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 2019.9.24author: vince'''import mathimport loggingimport numpy import randomimport 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, 0], [1, 1], [0, 1]]); Y = numpy.array([0, 1, 0, 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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