d = 3 # 输入节点个数 l = 1 # 输出节点个数 q = 2 * d + 1 # 隐层个数,采用经验公式2d+1隐藏层公式改成Leaky ReLU怎么改

将原来的隐藏层公式修改为 Leaky ReLU 函数可以直接在原有代码基础上进行修改，只需要将原来的激活函数 `sigmoid` 替换为 `leaky_relu` 即可。具体修改如下：

import numpy as np

def leaky_relu(x, alpha=0.01):
    return np.maximum(alpha * x, x)

class BPNeuralNetwork:
    def __init__(self, input_nodes, output_nodes, hidden_nodes):
        self.input_nodes = input_nodes
        self.output_nodes = output_nodes
        self.hidden_nodes = hidden_nodes
        self.w1 = np.random.randn(self.input_nodes, self.hidden_nodes)
        self.b1 = np.zeros((1, self.hidden_nodes))
        self.w2 = np.random.randn(self.hidden_nodes, self.output_nodes)
        self.b2 = np.zeros((1, self.output_nodes))

    def forward(self, X):
        self.z1 = np.dot(X, self.w1) + self.b1
        self.a1 = leaky_relu(self.z1)  # 将 sigmoid 替换为 leaky_relu
        self.z2 = np.dot(self.a1, self.w2) + self.b2
        self.y_hat = self.z2

    def backward(self, X, y, learning_rate=0.1):
        m = X.shape[0]
        delta_y_hat = self.y_hat - y
        delta_z2 = delta_y_hat
        delta_w2 = np.dot(self.a1.T, delta_z2)
        delta_b2 = np.sum(delta_z2, axis=0, keepdims=True)
        delta_a1 = np.dot(delta_z2, self.w2.T)
        delta_z1 = delta_a1 * np.where(self.z1 > 0, 1, 0.01)  # 使用 leaky ReLU 的导数
        delta_w1 = np.dot(X.T, delta_z1)
        delta_b1 = np.sum(delta_z1, axis=0)
        self.w2 -= learning_rate * delta_w2 / m
        self.b2 -= learning_rate * delta_b2 / m
        self.w1 -= learning_rate * delta_w1 / m
        self.b1 -= learning_rate * delta_b1 / m

    def train(self, X, y, epochs=1000, learning_rate=0.1):
        for i in range(epochs):
            self.forward(X)
            self.backward(X, y, learning_rate)
            loss = np.mean(np.square(self.y_hat - y))
            if i % 100 == 0:
                print(f"epoch {i}, loss {loss}")

在这里，将 `leaky_relu` 函数作为隐藏层的激活函数，使用了 `np.where` 函数实现了其导数，其中超参数 `alpha` 默认为 0.01。