BP神经网络是最简单的神经网络模型了，三层能够模拟非线性函数效果。

难点：

• 如何确定初始化参数？
• 如何确定隐含层节点数量？
• 迭代多少次？如何更快收敛？
• 如何获得全局最优解？

'''
neural networks 
 
created on 2019.9.24
author: vince
'''
import math
import logging
import numpy
import random
import matplotlib.pyplot as plt
 
'''
neural network
'''
class NeuralNetwork:
 
 def __init__(self,layer_nums,iter_num = 10000,batch_size = 1):
  self.__ILI = 0;
  self.__HLI = 1;
  self.__OLI = 2;
  self.__TLN = 3;
 
  if len(layer_nums) != self.__TLN:
   raise Exception("layer_nums length must be 3");
 
  self.__layer_nums = layer_nums; #array [layer0_num,layer1_num ...layerN_num]
  self.__iter_num = iter_num;
  self.__batch_size = batch_size;
 
 def train(self,X,Y):
  X = numpy.array(X);
  Y = numpy.array(Y);
 
  self.L = [];
  #initialize parameters
  self.__weight = [];
  self.__bias = [];
  self.__step_len = [];
  for layer_index in range(1,self.__TLN):
   self.__weight.append(numpy.random.rand(self.__layer_nums[layer_index - 1],self.__layer_nums[layer_index]) * 2 - 1.0);
   self.__bias.append(numpy.random.rand(self.__layer_nums[layer_index]) * 2 - 1.0);
   self.__step_len.append(0.3);
 
  logging.info("bias:%s" % (self.__bias));
  logging.info("weight:%s" % (self.__weight));
 
  for iter_index in range(self.__iter_num):
   sample_index = random.randint(0,len(X) - 1);
   logging.debug("-----round:%s,select sample %s-----" % (iter_index,sample_index));
   output = self.forward_pass(X[sample_index]);
   g = (-output[2] + Y[sample_index]) * self.activation_drive(output[2]);
   logging.debug("g:%s" % (g));
   for j in range(len(output[1])):
    self.__weight[1][j] += self.__step_len[1] * g * output[1][j];
   self.__bias[1] -= self.__step_len[1] * g;
 
   e = [];
   for i in range(self.__layer_nums[self.__HLI]):
    e.append(numpy.dot(g,self.__weight[1][i]) * self.activation_drive(output[1][i]));
   e = numpy.array(e);
   logging.debug("e:%s" % (e));
   for j in range(len(output[0])):
    self.__weight[0][j] += self.__step_len[0] * e * output[0][j];
   self.__bias[0] -= self.__step_len[0] * e;
 
   l = 0;
   for i in range(len(X)):
    predictions = self.forward_pass(X[i])[2];
    l += 0.5 * numpy.sum((predictions - Y[i]) ** 2);
   l /= len(X);
   self.L.append(l);
 
   logging.debug("bias:%s" % (self.__bias));
   logging.debug("weight:%s" % (self.__weight));
   logging.debug("loss:%s" % (l));
  logging.info("bias:%s" % (self.__bias));
  logging.info("weight:%s" % (self.__weight));
  logging.info("L:%s" % (self.L));
 
 def activation(self,z):
  return (1.0 / (1.0 + numpy.exp(-z)));
 
 def activation_drive(self,y):
  return y * (1.0 - y);
 
 def forward_pass(self,x):
  data = numpy.copy(x);
  result = [];
  result.append(data);
  for layer_index in range(self.__TLN - 1):
   data = self.activation(numpy.dot(data,self.__weight[layer_index]) - self.__bias[layer_index]);
   result.append(data);
  return numpy.array(result);
 
 def predict(self,x):
  return self.forward_pass(x)[self.__OLI];
 
def main():
 logging.basicConfig(level = logging.INFO,format = '%(asctime)s %(filename)s[line:%(lineno)d] %(levelname)s %(message)s',datefmt = '%a,%d %b %Y %H:%M:%S');
 
 logging.info("trainning begin.");
 nn = NeuralNetwork([2,2,1]);
 X = numpy.array([[0,0],[1,1],[0,1]]);
 Y = numpy.array([0,1,1]);
 nn.train(X,Y);
 
 logging.info("trainning end. predict begin.");
 for x in X:
  print(x,nn.predict(x));
 
 plt.plot(nn.L)
 plt.show();
 
if __name__ == "__main__":
 main();

具体收敛效果

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