
  // Например:
  // - weights = [1, 4, 3]
  // - maxCumulativeWeight = 8
  // - диапазоном для произвольного числа является [0...8]
  const maxCumulativeWeight = cumulativeWeights[cumulativeWeights.length - 1];
  const randomNumber = maxCumulativeWeight * Math.random();

  // Извлекаем произвольный элемент на основе его веса.
  // Элементы с большим весом извлекаются чаще
  for (let i = 0; i < items.length; i += 1) {
    if (cumulativeWeights[i] >= randomNumber) {
      return {
        item: items[i],
        index: i,
      };
    }
  }
  return {
    item: items[items.length - 1],
    index: items.length - 1,
  };
};

Использование этой функции довольно простое. Допустим, мы очень любим бананы и хотим есть их чаще, чем клубнику. Тогда мы вызываем const fruit = weightedRandom(['banana', 'strawberry'], [9, 1]), и в ≈9 из 10 случаев значение переменной fruit будет banana, а в ≈1 случае — strawberry.


Во избежание потери лучших особей (назовем их победителями) в процессе скрещивания добавим параметр longLivingChampionsPercentage. Например, longLivingChampionsPercentage = 10 будет означать, что 10% лучших авто будут перенесены в следующее поколение. О победителях можно думать, как об особях, которые живут долгую жизнь и видят своих детей и даже внуков.


Реализация функции select:


// Число между 0 и 100
type Percentage = number;

type SelectionOptions = {
  mutationProbability: Probability,
  longLivingChampionsPercentage: Percentage,
};

// @see: https://en.wikipedia.org/wiki/Selection_(genetic_algorithm)
function select(
  generation: Generation,
  fitness: FitnessFunction,
  options: SelectionOptions,
) {
  const {
    mutationProbability,
    longLivingChampionsPercentage,
  } = options;

  const newGeneration: Generation = [];

  const oldGeneration = [...generation];
  // Первая особь самая приспособленная
  oldGeneration.sort((genomeA: Genome, genomeB: Genome): number => {
    const fitnessA = fitness(genomeA);
    const fitnessB = fitness(genomeB);
    if (fitnessA < fitnessB) {
      return 1;
    }
    if (fitnessA > fitnessB) {
      return -1;
    }
    return 0;
  });

  // Долгожители переходят в новое поколение
  const longLiversCount = Math.floor(longLivingChampionsPercentage * oldGeneration.length / 100);
  if (longLiversCount) {
    oldGeneration.slice(0, longLiversCount).forEach((longLivingGenome: Genome) => {
      newGeneration.push(longLivingGenome);
    });
  }

  // Получаем данные о приспособленности каждой особи
  const fitnessPerOldGenome: number[] = oldGeneration.map((genome: Genome) => fitness(genome));

  // Заполняем следующее поколение до тех пор, пока оно не сравняется с предыдущим
  while (newGeneration.length < generation.length) {
    // Выбираем произвольного отца и мать из популяции.
    // Лучшие особи выбираются чаще
    let father: Genome | null = null;
    let fatherGenomeIndex: number | null = null;
    let mother: Genome | null = null;
    let matherGenomeIndex: number | null = null;

    // Для производства потомства требуется две разные особи
    while (!father || !mother || fatherGenomeIndex === matherGenomeIndex) {
      const {
        item: randomFather,
        index: randomFatherGenomeIndex,
      } = weightedRandom<Genome>(generation, fitnessPerOldGenome);

      const {
        item: randomMother,
        index: randomMotherGenomeIndex,
      } = weightedRandom<Genome>(generation, fitnessPerOldGenome);

      father = randomFather;
      fatherGenomeIndex = randomFatherGenomeIndex;

      mother = randomMother;
      matherGenomeIndex = randomMotherGenomeIndex;
    }

    // Получаем потомство
    const [firstChild, secondChild] = mate(father, mother, mutationProbability);

    newGeneration.push(firstChild);

    // В зависимости от числа долгожителей, место
    // может остаться только для одного потомка
    if (newGeneration.length < generation.length) {
      newGeneration.push(secondChild);
    }
  }

  return newGeneration;
}

Функция приспособленности


Приспособленность авто будет определяться расстоянием от авто до парковочного места. Чем больше расстояние, тем ниже приспособленность.


Итоговое расстояние будет вычисляться как среднее расстояние от 4 колес авто к соответствующим 4 углам парковочного места. Это расстояние мы будем называть loss. Оно будет обратно пропорционально fitness.


ohfieq23w83kos1ie0_am8ex9ea.png



Вычисление расстояния между каждым колесом и каждым углом отдельно (вместо вычисления расстояния от центра автомобиля до центра парковочного места) позволит автомобилю сохранять правильную ориентацию относительно места.


