神经网络反向传播神经网络反向传播求解w代码验证求解w，用python代码验证

神经网络的反向传播算法是一种用于训练深层模型的关键技术，它通过计算梯度来更新权重（w），使得损失函数逐渐减小。这个过程可以分为前向传播（预测）和反向传播（误差反传）。这里以简单的线性回归为例，展示如何用Python和NumPy实现一个一隐藏层的神经网络，并包含反向传播的部分：

import numpy as np

# 假设我们有输入数据 X (形状(m, n)) 和对应的标签 y (形状(m, 1))
def sigmoid(x):
    return 1 / (1 + np.exp(-x))

def forward(X, w1, b1, w2, b2):
    # 前向传播
    z1 = np.dot(X, w1) + b1
    a1 = sigmoid(z1)
    z2 = np.dot(a1, w2) + b2
    y_pred = sigmoid(z2)
    return y_pred, a1, z2

def cost_function(y_true, y_pred):
    m = len(y_true)
    J = -(1/m) * np.sum(y_true * np.log(y_pred) + (1 - y_true) * np.log(1 - y_pred))
    return J

# 假设已经初始化了权重 w1、b1、w2、b2
# 每次迭代我们要做的就是：
def backward(X, y_true, y_pred, a1, z2, hessian=False):
    m = X.shape[0]
    dz2 = y_pred - y_true
    dw2 = (1/m) * np.dot(a1.T, dz2)
    db2 = (1/m) * np.sum(dz2, axis=0, keepdims=True)

    if hessian:
        da1 = np.dot(dz2, w2)
        da1[a1 <= 0.5] *= -1
        dz1 = np.dot(da1, w2.T)
        dw1 = (1/m) * np.dot(X.T, dz1)
        db1 = (1/m) * np.sum(dz1, axis=0, keepdims=True)
        return dw1, db1, dw2, db2
    else:
        dz1 = np.dot(da1, dw2)
        dz1[a1 <= 0.5] *= -1
        dw1 = (1/m) * np.dot(X.T, dz1)
        db1 = (1/m) * np.sum(dz1, axis=0, keepdims=True)
        return dw1, db1

# 训练循环
learning_rate = 0.01
epochs = 1000
for epoch in range(epochs):
    y_pred, a1, z2 = forward(X, w1, b1, w2, b2)
    dw1, db1, dw2, db2 = backward(X, y_true, y_pred, a1, z2)

    # 更新权重
    w1 -= learning_rate * dw1
    b1 -= learning_rate * db1
    w2 -= learning_rate * dw2
    b2 -= learning_rate * db2

    # 打印一些进度信息
    if epoch % 100 == 0:
        print(f"Epoch {epoch+1}/{epochs}, Loss: {cost_function(y_true, y_pred)}")