附加动量的BP算法、学习率自适应算法和LM算法，其代表性的训练函数分别为

附加动量的BP算法的代表性训练函数为：

def train_with_momentum(x_train, y_train, learning_rate, momentum, num_epochs):
    # 初始化权重和偏置
    w = np.random.randn(x_train.shape[1], 1)
    b = np.random.randn()

    # 初始化动量
    v_w = np.zeros_like(w)
    v_b = 0

    # 训练
    for epoch in range(num_epochs):
        # 前向传播
        z = np.dot(x_train, w) + b
        a = sigmoid(z)

        # 计算损失和梯度
        loss = binary_cross_entropy(y_train, a)
        d_loss = a - y_train
        d_z = d_loss * sigmoid_derivative(a)

        # 计算动量
        v_w = momentum * v_w - learning_rate * np.dot(x_train.T, d_z)
        v_b = momentum * v_b - learning_rate * np.sum(d_z)

        # 更新权重和偏置
        w += v_w
        b += v_b

    return w, b

学习率自适应算法的代表性训练函数为：

def train_with_adaptive_learning_rate(x_train, y_train, num_epochs):
    # 初始化权重和偏置
    w = np.random.randn(x_train.shape[1], 1)
    b = np.random.randn()

    # 初始化学习率
    learning_rate = 0.1

    # 训练
    for epoch in range(num_epochs):
        # 前向传播
        z = np.dot(x_train, w) + b
        a = sigmoid(z)

        # 计算损失和梯度
        loss = binary_cross_entropy(y_train, a)
        d_loss = a - y_train
        d_z = d_loss * sigmoid_derivative(a)

        # 更新权重和偏置
        w -= learning_rate * np.dot(x_train.T, d_z)
        b -= learning_rate * np.sum(d_z)

        # 自适应学习率
        if epoch % 10 == 0:
            if loss < 0.1:
                learning_rate *= 0.5
            elif loss > 0.5:
                learning_rate *= 1.1

    return w, b

LM算法的代表性训练函数为：

def train_with_lm(x_train, y_train, num_epochs):
    # 初始化权重和偏置
    w = np.random.randn(x_train.shape[1], 1)
    b = np.random.randn()

    # 初始化阻尼因子
    mu = 0.1

    # 训练
    for epoch in range(num_epochs):
        # 前向传播
        z = np.dot(x_train, w) + b
        a = sigmoid(z)

        # 计算损失和梯度
        loss = binary_cross_entropy(y_train, a)
        d_loss = a - y_train
        d_z = d_loss * sigmoid_derivative(a)

        # 计算Hessian矩阵和雅可比矩阵
        H = np.dot(x_train.T, x_train) + mu * np.eye(x_train.shape[1])
        J = np.dot(x_train.T, d_z)

        # 求解线性方程组
        dw, db = np.linalg.solve(H, -J)

        # 更新权重和偏置
        w += dw.reshape(w.shape)
        b += db

        # 自适应阻尼因子
        if loss < 0.1:
            mu *= 0.5
        elif loss > 0.5:
            mu *= 2

    return w, b