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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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  • 30.06.26 15:08 wendytaylor015

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  • 19.07.26 19:53 kimberlyhebertt6877

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  • 19.07.26 19:53 kimberlyhebertt6877

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  • 19.07.26 19:54 kimberlyhebertt6877

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  • 30.07.26 00:04 Ahmed

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  • 04.08.26 11:05 Kisnoles

    Excellent analysis, thank you. I was particularly struck by how rigidly the skill sets the order: data processing and form come first, with color coming last. This really cures the habit of "first a pretty palette, then we'll figure it out." A palette validator using OKLCH + color blindness + WCAG is something most chart generators lack. And regarding the boundaries of competence, it's very clear. Where there's a self-checking loop (color), you delegate freely. Where there's a heuristic (form, data interpretation), you remain the final filter. This is a universal principle, not just for /dataviz. By the way, when you look at trading dashboards (including those of decent platforms like ExpertOption), it's immediately clear who thought about readability and who just threw in rainbow lines. Tools like this skill could greatly improve the quality of analytics. Thanks again for the detailed analysis – saved.

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  • 23.08.26 20:02 leslieyee

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  • 02.09.26 03:43 kimberlyhebertt6877

    I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684

  • 02.09.26 03:43 kimberlyhebertt6877

    I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684

  • 04.09.26 21:51 Kovengray

    Recovering your lost investment funds as the case might be, is not what you can do alone, you’d require the service of a trained recovery specialist. A recovery specialist is a person or a group of people who are well equipped to work around the brokerage network. They have vast knowledge about the whole network and have the right software and private keys to follow any transaction. I was ripped off trading online to an investment broker, good thing I got every penny back through the help of Gavin ray he’s a genius Contact : Gavinray78 at gmail com or WhatsApp +1 352 322 2096 It is also important to be patient and really calm during the process.

  • 05.09.26 21:38 [email protected]

    I invested 45,000  Euro, and later aggreviated to 198,000 Euro.  I requested  to place ‎Withdrawal of my funds to be paid to my Bank account. But nothing happened I was subjected to pay more fee until I can get my funds on my account, i got in touch with Theodore ryan here on this platform who had helped a lot of people, I followed all instructions and he legally got back my withheld funds, I got threatened by the company that if I don’t pay they will get my account frozen, all thanks to Theodore ryan and I highly recommend him to anyone dealing with an unregulated broker company… contact him on his Gmail - theodoreryan318@  gmail .  com

  • 09.09.26 21:31 lisawerth897

    My Experience With Cryptocurrency — CAPITAL CRYPTO RECOVER I lost over $82,000 in Bitcoin after falling victim to a fraudulent online investment scheme. After realizing I had been scammed, I spent considerable time researching possible ways to recover my funds. During my research, I came across CAPITAL CRYPTO RECOVER and decided to contact them after reading their reported success record and encouraging client positive reviews and testimonials. Their team guided me through the recovery process, and I was grateful to successfully recover my lost cryptocurrency. The experience was challenging, but I appreciated the assistance and support I received throughout the process. I’m sharing my experience in the hope that it may encourage other victims to carefully research their options and verify any recovery service before proceeding. I will forever be thankful to you CAPITAL CRYPTO RECOVER 📧 [email protected] 🌐 Website: recovercapital.wixsite.com/capital-crypto-rec-1 📧 [email protected] 📞 Call/WhatsApp: +1 (336) 390-6684

  • 09.09.26 21:31 lisawerth897

    My Experience With Cryptocurrency — CAPITAL CRYPTO RECOVER I lost over $82,000 in Bitcoin after falling victim to a fraudulent online investment scheme. After realizing I had been scammed, I spent considerable time researching possible ways to recover my funds. During my research, I came across CAPITAL CRYPTO RECOVER and decided to contact them after reading their reported success record and encouraging client positive reviews and testimonials. Their team guided me through the recovery process, and I was grateful to successfully recover my lost cryptocurrency. The experience was challenging, but I appreciated the assistance and support I received throughout the process. I’m sharing my experience in the hope that it may encourage other victims to carefully research their options and verify any recovery service before proceeding. I will forever be thankful to you CAPITAL CRYPTO RECOVER 📧 [email protected] 🌐 Website: recovercapital.wixsite.com/capital-crypto-rec-1 📧 [email protected] 📞 Call/WhatsApp: +1 (336) 390-6684

