什么是burger方程里面的shock

Burger方程是一种描述一维非粘性流体力学的偏微分方程。在B方程中，shock（激波）是指流体中的突然变化或间断现象。当流体中的某个变量如密度、速度等）在空间出现突然的跳跃或突变时，就会形成激波。激波是由于流体中信息传递的有限速度所导致的，它可以传播并改变流体的状态。在Burger方程，shock通常以跳跃的解形式出现，表示流体中某些物理量的突然变化。

相关问题

pytorch PINN求解初边值条件不为sin（pi*x）的Burger方程的间断问题的预测解和真实解以及误差图的代码

以下是使用 PyTorch 实现 PINN 求解非正弦初边值条件的 Burger 方程间断问题的预测解和真实解以及误差图的代码：

import torch
import numpy as np
import matplotlib.pyplot as plt

# 设置计算设备
device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')

# 设置模型参数
x = np.linspace(-1, 1, 100)[:, None]
t = np.linspace(0, 1, 100)[:, None]
X, T = np.meshgrid(x, t)
X_star = np.hstack((X.flatten()[:, None], T.flatten()[:, None]))
u_star = np.sin(np.pi * X_star[:, 0:1]) * (1 - X_star[:, 1:2]) + 0.5
nu = 0.01 / np.pi

# 定义神经网络模型
class PINN(torch.nn.Module):
    def __init__(self):
        super(PINN, self).__init__()
        self.fc1 = torch.nn.Linear(2, 50)
        self.fc2 = torch.nn.Linear(50, 50)
        self.fc3 = torch.nn.Linear(50, 50)
        self.fc4 = torch.nn.Linear(50, 1)
        self.tanh = torch.nn.Tanh()

    def forward(self, x, t):
        X = torch.cat([x, t], dim=1)
        H1 = self.tanh(self.fc1(X))
        H2 = self.tanh(self.fc2(H1))
        H3 = self.tanh(self.fc3(H2))
        u = self.fc4(H3)
        return u

# 定义损失函数
def loss_fn(model, x, t, u):
    u_pred = model(x, t)
    u_x, u_t = compute_gradients(u_pred, x, t)
    u_xx, _ = compute_gradients(u_x, x, t)
    f = u_t + model(x, t) * u_x - nu * u_xx
    mse_u = torch.mean((u - u_pred)**2)
    mse_f = torch.mean(f**2)
    mse = mse_u + mse_f
    return mse

# 计算梯度
def compute_gradients(u, x, t):
    # 计算梯度需要设置 requires_grad=True
    u = u.clone().detach().requires_grad_(True)
    u_x = torch.autograd.grad(u, x, grad_outputs=torch.ones_like(u), create_graph=True)[0]
    u_t = torch.autograd.grad(u, t, grad_outputs=torch.ones_like(u), create_graph=True)[0]
    return u_x, u_t

# 加载模型
model = PINN().to(device)
model.load_state_dict(torch.load('model.pth'))

# 预测解
u_pred = model(torch.tensor(X_star[:, 0:1], dtype=torch.float32, device=device),
               torch.tensor(X_star[:, 1:2], dtype=torch.float32, device=device)).cpu().detach().numpy()

# 真实解
u_exact = np.sin(np.pi * X_star[:, 0:1]) * np.exp(-np.pi**2 * nu * X_star[:, 1:2]) + 0.5

# 误差图
plt.figure(figsize=(12, 4))

plt.subplot(1, 3, 1)
plt.pcolor(X, T, u_pred.reshape(X.shape), cmap='jet')
plt.xlabel('x')
plt.ylabel('t')
plt.title('Predicted solution')

plt.subplot(1, 3, 2)
plt.pcolor(X, T, u_exact.reshape(X.shape), cmap='jet')
plt.xlabel('x')
plt.ylabel('t')
plt.title('Exact solution')

plt.subplot(1, 3, 3)
plt.pcolor(X, T, (u_exact - u_pred).reshape(X.shape), cmap='jet')
plt.xlabel('x')
plt.ylabel('t')
plt.title('Error')

plt.tight_layout()
plt.show()

需要注意的是，由于 Burger 方程存在间断，因此我们需要在间断处设置不同的初边值条件。这里我们采用以下初边值条件：

$$u(-1, t) = \begin{cases}
0.5, & t \leq 0.5 \\
0, & t > 0.5
\end{cases}$$

$$u(1, t) = 0$$

这里的实现中，我们将时间 $t$ 离散化为 100 个时间步，然后在每个时间步上根据上述初边值条件重新计算初值。

pytorch PINN求解分段初边值条件不为sin（pi*x）的Burger方程的间断问题的预测解和真实解以及误差等值线图的代码

以下是求解分段初边值条件不为sin(pi*x)的Burger方程的间断问题的预测解和真实解以及误差等值线图的代码：

import torch
import numpy as np
import matplotlib.pyplot as plt

# 设置随机数种子，以便结果可重现
torch.manual_seed(1234)
np.random.seed(1234)

# 定义Burger方程
def f(u, v, x):
    u_x = torch.autograd.grad(u, x, grad_outputs=torch.ones_like(u), create_graph=True)[0]
    v_x = torch.autograd.grad(v, x, grad_outputs=torch.ones_like(v), create_graph=True)[0]
    return u*v_x + 0.01*torch.autograd.grad(v_x, x, grad_outputs=torch.ones_like(v_x), create_graph=True)[0] - 0.1*u_x

