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人工神经网络
（Artificial Neural Network，ANN）是一种模仿生物神经网络的结构和功能的数学模型，其目的是通过学习和训练，在处理未知的输入数据时能够进行复杂的非线性映射关系，实现自适应的智能决策。可以说，ANN是人工智能算法中最基础、最核心的一种算法。
ANN模型的基本结构包含输入层、隐藏层和输出层。输入层接收输入数据，隐藏层负责对数据进行多层次、高维度的变换和处理，输出层对处理后的数据进行输出。ANN的训练过程是通过多次迭代，不断调整神经网络中各层的权重，从而使得神经网络能够对输入数据进行正确的预测和分类。
人工神经网络算法示例
接下来看看一个简单的人工神经网络算法示例：
import numpy as np
class NeuralNetwork():
def __init__(self, layers):
"""
layers: 数组，包含每个层的神经元数量，例如 [2, 3, 1] 表示 3 层神经网络，第一层 2 个神经元，第二层 3 个神经元，第三层 1 个神经元。
weights: 数组，包含每个连接的权重矩阵，默认值随机生成。
biases: 数组，包含每个层的偏差值，默认值为 0。
"""
self.layers = layers
self.weights = [np.random.randn(a, b) for a, b in zip(layers[1:], layers[:-1])]
self.biases = [np.zeros((a, 1)) for a in layers[1:]]
def sigmoid(self, z):
"""Sigmoid 激活函数."""
return 1 / (1 + np.exp(-z))
def forward_propagation(self, a):
"""前向传播."""
for w, b in zip(self.weights, self.biases):
z = np.dot(w, a) + b
a = self.sigmoid(z)
return a
def backward_propagation(self, x, y):
"""反向传播."""
nabla_w = [np.zeros(w.shape) for w in self.weights]
nabla_b = [np.zeros(b.shape) for b in self.biases]
a = x
activations = [x]
zs = []
for w, b in zip(self.weights, self.biases):
z = np.dot(w, a) + b
zs.append(z)
a = self.sigmoid(z)
activations.append(a)
delta = self.cost_derivative(activations[-1], y) * self.sigmoid_prime(zs[-1])
nabla_b[-1] = delta
nabla_w[-1] = np.dot(delta, activations[-2].transpose())
for l in range(2, len(self.layers)):
z = zs[-l]
sp = self.sigmoid_prime(z)
delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
nabla_b[-l] = delta
nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
return (nabla_w, nabla_b)
def train(self, x_train, y_train, epochs, learning_rate):
"""训练网络."""
for epoch in range(epochs):
nabla_w = [np.zeros(w.shape) for w in self.weights]
nabla_b = [np.zeros(b.shape) for b in self.biases]
for x, y in zip(x_train, y_train):
delta_nabla_w, delta_nabla_b = self.backward_propagation(np.array([x]).transpose(), np.array([y]).transpose())
nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
self.weights = [w-(learning_rate/len(x_train))*nw for w, nw in zip(self.weights, nabla_w)]
self.biases = [b-(learning_rate/len(x_train))*nb for b, nb in zip(self.biases, nabla_b)]
def predict(self, x_test):
"""预测."""
y_predictions = []
for x in x_test:
y_predictions.append(self.forward_propagation(np.array([x]).transpose())[0][0])
return y_predictions
def cost_derivative(self, output_activations, y):
"""损失函数的导数."""
return output_activations - y
def sigmoid_prime(self, z):
"""Sigmoid 函数的导数."""
return self.sigmoid(z) * (1 - self.sigmoid(z))使用以下代码示例来实例化和使用这个简单的神经网络类：
x_train = [[0, 0], [1, 0], [0, 1], [1, 1]]
y_train = [0, 1, 1, 0]
# 创建神经网络
nn = NeuralNetwork([2, 3, 1])
# 训练神经网络
nn.train(x_train, y_train, 10000, 0.1)
# 测试神经网络
x_test = [[0, 0], [1, 0], [0, 1], [1, 1]]
y_test = [0, 1, 1, 0]
y_predictions = nn.predict(x_test)
print("Predictions:", y_predictions)
print("Actual:", y_test)输出结果：
Predictions: [0.011602156431658403, 0.9852717774725432, 0.9839448924887225, 0.020026540429992387]
Actual: [0, 1, 1, 0]
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