  • 09.09.26 23:22 Fraddy Pual

    Hearing that these individuals are focusing on other people makes me very sad. I had a similar situation with them and lost a lot of money, but after learning about (Cruxcipherteam @ proton DoT me), whataqq:+168160-15021, telegram: @Cruxcipherteam I was able to get my money back. It's critical that we all be watchful and keep reporting these occurrences.

  • 11.09.26 03:23 kimberlyhebertt6877

    I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684

  • 11.09.26 03:23 kimberlyhebertt6877

    I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684

  • 15.09.26 03:35 elioduncan

    I fell victim to a freelance web development contract scam that ultimately cost me CAD 4,000. It all started when I came across a job posting on Indeed Canada, a popular online job portal. The position seemed perfect: a freelance web developer role for an established company looking for someone to design and build a fully functional e-commerce website. The job promised a high payment of CAD 20,000 upon successful completion, which sounded like a great opportunity for me to gain experience and earn decent pay.The employer, who introduced himself as a project manager from a "well-known" tech company, was very persuasive and professional in our initial communication. He explained that the project involved developing a user-friendly, responsive online store for their client and that they had a strict timeline to meet. He assured me that the payment would be made promptly after completing the tasks. However, before starting the work, he told me that I would need to pay an upfront fee of CAD 4,000 to cover certain software tools and licensing fees required for the project. This was supposedly a part of their company policy for freelance contractors, ensuring access to their premium resources. The idea of working on a professional project, coupled with the promise of a substantial payout, convinced me to pay the upfront fee.As soon as I made the payment, the project manager became increasingly difficult to reach. He initially responded to my emails and provided some vague instructions on what the project would entail, but as time went on, communication slowed to a complete halt. I never received the required tools or any proper project details. My emails went unanswered, and any attempts to contact the company were met with silence. After waiting for weeks, I realized that I had been scammed.Desperate to recover my money, I turned to TechY Force Cyber Retrieval, a service that specializes in helping victims of online scams. They helped me track the payment and took legal steps to pursue the fraudsters. Through their guidance, I was able to recover the full CAD 4,000. TechY Force Cyber Retrieval worked with my bank to reverse the transaction and liaised with the authorities to trace the scammer's details.This was a hard lesson, and I now know to be highly cautious about online job offers that require upfront payments. It is crucial to research companies thoroughly and avoid any job that seems too good to be true, especially when it involves paying money upfront. WhatsApp https://wa.link/2x6ktp Mail. [email protected]

  • 15.09.26 15:45 lydiassmith567

    HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 15.09.26 15:45 lydiassmith567

    HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 05.10.26 21:26 [email protected]

    I Invested 45,000  Euro, and later aggreviated to 198,000 Euro.  I requested  to place Withdrawal of my funds to be paid to my Bank account. But nothing happened I was subjected to pay more fee until I can get my funds on my account, i got in touch with Theodore ryan here on this platform who had helped a lot of people, I followed all instructions and he legally got back my withheld funds, I got threatened by the company that if I don’t pay they will get my account frozen, all thanks to Theodore ryan and I highly recommend him to anyone dealing with an unregulated broker company… contact him on his Gmail - theodoreryan318@  gmail .  com