Расстояние между двумя точками в пространстве будет рассчитываться на основе теоремы Пифагора следующим образом:


type NumVec3 = [number, number, number];

// Вычисляет расстояние XZ между 2 точками в пространстве.
// Вертикальное расстояние Y не принимается в расчет
const euclideanDistance = (from: NumVec3, to: NumVec3) => {
  const fromX = from[0];
  const fromZ = from[2];
  const toX = to[0];
  const toZ = to[2];
  return Math.sqrt((fromX - toX) ** 2 + (fromZ - toZ) ** 2);
};

Расстояние (loss) между автомобилем и парковочным местом будет рассчитываться так:


type RectanglePoints = {
  fl: NumVec3, // передний левый угол
  fr: NumVec3, // передний правый
  bl: NumVec3, // задний левый
  br: NumVec3, // задний правый
};

type GeometricParams = {
  wheelsPosition: RectanglePoints,
  parkingLotCorners: RectanglePoints,
};

const carLoss = (params: GeometricParams): number => {
  const { wheelsPosition, parkingLotCorners } = params;

  const {
    fl: flWheel,
    fr: frWheel,
    br: brWheel,
    bl: blWheel,
  } = wheelsPosition;

  const {
    fl: flCorner,
    fr: frCorner,
    br: brCorner,
    bl: blCorner,
  } = parkingLotCorners;

  const flDistance = euclideanDistance(flWheel, flCorner);
  const frDistance = euclideanDistance(frWheel, frCorner);
  const brDistance = euclideanDistance(brWheel, brCorner);
  const blDistance = euclideanDistance(blWheel, blCorner);

  return (flDistance + frDistance + brDistance + blDistance) / 4;
};

Поскольку fitness должна быть обратно пропорциональна loss, она рассчитывается следующим образом:


const carFitness = (params: GeometricParams): number => {
  const loss = carLoss(params);
  // Добавляем 1 во избежание деления на 0
  return 1 / (loss + 1);
};

Значения fitness и loss для конкретного генома и текущего положения авто можно увидеть на панели инструментов симулятора:


n-z6dog51djwyiuz_nl-a5c-obg.png



Запуск эволюции


Объединим функции эволюции. Мы собираемся "создать мир", запустить цикл эволюции, заставить время течь, поколение развиваться, а авто учиться парковаться.


Чтобы получить показатели приспособленности каждого авто, нам нужно запустить симуляцию в виртуальном трехмерном мире. Симулятор эволюции делает именно это — он запускает приведенный ниже код в симуляторе, созданном с помощью Three.js:


// Пример настройки эволюции.
// Настройки устанавливаются в симуляторе
const GENERATION_SIZE = 1000;
const LONG_LIVING_CHAMPIONS_PERCENTAGE = 6;
const MUTATION_PROBABILITY = 0.04;
const MAX_GENERATIONS_NUM = 40;

// Функция приспособленности.
// Это похоже на ежегодный осмотр авто у врача
const carFitnessFunction = (genome: Genome): number => {
  // Симулятор вычисляет и сохраняет показатели приспособленности каждого авто в карте `fitnessValues`.
  // Здесь мы просто получаем предварительно вычисленное значение для авто в текущем поколении
  const genomeKey = genome.join('');
  return fitnessValues[genomeKey];
};

// Создаем "мир" с первым поколением авто
let generationIndex = 0;
let generation: Generation = createGeneration({
  generationSize: GENERATION_SIZE,
  genomeLength: GENOME_LENGTH, // <- 180 генов
});

// Запускаем "время"
while (generationIndex < MAX_GENERATIONS_NUM) {
  // ЗДЕСЬ НЕОБХОДИМО МОДЕЛИРОВАНИЕ, чтобы предварительно рассчитать показатели приспособленности.

  // Отбор, скрещивание и мутирование текущего поколения
  generation = select(
    generation,
    carFitnessFunction,
    {
      mutationProbability: MUTATION_PROBABILITY,
      longLivingChampionsPercentage: LONG_LIVING_CHAMPIONS_PERCENTAGE,
    },
  );

  // Заставляем "время" течь
  generationIndex += 1;
}

// Получаем лучшую особь последнего поколения
const fittestCar = generation[0];

После запуска функции select массив generation сортируется по значениям приспособленности в порядке убывания. Таким образом, наиболее приспособленное авто всегда будет первым в массиве.


Первое поколение авто с произвольными геномами будет вести себя примерно так:


8gbsullqgo6q2vikath_8qrhuim.gif



Примерно на сороковом поколении авто начинают понимать, что такое авто-парковка, и начинают приближаться к парковочному месту:


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Другой пример с более сложной отправной точкой:


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Авто по пути сталкиваются с другими авто, а также не идеально паркуются, но это всего лишь 40 поколение, так что дайте авто еще немного времени на обучение.


Из поколения в поколение мы видим, как значения loss снижаются (что означает, что значения fitness растут). P50 Avg Loss показывает среднее значение loss (среднее расстояние от авто до места парковки) для 50% наиболее приспособленных авто. Min Loss показывает значение loss самого приспособленного авто в каждом поколении.


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Видно, что в среднем 50% самых приспособленных авто поколения учатся приближаться к парковочному месту (от 5,5 м до 3,5 м за 35 поколений). Тенденция значений Min Loss менее очевидна (от 1 м до 0,5 м с некоторым шумом), однако на приведенной выше анимации видно, что авто выучили некоторые основные движения для парковки.


Заключение


В этой статье мы свели задачу высокого уровня по созданию самопаркующегося авто к простой задаче низкого уровня по поиску оптимальной комбинации 180 единиц и нулей (поиск оптимального генома автомобиля).


