BP神经网络正反向传播流程图解与实现：从数学推导到代码实践

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BP 神经网络（Backpropagation Neural Network）是深度学习最基础的模型之一，它的核心思想就像教小孩子认字：先看正确答案（前向传播），再根据错误调整学习方法（反向传播）。举个简单例子，当网络把猫图片误判成狗时，它会自动计算每个参数该负多少责任，并通过链式法则逐层调整权重。

关键公式用 LaTeX 展示如下：

前向传播时第 l 层的输出：
$$a^{(l)} = \sigma(z^{(l)}) = \sigma(W^{(l)}a^{(l-1)} + b^{(l)})$$
损失函数对权重的偏导（链式法则）：
$$\frac{\partial L}{\partial W_{ij}^{(l)}} = \frac{\partial L}{\partial z_{i}^{(l)}} \cdot \frac{\partial z_{i}^{(l)}}{\partial W_{ij}^{(l)}} = \delta_{i}^{(l)} \cdot a_{j}^{(l-1)}$$
误差项反向传递：
$$\delta^{(l)} = ((W^{(l+1)})^T \delta^{(l+1)}) \odot \sigma'(z^{(l)})$$

import numpy as np

class NeuralNetwork:
    def __init__(self, layers):
        self.weights = [np.random.randn(y, x) for x, y in zip(layers[:-1], layers[1:])]
        self.biases = [np.random.randn(y, 1) for y in layers[1:]]

    def forward(self, x):
        # 前向传播计算图
        for w, b in zip(self.weights, self.biases):
            x = sigmoid(np.dot(w, x) + b)
        return x

    def backprop(self, x, y):
        # 反向传播核心代码
        nabla_w = [np.zeros(w.shape) for w in self.weights]
        nabla_b = [np.zeros(b.shape) for b in self.biases]

        # 前向传播缓存中间值
        activation = x
        activations = [x]
        zs = []
        for w, b in zip(self.weights, self.biases):
            z = np.dot(w, activation) + b
            zs.append(z)
            activation = sigmoid(z)
            activations.append(activation)

        # 输出层误差
        delta = (activations[-1] - y) * sigmoid_prime(zs[-1])
        nabla_b[-1] = delta
        nabla_w[-1] = np.dot(delta, activations[-2].T)

        # 反向传播误差
        for l in range(2, len(self.weights)+1):
            z = zs[-l]
            sp = sigmoid_prime(z)
            delta = np.dot(self.weights[-l+1].T, delta) * sp
            nabla_b[-l] = delta
            nabla_w[-l] = np.dot(delta, activations[-l-1].T)

        return nabla_w, nabla_b

标准流程图包含以下要素：