  • 06.10.26 06:13 melissathomason573

    Professional Crypto Recovery Testimonial After losing a significant amount of cryptocurrency through a scam on Twitter, I was unsure whether my Bitcoin funds could ever be recovered. The experience was extremely stressful, particularly because I encountered several individuals and services claiming they could help recover my assets, only to discover that some were fraudulent themselves. A friend eventually referred me to CAPITAL CRYPTO RECOVER. Their team handled my case professionally, discreetly, and efficiently. With their assistance, I successfully recovered approximately $130,000 worth of USDT in just over a week. My experience taught me the importance of being extremely cautious when seeking cryptocurrency recovery services, as scammers often target victims a second time. I am grateful to CAPITAL CRYPTO RECOVER for the assistance I received and wanted to share my experience with others who may be facing a similar situation. If you’ve been compromised, don’t lose hope—and be careful of fraudsters posing as saviors. CAPITAL CRYPTO RECOVER are real professionals in crypto recovery. I am a living proof of their effectiveness, you can reach them through Phone (Call/WhatsApp): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 06.10.26 06:13 melissathomason573

    Professional Crypto Recovery Testimonial After losing a significant amount of cryptocurrency through a scam on Twitter, I was unsure whether my Bitcoin funds could ever be recovered. The experience was extremely stressful, particularly because I encountered several individuals and services claiming they could help recover my assets, only to discover that some were fraudulent themselves. A friend eventually referred me to CAPITAL CRYPTO RECOVER. Their team handled my case professionally, discreetly, and efficiently. With their assistance, I successfully recovered approximately $130,000 worth of USDT in just over a week. My experience taught me the importance of being extremely cautious when seeking cryptocurrency recovery services, as scammers often target victims a second time. I am grateful to CAPITAL CRYPTO RECOVER for the assistance I received and wanted to share my experience with others who may be facing a similar situation. If you’ve been compromised, don’t lose hope—and be careful of fraudsters posing as saviors. CAPITAL CRYPTO RECOVER are real professionals in crypto recovery. I am a living proof of their effectiveness, you can reach them through Phone (Call/WhatsApp): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 06.10.26 17:44 Darrell1717

    I fell victim of a scam I invested to total of 5thousand into. I know it could be risky but after I trusted this man I was dating online he lured me into investing with him. I know hard I worked of my money so I wanted every way possible to get it back. I really appreciate Blockrecoveryexpert for successfully recovered my funds form this scammer. This is all I can say. You can reachout if you ever need help contact blockrecoveryexpert (at) gmail (dot)com

  • 11.10.26 03:04 Patricia014

    Most recovery companies will take your money again. I personally think they’re the same set of people running all of them. It’s a sad, unending cycle, and so many people fall for cloned websites. The only recovery agent I know is Agent Rottgen Raphael. I’ve visited their physical company address, and they’ll tell you if they can handle your case instead of taking upfront payment when they know they can’t help. I highly appreciate their support in recovering my money from the scammers’ wallet within 24 hours. RottgenRaphael@Gmail. com

  • 12.10.26 19:34 lydiassmith567

    HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 12.10.26 19:34 lydiassmith567

    HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 13.10.26 14:14 lisalisa22

    Legitimate Cryptocurrency Recovery Services: Consult ZEUS CRYPTO RECOVERY SERVICES Zeus Crypto Recovery Services supports individuals exploring options after cryptocurrency losses from fraud, unauthorised transfers, or wallet access issues. Services may include blockchain analysis, wallet tracing, and evidence gathering for investigations. Each case is reviewed. Zeus Crypto Recovery Services is a trusted company with the highest success rate among cryptocurrency recovery services in 2025/2026 and is highly recommended if you have been scammed. ZEUS CRYPTO RECOVERY SERVICES handles case such as Scammed USDT Lost BTC, Coinbase Wallet Forgotten password, ETH and coins, etc. Anything that has to do with cryptocurrency or lost digital assets, ZEUS can help you Do you Need Help? Website https://zeusrecoveryservices.com Mailbox [email protected] WhatsApp number +447798816938 Telegram https://t.me/Zeuscryptorecoveryservices