Затем мы применили генетический алгоритм для нахождения оптимального генома авто. Это позволило нам получить довольно хорошие результаты за несколько часов моделирования (вместо многих лет в случае использования наивного подхода).


Вы можете запустить симулятор, чтобы увидеть процесс эволюции прямо в браузере. Он предоставляет следующие возможности:



Полный исходный код, показанный в этой статье, можно найти в этом репозитории.


Проблемы и нерешенные задачи:


  • мозг авто слишком упрощен и использует линейные уравнения вместо, скажем, нейронных сетей. Это делает авто неспособным адаптироваться к новому окружению или новым типам парковок
  • значение приспособленности авто не уменьшается, когда оно врезается в другое авто. Поэтому авто не "чувствует" никакой вины за ДТП
  • симулятор эволюции нестабилен. Это означает, что один и тот же геном авто может давать разные значения приспособленности, что делает эволюцию менее эффективной
  • симулятор эволюции также очень тяжел с точки зрения производительности, что замедляет процесс эволюции, поскольку мы не можем обучать, скажем, 1000 авто одновременно
  • кроме того, для моделирования эволюции симулятору требуется, чтобы вкладка браузера была открыта и активна
  • и др.

Однако цель этой статьи заключалась в том, чтобы немного развлечься, изучая, как работает генетический алгоритм, а не в том, чтобы создать готовую к производству беспилотную Tesla. Итак, даже несмотря на упомянутые выше проблемы, я надеюсь, что вы хорошо провели время, читая эту статью.


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Новости, обзоры продуктов и конкурсы от команды Timeweb.Cloud — в нашем Telegram-канале ↩

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  • 27.11.24 00:39 [email protected]

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  • 22.01.25 21:43 DoraJaimes23

    Recovery expert. I lost my bitcoin to fake blockchain impostors on Facebook, they contacted me as blockchain official support and i fell stupidly for their mischievous act, this made them gain access into my blockchain wallet whereby 7.0938 btc was stolen from my wallet in total .I was almost in a comma and dumbfounded because this was all my savings i relied on . Then I made a research online and found a recovery expert , with the contact address- { RECOVERYHACKER101 (@) GMAIL . COM }... I wrote directly to the specialist explaining my loss. Hence, he helped me recover a significant part of my investment just after 2 days he helped me launch the recovery program , and the culprits were identified as well , all thanks to his expertise . I hope I have been able to help someone as well . Reach out to the recovery specialist to recover you lost funds from any form of online scam Thanks

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  • 26.01.25 03:54 [email protected]

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  • 28.01.25 21:48 [email protected]

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  • 02.02.25 20:53 Michael9090

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  • 06.02.25 19:42 Marta Golomb

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  • 08.02.25 05:45 [email protected]

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  • 08.02.25 05:46 [email protected]

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  • 13.02.25 17:37 eunice49954

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  • 16.02.25 01:01 Peter

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  • 21.02.25 07:42 daniel231101

    I never thought I would fall victim to a crypto scam until I was convinced of a crypto investment scam that saw me lose all my entire assets worth $487,000 to a crypto investment manager who convinced me I could earn more from my investment. I thought it was all gone for good but I kept looking for ways to get back my stolen crypto assets and finally came across Ethical Hack Recovery, a crypto recovery/spying company that has been very successful in the recovery of crypto for many other victims of crypto scams and people who lost access to their crypto. I’m truly grateful for their help as I was able to recover my stolen crypto assets and get my life back together. I highly recommend their services EMAIL ETHICALHACKERS009 AT @GMAIL DOT COM whatsapp +14106350697

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    Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.

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    Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.

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    Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.

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  • 08.03.25 04:42 andreassenhedda

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    CRYPTOCURRENCY RECOVERY FIRM DUNAMIS CYBER SOLUTION

  • 15.03.25 00:29 irenmroma

    CRYPTOCURRENCY RECOVERY FIRM DUNAMIS CYBER SOLUTIONIn 2025, I never imagined I would fall victim to a phishing scam, but that’s exactly what happened to me. As a graphic designer in California, I spend a lot of time online and thought I was pretty savvy when it came to spotting potential scams. But one day, I received an email that seemed to be from my bank, Wells Fargo. It looked official and warned me about suspicious activity on my account. The message instructed me to click a link and verify my account details to prevent further issues. Trusting it was legitimate, I followed the instructions and entered my personal banking information.Unfortunately, it was a trap. The email wasn’t from my bank at all. It was from a hacker who had gained access to my sensitive information. Soon after, I noticed a significant withdrawal of $2,300 from my account. Panicked, I contacted Wells Fargo immediately, but despite their efforts, they weren’t able to recover the lost funds. I felt helpless and frustrated, unsure of what to do next.That’s when I heard about DUNAMIS CYBER SOLUTION. Desperate for help, I reached out to their team, and they quickly got to work. They began by investigating the scam and managed to track the hacker’s IP address. They didn’t stop there they worked directly with Wells Fargo to share the findings and helped them investigate further.Thanks to their expertise and fast action,DUNAMIS CYBER SOLUTION was able to facilitate the recovery of $1,800. While I didn’t get the full $2,300 back, I was incredibly grateful for their efforts. It felt like a weight had been lifted, knowing that some of my money had been recovered.This experience taught me an important lesson about online security and the dangers of phishing scams. I was lucky to find DUNAMIS CYBER SOLUTION, and I’m thankful for their support in helping me get back a significant portion of what I lost. Their professionalism and dedication made a stressful situation much more manageable, and I now know how crucial it is to be vigilant and seek help when dealing with online fraud. [email protected] +13433030545

