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人工神经网络
（Artificial Neural Network，ANN）是一种模仿生物神经网络的结构和功能的数学模型，其目的是通过学习和训练，在处理未知的输入数据时能够进行复杂的非线性映射关系，实现自适应的智能决策。可以说，ANN是人工智能算法中最基础、最核心的一种算法。
ANN模型的基本结构包含输入层、隐藏层和输出层。输入层接收输入数据，隐藏层负责对数据进行多层次、高维度的变换和处理，输出层对处理后的数据进行输出。ANN的训练过程是通过多次迭代，不断调整神经网络中各层的权重，从而使得神经网络能够对输入数据进行正确的预测和分类。
人工神经网络算法示例
接下来看看一个简单的人工神经网络算法示例：
import numpy as np class NeuralNetwork(): def __init__(self, layers): """ layers: 数组，包含每个层的神经元数量，例如 [2, 3, 1] 表示 3 层神经网络，第一层 2 个神经元，第二层 3 个神经元，第三层 1 个神经元。 weights: 数组，包含每个连接的权重矩阵，默认值随机生成。 biases: 数组，包含每个层的偏差值，默认值为 0。 """ self.layers = layers self.weights = [np.random.randn(a, b) for a, b in zip(layers[1:], layers[:-1])] self.biases = [np.zeros((a, 1)) for a in layers[1:]] def sigmoid(self, z): """Sigmoid 激活函数.""" return 1 / (1 + np.exp(-z)) def forward_propagation(self, a): """前向传播.""" for w, b in zip(self.weights, self.biases): z = np.dot(w, a) + b a = self.sigmoid(z) return a def backward_propagation(self, x, y): """反向传播.""" nabla_w = [np.zeros(w.shape) for w in self.weights] nabla_b = [np.zeros(b.shape) for b in self.biases] a = x activations = [x] zs = [] for w, b in zip(self.weights, self.biases): z = np.dot(w, a) + b zs.append(z) a = self.sigmoid(z) activations.append(a) delta = self.cost_derivative(activations[-1], y) * self.sigmoid_prime(zs[-1]) nabla_b[-1] = delta nabla_w[-1] = np.dot(delta, activations[-2].transpose()) for l in range(2, len(self.layers)): z = zs[-l] sp = self.sigmoid_prime(z) delta = np.dot(self.weights[-l+1].transpose(), delta) * sp nabla_b[-l] = delta nabla_w[-l] = np.dot(delta, activations[-l-1].transpose()) return (nabla_w, nabla_b) def train(self, x_train, y_train, epochs, learning_rate): """训练网络.""" for epoch in range(epochs): nabla_w = [np.zeros(w.shape) for w in self.weights] nabla_b = [np.zeros(b.shape) for b in self.biases] for x, y in zip(x_train, y_train): delta_nabla_w, delta_nabla_b = self.backward_propagation(np.array([x]).transpose(), np.array([y]).transpose()) nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)] nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)] self.weights = [w-(learning_rate/len(x_train))*nw for w, nw in zip(self.weights, nabla_w)] self.biases = [b-(learning_rate/len(x_train))*nb for b, nb in zip(self.biases, nabla_b)] def predict(self, x_test): """预测.""" y_predictions = [] for x in x_test: y_predictions.append(self.forward_propagation(np.array([x]).transpose())[0][0]) return y_predictions def cost_derivative(self, output_activations, y): """损失函数的导数.""" return output_activations - y def sigmoid_prime(self, z): """Sigmoid 函数的导数.""" return self.sigmoid(z) * (1 - self.sigmoid(z))
使用以下代码示例来实例化和使用这个简单的神经网络类：
x_train = [[0, 0], [1, 0], [0, 1], [1, 1]]
y_train = [0, 1, 1, 0]
# 创建神经网络
nn = NeuralNetwork([2, 3, 1])
# 训练神经网络
nn.train(x_train, y_train, 10000, 0.1)
# 测试神经网络
x_test = [[0, 0], [1, 0], [0, 1], [1, 1]]
y_test = [0, 1, 1, 0]
y_predictions = nn.predict(x_test)
print("Predictions:", y_predictions)
print("Actual:", y_test)输出结果：
Predictions: [0.011602156431658403, 0.9852717774725432, 0.9839448924887225, 0.020026540429992387]
Actual: [0, 1, 1, 0]