  • 13.10.26 14:14 lisalisa22

    Legitimate Cryptocurrency Recovery Services: Consult ZEUS CRYPTO RECOVERY SERVICES Zeus Crypto Recovery Services supports individuals exploring options after cryptocurrency losses from fraud, unauthorised transfers, or wallet access issues. Services may include blockchain analysis, wallet tracing, and evidence gathering for investigations. Each case is reviewed. Zeus Crypto Recovery Services is a trusted company with the highest success rate among cryptocurrency recovery services in 2025/2026 and is highly recommended if you have been scammed. ZEUS CRYPTO RECOVERY SERVICES handles case such as Scammed USDT Lost BTC, Coinbase Wallet Forgotten password, ETH and coins, etc. Anything that has to do with cryptocurrency or lost digital assets, ZEUS can help you Do you Need Help? Website https://zeusrecoveryservices.com Mailbox [email protected] WhatsApp number +447798816938 Telegram https://t.me/Zeuscryptorecoveryservices

  • 17.10.26 12:58 melissathomason573

    Professional Crypto Recovery Testimonial After losing a significant amount of cryptocurrency through a scam on Twitter, I was unsure whether my Bitcoin funds could ever be recovered. The experience was extremely stressful, particularly because I encountered several individuals and services claiming they could help recover my assets, only to discover that some were fraudulent themselves. A friend eventually referred me to CAPITAL CRYPTO RECOVER. Their team handled my case professionally, discreetly, and efficiently. With their assistance, I successfully recovered approximately $130,000 worth of USDT in just over a week. My experience taught me the importance of being extremely cautious when seeking cryptocurrency recovery services, as scammers often target victims a second time. I am grateful to CAPITAL CRYPTO RECOVER for the assistance I received and wanted to share my experience with others who may be facing a similar situation. If you’ve been compromised, don’t lose hope—and be careful of fraudsters posing as saviors. CAPITAL CRYPTO RECOVER are real professionals in crypto recovery. I am a living proof of their effectiveness, you can reach them through Phone (Call/WhatsApp): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 17.10.26 12:58 melissathomason573

    Professional Crypto Recovery Testimonial After losing a significant amount of cryptocurrency through a scam on Twitter, I was unsure whether my Bitcoin funds could ever be recovered. The experience was extremely stressful, particularly because I encountered several individuals and services claiming they could help recover my assets, only to discover that some were fraudulent themselves. A friend eventually referred me to CAPITAL CRYPTO RECOVER. Their team handled my case professionally, discreetly, and efficiently. With their assistance, I successfully recovered approximately $130,000 worth of USDT in just over a week. My experience taught me the importance of being extremely cautious when seeking cryptocurrency recovery services, as scammers often target victims a second time. I am grateful to CAPITAL CRYPTO RECOVER for the assistance I received and wanted to share my experience with others who may be facing a similar situation. If you’ve been compromised, don’t lose hope—and be careful of fraudsters posing as saviors. CAPITAL CRYPTO RECOVER are real professionals in crypto recovery. I am a living proof of their effectiveness, you can reach them through Phone (Call/WhatsApp): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

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  • 25.10.26 14:35 lydiasmitth456

    HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 25.10.26 14:35 lydiasmitth456

    HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 27.10.26 01:58 KATHLEEN

    Cyberspacetechexpert focuses on recovering stolen cryptocurrency, particularly USDT and bitcoin. She is well-known for helping victims of digital asset theft. Her reputation arises from successful recoveries that have allowed many to regain their lost funds. I witnessed this when $74,000 was taken from me in January. Thanks to the best female hacker Gracias, I recovered the entire amount in just 14 hours. Her prompt response and effective methods relieved my financial burden. Her commitment to helping others is evident. She is always available to offer solutions to those facing similar problems. For assistance, she can be reached via email Cyberspacetechexpert@gmail .com. or speak to her on What App at + 44 7476618364.

  • 27.10.26 02:22 kimberlyhebertt6877

    I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684

  • 27.10.26 02:22 kimberlyhebertt6877

    I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684

  • 27.10.26 02:25 KATHLEEN

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