BP神经网络实现mnist

MNIST是一个手写数字的图像数据集，BP神经网络可以用于分类这些图像。下面是一个简单的Python代码实现MNIST数据集的分类任务，使用了一个三层的BP神经网络。

首先，我们需要导入必要的库和MNIST数据集：

import numpy as np
from sklearn.datasets import fetch_openml

mnist = fetch_openml('mnist_784')
X = mnist.data.astype('float32')
y = mnist.target.astype('int32')

然后，我们将数据集划分为训练集和测试集：

num_train = 60000
X_train, X_test = np.split(X, [num_train])
y_train, y_test = np.split(y, [num_train])

接下来，我们定义BP神经网络的结构和超参数：

input_size = 784
hidden_size = 256
output_size = 10
learning_rate = 0.1
num_epochs = 20
batch_size = 100

然后，我们定义激活函数和BP神经网络的前向和反向传播函数：

def sigmoid(x):
    return 1 / (1 + np.exp(-x))

def forward(x, W1, b1, W2, b2):
    z1 = np.dot(x, W1) + b1
    a1 = sigmoid(z1)
    z2 = np.dot(a1, W2) + b2
    a2 = np.exp(z2) / np.sum(np.exp(z2), axis=1, keepdims=True)
    return a1, a2

def backward(x, y, a1, a2, W1, b1, W2, b2):
    m = x.shape[0]
    dz2 = a2 - y
    dW2 = np.dot(a1.T, dz2) / m
    db2 = np.sum(dz2, axis=0, keepdims=True) / m
    dz1 = np.dot(dz2, W2.T) * a1 * (1 - a1)
    dW1 = np.dot(x.T, dz1) / m
    db1 = np.sum(dz1, axis=0, keepdims=True) / m
    return dW1, db1, dW2, db2

接下来，我们初始化BP神经网络的参数：

W1 = np.random.randn(input_size, hidden_size) * 0.01
b1 = np.zeros((1, hidden_size))
W2 = np.random.randn(hidden_size, output_size) * 0.01
b2 = np.zeros((1, output_size))

然后，我们开始训练BP神经网络：

num_batches = num_train // batch_size
for epoch in range(num_epochs):
    for batch in range(num_batches):
        start = batch * batch_size
        end = start + batch_size
        x_batch = X_train[start:end]
        y_batch = np.zeros((batch_size, output_size))
        y_batch[np.arange(batch_size), y_train[start:end]] = 1
        a1, a2 = forward(x_batch, W1, b1, W2, b2)
        dW1, db1, dW2, db2 = backward(x_batch, y_batch, a1, a2, W1, b1, W2, b2)
        W1 -= learning_rate * dW1
        b1 -= learning_rate * db1
        W2 -= learning_rate * dW2
        b2 -= learning_rate * db2
    _, a2 = forward(X_train, W1, b1, W2, b2)
    train_loss = -np.mean(np.sum(np.log(a2) * (y_train == np.arange(output_size)), axis=1))
    _, a2 = forward(X_test, W1, b1, W2, b2)
    test_loss = -np.mean(np.sum(np.log(a2) * (y_test == np.arange(output_size)), axis=1))
    print('Epoch %d, train loss: %.4f, test loss: %.4f' % (epoch+1, train_loss, test_loss))

最后，我们可以使用训练好的BP神经网络对测试集进行分类：

_, a2 = forward(X_test, W1, b1, W2, b2)
y_pred = np.argmax(a2, axis=1)
accuracy = np.mean(y_pred == y_test)
print('Accuracy: %.4f' % accuracy)

这个简单的BP神经网络可以得到约96%的准确率。需要注意的是，这只是一个简单的示例，实际上，为了得到更好的性能，需要进行更多的优化和改进。