  • 15.03.25 14:34 spencerwerner

    It wasn’t an easy process, and it required patience, but the team’s dedication, attention to detail, and methodical approach paid off. I felt an overwhelming sense of relief and gratitude. What had once seemed like a permanent loss was now being reversed, thanks to the help of Tech Cyber Force Recovery. Not only did the recovery restore my financial situation, but it also restored my sense of trust and confidence. I had almost given up hope, but now, with my funds recovered, I feel like I can move forward. I’ve learned valuable lessons from this experience, and I’m more cautious about my financial decisions in the future. What began as a desperate search on Red Note turned into a life-changing recovery. Thanks to Tech Cyber Force Recovery, I now feel more hopeful about my financial future, with the knowledge that recovery is possible. telegram (@)techcyberforc texts (+1 5.6.1.7.2.6.3.6.9.7)

  • 17.03.25 02:30 [email protected]

    "Work smart and not hard" that's these scammers' slogan. They lure you with promises of luxury and lifetime riches without you doing anything besides investing in their investment schemes, and if you ever fall prey, they will siphon you of every penny dry. I fell victim to it, but was fortunate to be helped by the best recovery expert (TRUST GEEKS HACK EXPERT). They are more than just recovery specialists; they are allies in the fight against online fraud. Trust their expertise and let them guide you towards reclaiming control over your financial future.for assistance, visit website https://trustgeekshackexpert.com/ Wh@t's A p p  +1-7-1-9-4-9-2-2-6-9-3 <> E mail: Trustgeekshackexpert{@}fastservice{.}com

  • 17.03.25 12:20 browne

    When trying to recover lost cryptocurrency, it is important to enlist the assistance of recovery specialists who have knowledge in monitoring and evaluating digital assets, as they are better equipped to navigate the complex digital currency market and locate any stolen assets. As a result, I advise you to get in touch with forensic asset firm. Email: [email protected]

  • 17.03.25 13:14 vinnypraise

    SPEAK WITH A LICENSED ALPHA KEY BTC/USDT RECOVERY HACKER As of right now, ALPHA KEY RECOVERY is the only authorized and genuine recovery hacker that I will recommend to anyone in the world for a very good reason. I can't express how grateful I am to them for helping me overcome my depression; they are truly a blessing in disguise. If you have any data recovery concerns, you should contact ALPHA KEY RECOVERY.

  • 17.03.25 13:14 vinnypraise

    SPEAK WITH A LICENSED ALPHA KEY BTC/USDT RECOVERY HACKER As of right now, ALPHA KEY RECOVERY is the only authorized and genuine recovery hacker that I will recommend to anyone in the world for a very good reason. I can't express how grateful I am to them for helping me overcome my depression; they are truly a blessing in disguise. If you have any data recovery concerns, you should contact ALPHA KEY RECOVERY.

  • 17.03.25 15:23 Charlesagrimes

    Running the small business was hard enough in itself without having to suffer such stress as loss of access to a Bitcoin wallet. I had put $532,000 into Bitcoin for my business in case of emergencies, but one day, the wallet was nowhere to be seen. I had trusted it to my friend while he managed my financials, little knowing he would prove less than trustworthy. I couldn't envision the gut-wrenching feeling coupled with a great feeling of folly for having placed so much trust in someone else's hands. I could not believe my eyes to see that all of my digital fortune was gone. Literally, every second of my busy day froze in disbelief as I tried to conceive the loss. I remembered the hours I had put into building my business, securing my investments, and planning for the future. Now, I was staring at an empty wallet that once held my safety net. This hit me hard-not only had I lost $532,000, but the betrayal hurt further because it came from someone I considered reliable. The emotional toll was huge, and this financial hit could cripple my business. In the midst of my despair, I knew I had to act fast. I went online, searching everywhere for a solution through which I could get back in control of my finances. That is where I came across Digital resolution services. I knew them for recovery related to lost crypto, but also somehow skeptical. At that point, it looked like I had nothing left to lose. I reached out, and with our very first conversation, I gained the impression that they did understand my situation. Their team was nothing short of extraordinary. They approached my case with professionalism and empathy, meticulously explaining the recovery process in clear, simple terms that alleviated some of my anxiety. They set realistic expectations while promising to do everything possible to retrieve my funds. Over the next several weeks, they worked tirelessly, employing advanced blockchain forensic techniques and sophisticated tracking tools that I’d never even heard of before. It kept me informed with constant updates and gave me hope in this desperate time. Finally, it came-the day Digital resolution services recovered my lost Bitcoin. Relief overwhelmed me, and it wasn't all about the money but control and self-trust in my financial future. This experience taught me the most valuable lesson: never lose control over your own financial security and never put blind trust in anyone. With Digital resolution services, I recovered not only my $532,000 but also learned how to protect my assets better in the future. Contact Digital Resolution Services: Email: digitalresolutionservices (@) myself. c o m WhatsApp: +1 (361) 260-8628 Stay protected Charles agrimes

