3. 使用pytorch，从零开始实现多层感知机，其中数据集选择mnist

使用PyTorch从头开始实现一个多层感知机（Multilayer Perceptron, MLP）来处理MNIST数据集，需要按照以下步骤：

1. **导入库**:
首先，导入所需的库，如`torch`, `torch.nn`, `torch.optim`, 和 `torchvision.datasets`。

import torch
import torch.nn as nn
import torch.optim as optim
from torchvision import datasets, transforms

2. **预处理数据**:
加载并标准化MNIST数据集，将像素值归一化到0到1之间。

transform = transforms.Compose([transforms.ToTensor(),
                                transforms.Normalize((0.1307,), (0.3081,))
])

train_dataset = datasets.MNIST(root='./data', train=True, download=True, transform=transform)
test_dataset = datasets.MNIST(root='./data', train=False, download=True, transform=transform)

train_loader = torch.utils.data.DataLoader(train_dataset, batch_size=64, shuffle=True)
test_loader = torch.utils.data.DataLoader(test_dataset, batch_size=64, shuffle=False)

3. **构建模型**:
设计一个多层神经网络，包括输入层、隐藏层和输出层。

class MLP(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(MLP, self).__init__()
        self.fc1 = nn.Linear(input_dim, hidden_dim)
        self.relu = nn.ReLU()
        self.fc2 = nn.Linear(hidden_dim, hidden_dim)
        self.fc3 = nn.Linear(hidden_dim, output_dim)

    def forward(self, x):
        out = self.fc1(x)
        out = self.relu(out)
        out = self.fc2(out)
        out = self.relu(out)
        out = self.fc3(out)
        return out

input_dim = 28 * 28  # MNIST图片大小
hidden_dim = 512
output_dim = 10  # 输出类别数
net = MLP(input_dim, hidden_dim, output_dim)

4. **设置损失函数和优化器**:
选择适当的损失函数（交叉熵），并配置优化器（例如Adam）。

criterion = nn.CrossEntropyLoss()
optimizer = optim.Adam(net.parameters(), lr=0.001)

5. **训练模型**:
在训练循环中，前向传播、反向传播以及更新权重。

num_epochs = 10
device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
net.to(device)

for epoch in range(num_epochs):
    for images, labels in train_loader:
        images, labels = images.to(device), labels.to(device)

        optimizer.zero_grad()
        outputs = net(images)
        loss = criterion(outputs, labels)
        loss.backward()
        optimizer.step()

6. **评估模型**:
测试模型在测试集上的性能。

correct_count, total = 0, 0
with torch.no_grad():
    for images, labels in test_loader:
        images, labels = images.to(device), labels.to(device)
        outputs = net(images)
        _, predicted_labels = torch.max(outputs.data, 1)
        total += labels.size(0)
        correct_count += (predicted_labels == labels).sum().item()

accuracy = correct_count / total
print(f"Accuracy after training: {accuracy * 100:.2f}%")