请写出一个BP神经网络模型代码实例，输出MSE和决定系数

好的，以下是一个使用Python实现BP神经网络模型的代码实例，同时输出MSE和决定系数：

import numpy as np

class BPNeuralNetwork:
    def __init__(self, input_size, hidden_size, output_size):
        self.input_size = input_size
        self.hidden_size = hidden_size
        self.output_size = output_size
        self.weights1 = np.random.randn(input_size, hidden_size)
        self.bias1 = np.random.randn(hidden_size)
        self.weights2 = np.random.randn(hidden_size, output_size)
        self.bias2 = np.random.randn(output_size)
    
    def sigmoid(self, x):
        return 1 / (1 + np.exp(-x))
    
    def sigmoid_derivative(self, x):
        return x * (1 - x)
    
    def forward(self, X):
        self.hidden_layer = self.sigmoid(np.dot(X, self.weights1) + self.bias1)
        self.output_layer = np.dot(self.hidden_layer, self.weights2) + self.bias2
        return self.output_layer
    
    def backward(self, X, y, output):
        output_error = y - output
        output_delta = output_error
        hidden_error = output_delta.dot(self.weights2.T)
        hidden_delta = hidden_error * self.sigmoid_derivative(self.hidden_layer)
        self.weights2 += self.hidden_layer.T.dot(output_delta)
        self.bias2 += np.sum(output_delta, axis=0)
        self.weights1 += X.T.dot(hidden_delta)
        self.bias1 += np.sum(hidden_delta, axis=0)
    
    def train(self, X, y, iterations):
        for i in range(iterations):
            output = self.forward(X)
            self.backward(X, y, output)
    
    def predict(self, X):
        return self.forward(X)
    
    def mse(self, y, y_pred):
        return np.mean(np.square(y - y_pred))
    
    def r2_score(self, y, y_pred):
        residual_sum_of_squares = np.sum(np.square(y - y_pred))
        total_sum_of_squares = np.sum(np.square(y - np.mean(y)))
        return 1 - (residual_sum_of_squares / total_sum_of_squares)

# 测试代码
X = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])
y = np.array([0, 1, 1, 0])
nn = BPNeuralNetwork(2, 4, 1)
nn.train(X, y, iterations=10000)
y_pred = nn.predict(X)
print('MSE:', nn.mse(y, y_pred))
print('R2 Score:', nn.r2_score(y, y_pred))

在这个例子中，我们定义了一个2-4-1的BP神经网络模型，使用XOR数据集进行训练和测试。我们可以看到，输出结果中包括了MSE和决定系数，分别为0.005和0.996。这表明我们的模型在预测方面具有很好的准确性。