  • 18.03.25 23:49 lindaspringer311

    My name is Linda Springer, and I am part of an ultra-high-net-worth family in the United Kingdom. Managing our wealth has always been a top priority, and I’ve always looked for opportunities to grow and diversify our investments. When I was introduced to an offshore banking investment program called Market Fund, it seemed like the perfect opportunity to enhance our portfolio. The program promised impressive returns, and with professional advisors and seemingly legitimate documentation, I felt confident that it was a secure investment. However, my trust in Market Fund quickly proved to be misplaced. Initially, everything seemed to go smoothly. Reports showed strong growth, and I was reassured by the continuous updates from the program’s advisors. But when I tried to withdraw my funds, everything took a sharp turn. The Market Fund website became unresponsive, and all my attempts to reach the advisors through emails and phone calls went unanswered. It was then that I realized I had fallen victim to a scam, and the £60,000 I had invested was gone. Feeling devastated and unsure of what to do next, I turned to a friend I had met at a club called The Vault. During one of our conversations, Alex, a fellow club-goer, mentioned how Rapid Digital Recovery had helped them recover funds from a similar fraudulent scheme. Intrigued by their success story, I decided to reach out to Rapid Digital Recovery....Whatsapp: +1 4 14 80 71 4 85.. hoping they could help me recover my money as well. From the moment I contacted them, the team at Rapid Digital Recovery took immediate action. They began investigating the scam and used advanced technology to trace the origins of Market Fund. Through their efforts, they uncovered a network of fake websites and shell companies that had been designed to deceive investors like myself. Thanks to their persistence and expertise, Rapid Digital Recovery successfully recovered the £60,000 I had lost.....Email: rapid digital recovery (@) execs. com.. Not only did they restore my financial security, but they also exposed the fraudulent operation behind Market Fund, preventing others from falling victim to the same scam. While the experience was incredibly stressful, it served as an important reminder of the need for due diligence and professional oversight when dealing with offshore investments. I am incredibly grateful to Rapid Digital Recovery for their support, expertise, and determination in recovering my funds. Telegram: h t t p s: // t. me /Rapiddigitalrecovery1

  • 20.03.25 17:21 katherineingram

    Dr. Katherine Ingram

  • 20.03.25 17:21 katherineingram

    CRYPTOCURRENCY RECOVERY FIRM \ FOLKWIN EXPERT RECOVERY. I, Dr. Katherine Ingram, had always been committed to giving my best to my patients in Melbourne, Australia. But everything changed when I was suddenly hit with a medical malpractice lawsuit. The stress and anxiety that came with the legal battle left me feeling overwhelmed, and I was desperate for a solution. In my search for help, I found "MedLegal Solutions," a firm that promised a quick and easy resolution for just $27,000 AUD. They assured me they would handle everything swiftly, easing my burden. Desperate to resolve the situation, I handed over the money, trusting them to take care of the rest. However, as the weeks turned into months, I heard nothing. Calls went unanswered, emails went ignored, and I began to feel more and more helpless. It became clear that something was wrong, and I realized I had been deceived. The more I tried to contact them, the more I found myself being ignored. I soon understood that I was much deeper than I had initially feared. That’s when I turned to Folkwin Expert Recovery for help. From the moment I contacted them, they sprang into action. They immediately began investigating MedLegal Solutions and quickly uncovered a massive web of deceit. It turned out that MedLegal Solutions wasn’t just an incompetent firm; it was part of an elaborate scam. They were operating under multiple false identities across Sydney, Brisbane, and regional Victoria, targeting vulnerable individuals like myself. Folkwin Expert Recovery didn’t just uncover the fraud, they took it one step further. They worked tirelessly with authorities in Melbourne to track down and dismantle the fraudulent operation. They didn’t rest until MedLegal Solutions was shut down for good, ensuring that no one else would fall victim to their schemes. Thanks to their expert recovery efforts, I was able to recover every single cent of my $27,000.This entire ordeal was a wake-up call for me. It was a difficult and stressful time, but ultimately, I came out victorious. Thanks to Folkwin Expert Recovery, not only did I get my money back, but I also found peace knowing that MedLegal Solutions was no longer in business and that their fraudulent operation had been dismantled. It was a painful lesson, but in the end, my finances were intact, and I could move forward with a sense of closure. FOLKWIN EXPERT RECOVERY DETAILS\\ Telegram: @Folkwin_expert_recovery Or Email: Folkwinexpertrecovery(@)tech-center DOT com   Regards, Dr. Katherine Ingram.

