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最新推荐文章于 2025-08-19 13:45:29 发布

原创最新推荐文章于 2025-08-19 13:45:29 发布 · 416 阅读

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#机器学习 #神经网络 #深度学习

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本文介绍了神经网络的前向传播过程，从输入层到隐含层再到输出层的计算步骤。接着详细阐述了反向传播中的损失函数和权重更新，特别是针对隐含层到输出层以及输入层到隐含层的权值更新策略。最后，文章通过TensorFlow展示了神经网络的代码实现。

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$x_{0}=1$ $x_{1}=0.05$ $x_{2}=0.10$

$v_{01}=0.35$ $v_{11}=0.15$ $v_{21}=0.20$

$v_{02}=0.35$ $v_{12}=0.25$ $v_{22}=0.30$

$w_{01}=0.60$ $w_{11}=0.40$ $w_{21}=0.45$

$w_{02}=0.60$ $w_{12}=0.50$ $w_{22}=0.55$

$y_{1}=0.01$ $y_{2}=0.99$

一. 前向传播

输入层->隐含层

$h_{1}$ 的输入: $net_{h_{1}}=x_{0}v_{01}+x_{1}v_{11}+x_{2}v_{21}=0.3775$

$h_{1}$ 的输出: $out_{h_{1}}=\frac{1}{1+e^{-net_{h_{1}}}}\approx 0.5932699921$

$h_{2}$ 的输入: $net_{h_{2}}=x_{0}v_{02}+x_{1}v_{12}+x_{2}v_{22}=0.3925$

$h_{2}$ 的输出: $out_{h_{2}}=\frac{1}{1+e^{-net_{h_{2}}}}\approx 0.5968843783$

2. 隐含层->输出层

$y_{1}$ 的输入: $net_{y_{1}}=h_{0}w_{01}+h_{1}w_{11}+h_{2}w_{21}\approx 1.1059059671$

$y_{1}$ 的输出: $out_{y_{1}}=\frac{1}{1+e^{-net_{y_{1}}}}\approx 0.7513650696$

$y_{2}$ 的输入: $net_{y_{2}}=h_{0}w_{02}+h_{1}w_{12}+h_{2}w_{22}\approx 1.2249214041$

$y_{2}$ 的输出: $out_{y_{2}}=\frac{1}{1+e^{-net_{y_{2}}}}\approx 0.7729284653$

二. 反向传播

损失函数(均方误差): $E_{total}=\frac{1}{2}\sum_{i=1}^{m}{(target_{y_{i}}-out_{y_{i}})^2}$
隐含层->输出层的权值更新

以 $w_{11}$ 为例, 用整体误差对 $w_{11}$ 求偏导

$\frac{\partial E_{total}}{\partial w_{11}}=\frac{\partial E_{y_{1}}}{\partial w_{11}}=\frac{\partial E_{y_{1}}}{\partial out_{y_{1}}}*\frac{\partial out_{y_{1}}}{\partial net_{y_{1}}}*\frac{\partial net_{y_{1}}}{\partial w_{11}}$

$\frac{\partial E_{total}}{\partial w_{11}}=\frac{\partial E_{y_{1}}}{\partial w_{11}}=(out_{y_{1}}-target_{y_{1}})*(out_{y_{1}}(1-out_{y_{1}}))*h_{1}$

$w_{11}^{+}=w_{11}-\eta \frac{\partial E_{total}}{\partial w_{11}}=0.40-0.5*0.0821670406=0.3589164797$

$w_{21}^{+}=w_{21}-\eta \frac{\partial E_{total}}{\partial w_{21}}=0.45-0.5*0.0826676278=0.4086661861$

3. 输入层->隐含层的权值更新

以 $v_{11}$ 为例, 用整体误差对 $v_{11}$ 求偏导

$\frac{\partial E_{total}}{\partial v_{11}}=\frac{\partial E_{y_{1}}}{\partial v_{11}}+\frac{\partial E_{y_{2}}}{\partial v_{11}}$

$\frac{\partial E_{y_{1}}}{\partial v_{11}}=\frac{\partial E_{y_{1}}}{\partial out_{y_{1}}}*\frac{\partial out_{y_{1}}}{\partial net_{y_{1}}}*\frac{\partial net_{y_{1}}}{\partial out_{h_{1}}}*\frac{\partial out_{h_{1}}}{\partial net_{h_{1}}}*\frac{\partial net_{h_{1}}}{\partial v_{11}}$

$\frac{\partial E_{y_{2}}}{\partial v_{11}}=\frac{\partial E_{y_{2}}}{\partial out_{y_{2}}}*\frac{\partial out_{y_{2}}}{\partial net_{y_{2}}}*\frac{\partial net_{y_{2}}}{\partial out_{h_{1}}}*\frac{\partial out_{h_{1}}}{\partial net_{h_{1}}}*\frac{\partial net_{h_{1}}}{\partial v_{11}}$

$\frac{\partial E_{y_{1}}}{\partial v_{11}}=(out_{y_{1}}-target_{y_{1}})*(out_{y_{1}}(1-out_{y_{1}}))*w_{11}*(out_{h_{1}}(1-out_{h_{1}}))*x_{1}$

$\frac{\partial E_{y_{2}}}{\partial v_{11}}=(out_{y_{2}}-target_{y_{2}})*(out_{y_{2}}(1-out_{y_{2}}))*w_{21}*(out_{h_{1}}(1-out_{h_{1}}))*x_{1}$

$v_{11}^{+}=v_{11}-\eta \frac{\partial E_{total}}{\partial v_{11}}=0.15-0.5*(0.0006683960+0.0007519455)=0.1492898293$

三. TensorFlow代码实现

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

import math
import numpy as np
import tensorflow as tf

sess = tf.InteractiveSession()

x_input = tf.constant([[0.05, 0.10]], dtype=tf.float32)
y_input = tf.constant([[0.01, 0.99]], dtype=tf.float32)

biases1 = tf.Variable(tf.constant([[0.35, 0.35]], dtype=tf.float32))
weights1 = tf.Variable(tf.constant([[0.15, 0.25], [0.20, 0.30]], dtype=tf.float32))

biases2 = tf.Variable(tf.constant([[0.60, 0.60]], dtype=tf.float32))
weights2 = tf.Variable(tf.constant([[0.40, 0.50], [0.45, 0.55]], dtype=tf.float32))

h_layer1 = tf.nn.sigmoid(tf.matmul(x_input, weights1) + biases1)
y_output = tf.nn.sigmoid(tf.matmul(h_layer1, weights2) + biases2)

loss_mse = tf.reduce_mean(tf.square(y_input - y_output))

tf.global_variables_initializer().run()

for step in range(401):
    tf.train.GradientDescentOptimizer(0.50).minimize(loss_mse).run()
    if step % 20 == 0:
        print(step, y_output.eval())

sess.close()