神经网络正向传播和反向传播计算示例：双隐层Sigmoid激活函数

(1) 输入层到H1层： z1 = 1 * 1 + (-1) * (-2) + 1 * (-1) + 1 * 1 + 1 = 3 a1 = sigmoid(z1) = 0.9526 z2 = 1 * 2 + (-1) * (-1) + 1 * (-2) + 1 * 0 + 0 = 3 a2 = sigmoid(z2) = 0.9526

H1层到H2层： z3 = 0 * 2 + 0 * (-1) + 0 * (-2) + 0 * 1 + 0 = 0 a3 = sigmoid(z3) = 0.5 z4 = 0 * 2 + 0 * (-1) + 0 * (-2) + 0 * (-1) + 0 = 0 a4 = sigmoid(z4) = 0.5

H2层到输出层： z5 = (-2) * 3 + 2 * 1 + 0 * (-1) + 0 * (-1) + 2 = -4 a5 = sigmoid(z5) = 0.0179 z6 = 2 * 3 + 0 * 1 + (-2) * (-1) + 0 * 4 + 2 = 8 a6 = sigmoid(z6) = 0.9997

(2) 输出层的误差为： E = (0.5 - 0.0179)^2 + (1 - 0.9997)^2 = 0.964

为了优化该样本的误差，可以使用反向传播算法来更新网络参数。具体步骤如下：

计算输出层的误差项： δ5 = (y5 - a5) * a5 * (1 - a5) = (0.5 - 0.0179) * 0.0179 * (1 - 0.0179) = 0.0084 δ6 = (y6 - a6) * a6 * (1 - a6) = (1 - 0.9997) * 0.9997 * (1 - 0.9997) = 0.0001
计算H2层的误差项： δ3 = δ5 * w5 * a3 * (1 - a3) = 0.0084 * (-2) * 0.5 * (1 - 0.5) = -0.0021 δ4 = δ6 * w6 * a4 * (1 - a4) = 0.0001 * 2 * 0.5 * (1 - 0.5) = 0.000025
更新H2层到输出层的权重和偏置： w5 = w5 + η * δ5 * a3 = 3 - 0.1 * 0.0084 * 0.5 = 2.9986 w6 = w6 + η * δ6 * a4 = -1 - 0.1 * 0.0001 * 0.5 = -1.000005 b2 = b2 + η * δ5 = 2 - 0.1 * 0.0084 = 1.99916
更新H1层到H2层的权重和偏置： w3 = w3 + η * δ3 * a2 = -1 - 0.1 * (-0.0021) * 0.9526 = -1.0002 w4 = w4 + η * δ4 * a2 = 1 - 0.1 * 0.000025 * 0.9526 = 0.999997 b1 = b1 + η * δ3 = 1 - 0.1 * (-0.0021) = 1.00021
更新输入层到H1层的权重和偏置： w1 = w1 + η * δ1 * x1 = 1 - 0.1 * (-0.0021) * 1 = 1.00021 w2 = w2 + η * δ2 * x2 = -2 - 0.1 * (-0.0021) * (-1) = -1.99979 b1 = b1 + η * δ1 = 1 - 0.1 * (-0.0021) = 1.00021 b2 = b2 + η * δ2 = 2 - 0.1 * (-0.0021) = 2.00021

其中，η为学习率，可以根据实际情况进行调整。