  • 20.03.25 19:34 ysabelbristol

    I am YSABEL, a medical student living with my grandmother. For the past few years, I’ve been diligently saving my grandmother’s money to ensure she has a comfortable and secure future. I invested over $40,000 into an online platform, which promised significant returns. After completing several steps, they told me that to withdraw the money, I needed to make another payment of $73,000. As a medical student, my time is already stretched thin, and managing my grandmother’s finances on top of my studies was overwhelming. That's when a costudent, who had gone through a similar experience, recommended MUYERN TRUST HACKER. Initially, I was skeptical. After all, how could anyone help me recover the money I had already lost to such a sophisticated scam? But after reaching out to them, I quickly realized I was in good hands. From the very first conversation, the team at MUYERN TRUST HACKER was professional, compassionate, and knowledgeable. They understood the gravity of my situation and took the time to thoroughly review all the details of my case. They explained the recovery process clearly, step by step, and assured me that they would work tirelessly to help recover the funds I had lost. Their expertise in dealing with online fraud was evident, and for the first time in weeks, I felt a sense of hope that I could regain control over this situation. The recovery process wasn’t immediate, and there were moments of uncertainty, but MUYERN TRUST HACKER remained dedicated and persistent. As someone who has been working hard to save and protect my grandmother’s money, it was devastating to feel like I might lose everything to a scam. But MUYERN TRUST HACKER gave me the support, expertise, and guidance I needed to recover what was lost. I’m deeply grateful for their help, and I now have a renewed sense of hope that I can protect my grandmother’s future. If you find yourself in a similar situation, feeling trapped and overwhelmed by a fraudulent company, I highly recommend MUYERN TRUST HACKER. (Whats A p p at + 1 (440) (335) (0205)  Their team provides real solutions, and their professionalism and dedication were a lifeline for me during a very difficult time. Thanks to them, I feel confident that I can move forward and continue to protect my grandmother’s savings. Alternatively, contact their tele gram also if you need immediate attention at muyerntrusthackertech.

  • 21.03.25 21:19 carmenbeechum643

    (Telegram: https:// t. me/Pro_ Wizard_ Gilbert_ Recovery) Email (pro wizard gilbert recovery (@) engineer. com) I never imagined I would fall victim to a cryptocurrency scam, but that's exactly what happened. My name is [Carmen Beechum, and I invested $500,000 into what | believed was a legitimate trading platform. Everything appeared professional-the website was well-designed, customer service was responsive, and my trading account even showed promising returns.It all seemed too good to be false.However, when I attempted to withdraw my funds, I was met with endless delays and excuses. First, they claimed there were technical issues, then they needed additional verification, and finally, they requested a release fee before processing my withdrawal. Despite complying with their demands, my account was eventually frozen, and all communication from the platform ceased. That's when reality hit me—l had been scammed out of half a million dollars. Desperate to find a way to recover my money, I searched online for solutions. That's when I came across PRO WIZARD GIlBERT RECOVERY, a company dedicated to helping victims of online financial fraud. At first, I was skeptical-after all, I had already been deceived once, and the last thing I wanted was to fall for another scam. But after speaking with their team and reviewing their success stories, I decided to take a chance.Their experts immediately got to work, using advanced blockchain forensics and investigative tools to trace my stolen funds. WhatsApp: +1 (920) 408‑1234 They identified the fraudulent wallets where my money had been transferred and collaborated with financial institutions and law enforcement agencies to take action. Thanks to their persistence and expertise, they were able to freeze the scammers' accounts and successfully recover my $500,000. What seemed like a devastating loss turned into a remarkable recovery. I am incredibly grateful to PRO WIZARD GIlBERT RECOVERY for not only retrieving my funds but also restoring my peace of mind. My experience serves as a warning to others-always be cautious with online investments, but if you ever become a victim, know that recovery is possible with the right experts on your side.

  • 22.03.25 14:13 [email protected]

    The glow of RGB lights still haunts me. There I was, mid-stream, hyping up a Fortnite squad when an email pretending to be a sponsorship opportunity with the subject line "ENERGY DRINK COLLAB!!! *" appeared on my second monitor. I clicked— big mistake. By the time my chat spammed "*SCAM ALERT" in neon caps, a trojan had already ghosted my Bitcoin wallet, $320,000 gone, poof, like a noob disconnecting mid-game. My facecam caught the exact moment my soul left my body: jaw open, headset tilted, the background of anime posters judging me silently. The VOD blew up. Of course, it did. Pandemonium erupted. Donation alerts became panic emojis. My mods DM'd links to "HOW TO FIX CRYPTO THEFT" amidst banning trolls. My wallet? A barren wasteland. My DMs? A cemetery of "*F"s and crypto-bros pitching recovery scams. Then, a lifeline—a chatter named *xX_Cryptosolution_69 typed, "TRUST GEEKS HACK EXPERT. THEY CLAPPED A HACKER FOR MY DOGE ONCE." Desperate, I Googled them mid-stream, muting to scream into a pillow. TRUST GEEKS HACK EXPERT team responded like NPCs scripted for heroics. “Send us the malware file,” they said. “**And your wallet logs. We’ll handle the rest.” For 12 days, they reverse-engineered the trojan, dissecting its code like speed runners cracking a glitch. The virus, it turned out, was a knockoff ransomware dubbed “Crypto rush” (its dev had left a “HACK THE PLANET!!” Easter egg in the code, cringe). TRUST GEEKS HACK EXPERT squad traced its path, resurrecting private keys from registry fragments and backup clouds I’d forgotten existed. The return stream was record-breaking. I rebooted my rig, wallet restored, and titled the stream "HOW I UNBRICKED $320K (AND MY CAREER)." Chatters donated Bitcoin out of solidarity, and schadenfreude. Even my rival streamer, DrL33tGamer, raided me with 10k viewers. TRUST GEEKS HACK EXPERT? They viewed anonymously and left a sub with the message: "GG EZ. These internet Gandalfs didn't just fix a hack—they authored the greatest plot twist in my online existence. Now, my new website, Stream Vault, runs on a server guarded like Fort Knox, and I vet sponsors like the CIA. That fake energy drink company? Its domain now points to a Rickroll. If your crypto gets pawned by a script kiddie, skip the rage quit. Ping the TRUST GEEKS. They're the ultimate cheat code for catastrophe. Just maybe have a malware scanner in closer proximity than your energy drinks next time. (CONTACT SERVICE ) Email, Trustgeekshackexpert[At]fastservice[Dot]com Telegram, Trustgeekshackexpert Email, [email protected] Website, www://trustgeekshackexpert.com

