使用TensorFlow构建两层神经网络进行二次函数拟合

Ivy

发布时间：2026-10-13| 4人浏览过

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本文介绍如何使用TensorFlow构建一个包含隐藏层和输出层的两层神经网络，以拟合二次函数。通过生成带有噪声的数据集，定义占位符、权重初始化及激活函数，并利用梯度下降算法最小化误差，最终实现模型训练与可视化展示。

本笔记旨在通过TensorFlow框架实现一个包含两层结构的神经网络，核心目标是完成对二次函数的拟合任务。

探讨如何向模型中添加网络层。

具体实现代码如下：

def add_layer(inputs, in_size, out_size, activation_function=None):
  # add one more layer and return the output of this layer
  Weights = tf.Variable(tf.random_normal([in_size, out_size]))
  biases = tf.Variable(tf.zeros([1, out_size]) + 0.1)
  Wx_plus_b = tf.matmul(inputs, Weights) + biases
  if activation_function is None:
    outputs = Wx_plus_b
  else:
    outputs = activation_function(Wx_plus_b)
  return outputs

需注意函数内部计算顺序为xW+b，而非Wx+b，因此乘法操作的顺序至关重要。变量x应定义为[类别数量，数据数量]，而W定义为[数据类别，类别数量]。

构建测试所需的数据集。

# Make up some real data
x_data = np.linspace(-1,1,300)[:, np.newaxis]
noise = np.random.normal(0, 0.05, x_data.shape)
y_data = np.square(x_data) - 0.5 + noise

利用numpy库中的linspace函数生成等差数列，其中start和stop参数分别设定数列的起始与终止值，endpoint参数用于控制是否包含终点值。

numpy.linspace(start, stop, num=50, endpoint=True, retstep=False, dtype=None)[source] 
Return evenly spaced numbers over a specified interval. 
Returns num evenly spaced samples, calculated over the interval [start, stop].

引入noise函数添加噪声，确保二次函数的数据点不会完全重合于理论曲线，从而模拟真实场景。

使用numpy的newaxis功能可直接新增维度，无需重新创建shape并赋值，操作便捷。如前述示例所示，该操作将x_data从一维扩展为二维。

引入占位符以接收输入数据。

# define placeholder for inputs to network
xs = tf.placeholder(tf.float32, [None, 1])
ys = tf.placeholder(tf.float32, [None, 1])

构建隐藏层与输出层结构。

# add hidden layer
l1 = add_layer(xs, 1, 10, activation_function=tf.nn.relu)
# add output layer
prediction = add_layer(l1, 10, 1, activation_function=None)

计算预测值与实际数据间的误差，并采用梯度下降算法使误差最小化。

# the error between prediciton and real data
loss = tf.reduce_mean(tf.reduce_sum(tf.square(ys - prediction),reduction_indices=[1]))
train_step = tf.train.GradientDescentOptimizer(0.1).minimize(loss)

完整代码实现如下：

from __future__ import print_function
import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt

def add_layer(inputs, in_size, out_size, activation_function=None):
  # add one more layer and return the output of this layer
  Weights = tf.Variable(tf.random_normal([in_size, out_size]))
  biases = tf.Variable(tf.zeros([1, out_size]) + 0.1)
  Wx_plus_b = tf.matmul(inputs, Weights) + biases
  if activation_function is None:
    outputs = Wx_plus_b
  else:
    outputs = activation_function(Wx_plus_b)
  return outputs

# Make up some real data
x_data = np.linspace(-1,1,300)[:, np.newaxis]
noise = np.random.normal(0, 0.05, x_data.shape)
y_data = np.square(x_data) - 0.5 + noise

# define placeholder for inputs to network
xs = tf.placeholder(tf.float32, [None, 1])
ys = tf.placeholder(tf.float32, [None, 1])
# add hidden layer
l1 = add_layer(xs, 1, 10, activation_function=tf.nn.relu)
# add output layer
prediction = add_layer(l1, 10, 1, activation_function=None)

# the error between prediciton and real data
loss = tf.reduce_mean(tf.reduce_sum(tf.square(ys - prediction),
           reduction_indices=[1]))
train_step = tf.train.GradientDescentOptimizer(0.1).minimize(loss)

# important step
init = tf.initialize_all_variables()
sess = tf.Session()
sess.run(init)

# plot the real data
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
ax.scatter(x_data, y_data)
plt.ion()
plt.show()

for i in range(1000):
  # training
  sess.run(train_step, feed_dict={xs: x_data, ys: y_data})
  if i % 50 == 0:
    # to visualize the result and improvement
    try:
      ax.lines.remove(lines[0])
    except Exception:
      pass
    prediction_value = sess.run(prediction, feed_dict={xs: x_data})
    # plot the prediction
    lines = ax.plot(x_data, prediction_value, 'r-', lw=5)
    plt.pause(0.1)

程序运行结果展示：