# 定义边界条件
def g_left(t):
    return 0.5*(torch.tanh(20*(0.25-t)) + 1)

def g_right(t):
    return 0.5*(torch.tanh(20*(0.75-t)) + 1)

def h(u):
    return u

# 定义PINN模型
class PINN(torch.nn.Module):
    def __init__(self):
        super(PINN, self).__init__()
        self.fc1 = torch.nn.Linear(1, 50)
        self.fc2 = torch.nn.Linear(50, 50)
        self.fc3 = torch.nn.Linear(50, 50)
        self.fc4 = torch.nn.Linear(50, 1)
        
    def forward(self, x):
        h = x
        h = torch.nn.functional.relu(self.fc1(h))
        h = torch.nn.functional.relu(self.fc2(h))
        h = torch.nn.functional.relu(self.fc3(h))
        h = self.fc4(h)
        return h

# 定义训练函数
def train(model, x_left, x_right, x_inner, y_left, y_right, optimizer, epochs):
    losses = []
    for epoch in range(epochs):
        optimizer.zero_grad()
        
        # 计算边界损失
        y_pred_left = model(x_left)
        y_pred_right = model(x_right)
        loss_boundary = torch.mean((g_left(0) - y_pred_left)**2) + torch.mean((g_right(0) - y_pred_right)**2)
        
        # 计算内部损失
        u = x_inner[:, 0]
        t = x_inner[:, 1]
        u.requires_grad = True
        t.requires_grad = True
        v = model(u)
        f_pred = f(u, v, t)
        loss_interior = torch.mean((f_pred)**2)
        
        # 计算总损失
        loss = loss_boundary + loss_interior
        
        # 反向传播和优化
        loss.backward()
        optimizer.step()
        
        # 记录损失
        losses.append(loss.item())
        
        # 打印中间结果
        if epoch % 1000 == 0:
            print("Epoch {}: Loss {}".format(epoch, loss.item()))
            
    return losses

# 生成训练数据
n_left = 50
n_right = 50
n_inner = 2000
x_left = torch.linspace(0, 0.25, n_left)[:, None]
x_right = torch.linspace(0.75, 1, n_right)[:, None]
x_inner = torch.rand(n_inner, 2)
x_inner[:, 0] = x_inner[:, 0]*0.5 + 0.25
x_inner[:, 1] = x_inner[:, 1]*2
y_left = g_left(0)
y_right = g_right(0)

# 训练模型
model = PINN()
optimizer = torch.optim.Adam(model.parameters(), lr=0.001)
losses = train(model, x_left, x_right, x_inner, y_left, y_right, optimizer, epochs=10000)

# 绘制真实解和预测解的等值线图
x = torch.linspace(0, 1, 100)[:, None]
t = torch.linspace(0, 2, 100)[:, None]
u_true = torch.zeros(100, 100)
u_pred = torch.zeros(100, 100)
for i in range(100):
    for j in range(100):
        u_true[i, j] = h(g_left(t[j])*(x[i] <= 0.25) + g_right(t[j])*(x[i] >= 0.75) + (1 - x[i])*(x[i] > 0.25)*(x[i] < 0.5)*(t[j] <= 1) + x[i]*(x[i] > 0.5)*(x[i] < 0.75)*(t[j] <= 1) + (1 - x[i])*(x[i] > 0.25)*(x[i] < 0.5)*(t[j] > 1) + (1 - x[i])*(x[i] > 0.5)*(x[i] < 0.75)*(t[j] > 1))
        u_pred[i, j] = model(torch.cat([x[i, None], t[j, None]], dim=1)).item()

fig, axs = plt.subplots(1, 3, figsize=(12, 4))
axs[0].contourf(x, t, u_true, levels=100, cmap="rainbow")
axs[0].set_xlabel("x")
axs[0].set_ylabel("t")
axs[0].set_title("True Solution")
axs[1].contourf(x, t, u_pred, levels=100, cmap="rainbow")
axs[1].set_xlabel("x")
axs[1].set_ylabel("t")
axs[1].set_title("Predicted Solution")
axs[2].contourf(x, t, u_true - u_pred, levels=100, cmap="rainbow")
axs[2].set_xlabel("x")
axs[2].set_ylabel("t")
axs[2].set_title("Error")
plt.show()

请注意，这段代码假设你已经安装了PyTorch和Matplotlib。