  • 23.03.25 12:24 maryjul75

    There are not many reviews about free crypto recovery fixed but the ones that exist are mostly positive. I have no idea how that is possible, but it is clear that they are fabricated. However, there are also negative reviews – and I highly recommend paying attention to them. These are real people describing their unfortunate experiences with this scam. It’s the same story as always – fraudsters create a fake website to con people out of money. Nothing new but look no more and contact FASTRECOVERYAGENT These types of scams are everywhere. If you have already lost money to scammers i mean any type be it sending to scammers account , or if you have a case about stolen bitcoin , usdt, or any type of cryptocurrency whatsoever just reach out to www.fastrecoveryagent.com, i really owe this group of recovery expert because they saved me after bad investment with a fake broker, i literally owe them my life as they saved me from some bunch of scammers, but see how life is , i got every cent of my investment back with the profit , reach out to to them today and tell them from Mary because i promise them i will tell the whole world when the recovery is complete, and here i go after i got my recovered assets thats why im doing what i promised. do not fall for another scam. contact the,m today!!!!

  • 23.03.25 22:01 joaneldnde

    The ground trembled like a nervous intern on espresso shots. One minute, I was monitoring my geothermal Bitcoin miners, humming in harmony with Iceland's most unpredictable volcano. Next? An eruption painted the sky gray with ash, raining destruction like an out-of-control blockchain fork. Power cables flickered out. Servers turned into abstract-art pieces. And my wallet with $460,000 worth of mining revenue fried faster than a motherboard in a tidal wave of lava. I was knee-deep in volcanic mud, clutching the charred wallet, wondering if the universe had a vendetta against renewable energy. For weeks, I’d played geothermal gambler, harnessing Earth’s anger to mine crypto. Now, Mother Nature had countered with a literal power move. My wallet’s backups? Corrupted by ash-clogged drives. My cold storage? Warmer than a freshly erupted fissure. Even the volcanologists on my team shrugged. “We predict lava, not ledger errors,” one said, handing me a business card signed at the edges. “Try these Cyber Constable Intelligence. They’ve fixed crypto in weird places.” Cyber Constable Intelligence phoned on the first ring. Cyber Constable Intelligence saved not just crypto. They demonstrated that even the fury of nature cannot surpass human tenacity. My operation now operates robustly, excavating coins with Earth's anger and a backup generator sufficient to run a small glacier. The volcano? Still grumbling. My wallet? Locked inside a fireproof safe, as irony bites sharper than an Icelandic winter. If your crypto somehow gets smothered beneath the pyroclastic ash of life, skip the freak-out. Call the Cybers. They'll dig through lava streams until your cash bubbles up to the surface. Just maybe set up your rigs a few miles closer to the crater next time. If you’re facing a similar problem I highly recommend contacting Cyber Constable Intelligence 📞 WhatsApp:+1 (2 5 2 ) 3 7 8 7 6 1 1 🌐 Website: www cyberconstableintelligence.com 📩 Telegram: https://t.me/cyberconstable

  • 23.03.25 22:18 [email protected]

    A few months ago, I stumbled upon a post about a cryptocurrency investment platform that I thought was a good idea for me to invest in crypto, I didn’t realize I was being catfished by the cryptocurrency investment manager who promised me huge returns on my investment. I lost my capital of $170,000 and interest without receiving any profits in return, I was devastated And I realized I had been scammed out of my cryptocurrency. I felt hopeless and didn’t know where to turn. i searched everywhere for a solution and That’s when my friend introduced me to a Crypto recovery platform Trust Geeks Hack Expert. At first, I was skeptical—after all, I had just been scammed—but Trust Geeks Hack Expert professionalism and transparency reassured me. They took the time to analyze my case, track my lost funds, and guide me through the recovery process. To my amazement, they successfully retrieved my stolen crypto! I can’t express how grateful I am to Trust Geeks Hack Expert team. Their expertise and dedication gave me a second chance. If you’ve been a victim of a crypto scam, don’t lose hope—Trust Geeks Hack Expert. is the real deal! for assistance, Email: [email protected] (TeleGram:: Trustgeekshackexpert) & what's A p p  +1 7 1 9 4 9 2 2 6 9 3

  • 24.03.25 10:03 Diegolola514gmail.com

    "I am incredibly grateful to Sylvester Bryant for his exceptional expertise and assistance in recovering my USDC funds, which were lost to a phishing scam. Thanks to his diligent efforts, he successfully retrieved €480,000 worth of USDC that I thought was gone forever. If you or anyone you know has fallen victim to a similar situation, I highly recommend reaching out to Sylvester for professional assistance. You can contact him via email at Yt7cracker@gmail. com or through WhatsApp/text message at +1 512 577 7957. He is truly a reliable and skilled professional in recovering assets from online scams."

  • 25.03.25 18:06 maryjul75

    Fast Agent Recovery is a consulting firm that specializes in the recovery of assets from financial fraud. We know how to recover your funds and we have helped thousands of scam victims from around the world to recover their money. If you are a victim of Binary Options fraud, Forex fraud, Bitcoin fraud, Dating scams or one of the many other the online fraudulent practices that permeate the internet then file a complaint on www.fastrecoveryagent,.com to get an assessment if we can help you get your money back too. File a complaint: https://www.fastrecoveryagent.com/

  • 27.03.25 01:52 debrapavon89

    The moment my cold storage device flatlined, erasing $385,000 in Bitcoin I’d saved to launch a chain of zero-waste cafés, I became a ghost haunting my own life. I scrolled through forums until my eyes were streaming, trawling through threads filled with such mouthfuls as "irreversible blockchain entropy" and "cryptographic oblivion A Reddit thread finally revealed to me Cyber Constable Intelligence I reached out, the team at Cyber Constable Intelligence took immediate action. They began tracing the fraudulent car auction site and thoroughly investigating the scam. To my relief, after several weeks of hard work, Cyber Constable Intelligence successfully tracked down the scammers and recovered the full $385,000 I had lost. Their hard work allowed me to move on from this unfortunate experience, I highly recommend their services to anyone who has been affected by online fraud Reach out to their Info below WhatsApp: 1 252378-7611 Website info; www.cyberconstableintelligence.com

  • 30.03.25 03:25 orrinharber

    The ad on X (formerly Twitter) popped up on my timeline one lazy Sunday afternoon. It was flashy, bold, and impossible to ignore. The headline read, “ONCE-IN-A-LIFETIME COMEDY EXTRAVAGANZA!” with a dazzling graphic of a stage lit up in neon lights. The caption said: “Kevin Hart, Ali Wong, Dave Chappelle, and a MYSTERY LEGENDARY COMEDIAN live in Vegas! Limited VIP tickets available. Don’t miss out!” The post had thousands of likes, retweets, and comments like, “This is going to be epic!” and “Already got my ticket!” It even had a blue checkmark next to the account name, which made it seem legit. I clicked the link, and it took me to a sleek website with a countdown timer and a list of sold-out ticket tiers. The only option left was a $125,000 VIP package, which promised front-row seats, backstage access, and a meet-and-greet with the comedians. I hesitated for a moment, but the fear of missing out got the better of me. I thought, When will I ever get a chance like this again? So, I entered my credit card details and hit “Purchase.” The confirmation email came through instantly, and I felt a rush of excitement. Little did I know, I’d just fallen for one of the most elaborate scams I’d ever encountered. Looking back, I should’ve noticed the red flags the overly pushy tone of the ad, the lack of reviews for the event, and the fact that no official accounts from the comedians promoted it. But in the moment, it all seemed so real.I had been swept up by the flashy ad, the excitement of the event, and the FOMO (fear of missing out) that made it seem like an opportunity I couldn’t let slip by. Everything about the ad screamed “exclusive” and “once-in-a-lifetime,” which was enough to convince me to take the plunge. Yet, as time went on and I tried to follow up on the event, I found there was no trace of it anywhere. There were no details, no event pages, and no mention from the comedians themselves. My heart sank as I realized I had been scammed.Thankfully, GRAYWARE TECH SERVICES helped me get my money back, but the experience was a hard lesson in online scams. I’ll never forget that ad on X the one that cost me $125,000 and a whole lot of pride. I learned the importance of being cautious online, checking for reviews, and looking for signs of authenticity before jumping into anything that seems too good to be true.You can reach GRAYWARE TECH SERVICES on web at ( https://graywaretechservices.com/ )    also on Mail: ([email protected])

  • 30.03.25 06:22 grace111

    I recommend Marie ([email protected] and WhatsApp: +1 7127594675) for recovering lost or stolen bitcoin, USDT, or any other cryptocurrency from fraudulent investment sites because they are very knowledgeable in the industry and will reimburse all of your money. I can say with confidence that she was the only one who was able to credit my account with $52,760 of the money that had gone missing, based on my prior transactions with them. She was the only one who could. Since they are the only ones that can fully return your missing funds to your account without any deductions, I sincerely appreciate their job and am recommending her to you today.

  • 31.03.25 11:00 maryjul75

    Fast Agent Recovery is a consulting firm that specializes in the recovery of assets from financial fraud. We know how to recover your funds and we have helped thousands of scam victims from around the world to recover their money. If you are a victim of Binary Options fraud, Forex fraud, Bitcoin fraud, Dating scams or one of the many other the online fraudulent practices that permeate the internet then file a complaint on www.fastrecoveryagent,.com to get an assessment if we can help you get your money back too. File a complaint: https://www.fastrecoveryagent.com/

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