Software

Please generate PyTorch-based code to simulate the wake flow behind a two-dimensional cylinder using PINN.

import torch
import torch.nn as nn
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from tqdm import tqdm

# デバイスの設定
device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')
print(f"Using device: {device}")

# ニューラルネットワークの定義
class PINN(nn.Module):
    def __init__(self, layers):
        super(PINN, self).__init__()
        self.layers = nn.ModuleList()
        
        for i in range(len(layers) - 1):
            self.layers.append(nn.Linear(layers[i], layers[i+1]))
            
    def forward(self, x, y, t):
        # 入力の正規化
        inputs = torch.cat([x, y, t], dim=1)
        
        for i in range(len(self.layers) - 1):
            inputs = torch.tanh(self.layers[i](inputs))
        
        # 最終層(活性化関数なし)
        output = self.layers[-1](inputs)
        
        # 速度と圧力を出力
        u = output[:, 0:1]  # x方向速度
        v = output[:, 1:2]  # y方向速度
        p = output[:, 2:3]  # 圧力
        
        return u, v, p

# Navier-Stokes方程式の残差計算
class NavierStokesPINN:
    def __init__(self, layers, Re=100):
        self.model = PINN(layers).to(device)
        self.Re = Re  # レイノルズ数
        self.nu = 1.0 / Re  # 動粘性係数
        
        # 最適化アルゴリズム
        self.optimizer = torch.optim.Adam(self.model.parameters(), lr=1e-3)
        self.scheduler = torch.optim.lr_scheduler.StepLR(
            self.optimizer, step_size=5000, gamma=0.9
        )
        
        self.loss_history = []
        
    def net_NS(self, x, y, t):
        """Navier-Stokes方程式の残差を計算"""
        x.requires_grad_(True)
        y.requires_grad_(True)
        t.requires_grad_(True)
        
        u, v, p = self.model(x, y, t)
        
        # 1階微分
        u_x = torch.autograd.grad(u, x, grad_outputs=torch.ones_like(u), 
                                   create_graph=True)[0]
        u_y = torch.autograd.grad(u, y, 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]
        
        v_x = torch.autograd.grad(v, x, grad_outputs=torch.ones_like(v), 
                                   create_graph=True)[0]
        v_y = torch.autograd.grad(v, y, grad_outputs=torch.ones_like(v), 
                                   create_graph=True)[0]
        v_t = torch.autograd.grad(v, t, grad_outputs=torch.ones_like(v), 
                                   create_graph=True)[0]
        
        p_x = torch.autograd.grad(p, x, grad_outputs=torch.ones_like(p), 
                                   create_graph=True)[0]
        p_y = torch.autograd.grad(p, y, grad_outputs=torch.ones_like(p), 
                                   create_graph=True)[0]
        
        # 2階微分
        u_xx = torch.autograd.grad(u_x, x, grad_outputs=torch.ones_like(u_x), 
                                    create_graph=True)[0]
        u_yy = torch.autograd.grad(u_y, y, grad_outputs=torch.ones_like(u_y), 
                                    create_graph=True)[0]
        
        v_xx = torch.autograd.grad(v_x, x, grad_outputs=torch.ones_like(v_x), 
                                    create_graph=True)[0]
        v_yy = torch.autograd.grad(v_y, y, grad_outputs=torch.ones_like(v_y), 
                                    create_graph=True)[0]
        
        # Navier-Stokes方程式(運動量保存則)
        f_u = u_t + u * u_x + v * u_y + p_x - self.nu * (u_xx + u_yy)
        f_v = v_t + u * v_x + v * v_y + p_y - self.nu * (v_xx + v_yy)
        
        # 連続の式(質量保存則)
        f_cont = u_x + v_y
        
        return f_u, f_v, f_cont, u, v, p
    
    def loss_function(self, x_domain, y_domain, t_domain,
                     x_inlet, y_inlet, t_inlet,
                     x_outlet, y_outlet, t_outlet,
                     x_wall_top, y_wall_top, t_wall_top,
                     x_wall_bottom, y_wall_bottom, t_wall_bottom,
                     x_cylinder, y_cylinder, t_cylinder,
                     x_init, y_init, t_init):
        """損失関数の計算"""
        
        # 領域内でのNavier-Stokes方程式の残差
        f_u, f_v, f_cont, _, _, _ = self.net_NS(x_domain, y_domain, t_domain)
        loss_pde = torch.mean(f_u**2) + torch.mean(f_v**2) + torch.mean(f_cont**2)
        
        # 入口境界条件(一様流)
        u_inlet, v_inlet, _ = self.model(x_inlet, y_inlet, t_inlet)
        loss_inlet = torch.mean((u_inlet - 1.0)**2) + torch.mean(v_inlet**2)
        
        # 出口境界条件(圧力ゼロ、速度の勾配ゼロに近似)
        u_outlet, v_outlet, p_outlet = self.model(x_outlet, y_outlet, t_outlet)
        loss_outlet = torch.mean(p_outlet**2)
        
        # 上下壁境界条件(自由滑り)
        u_wall_t, v_wall_t, _ = self.model(x_wall_top, y_wall_top, t_wall_top)
        u_wall_b, v_wall_b, _ = self.model(x_wall_bottom, y_wall_bottom, t_wall_bottom)
        loss_wall = torch.mean(v_wall_t**2) + torch.mean(v_wall_b**2)
        
        # 円柱表面境界条件(無滑り条件)
        u_cyl, v_cyl, _ = self.model(x_cylinder, y_cylinder, t_cylinder)
        loss_cylinder = torch.mean(u_cyl**2) + torch.mean(v_cyl**2)
        
        # 初期条件
        u_init, v_init, _ = self.model(x_init, y_init, t_init)
        loss_init = torch.mean((u_init - 1.0)**2) + torch.mean(v_init**2)
        
        # 総損失
        loss = loss_pde + 10.0 * (loss_inlet + loss_outlet + loss_wall + 
                                   loss_cylinder + loss_init)
        
        return loss, loss_pde, loss_inlet, loss_outlet, loss_wall, loss_cylinder
    
    def train(self, epochs, x_domain, y_domain, t_domain,
              x_inlet, y_inlet, t_inlet,
              x_outlet, y_outlet, t_outlet,
              x_wall_top, y_wall_top, t_wall_top,
              x_wall_bottom, y_wall_bottom, t_wall_bottom,
              x_cylinder, y_cylinder, t_cylinder,
              x_init, y_init, t_init):
        """モデルの訓練"""
        
        self.model.train()
        
        pbar = tqdm(range(epochs), desc="Training")
        for epoch in pbar:
            self.optimizer.zero_grad()
            
            loss, loss_pde, loss_inlet, loss_outlet, loss_wall, loss_cylinder = \
                self.loss_function(x_domain, y_domain, t_domain,
                                 x_inlet, y_inlet, t_inlet,
                                 x_outlet, y_outlet, t_outlet,
                                 x_wall_top, y_wall_top, t_wall_top,
                                 x_wall_bottom, y_wall_bottom, t_wall_bottom,
                                 x_cylinder, y_cylinder, t_cylinder,
                                 x_init, y_init, t_init)
            
            loss.backward()
            self.optimizer.step()
            self.scheduler.step()
            
            self.loss_history.append(loss.item())
            
            if epoch % 100 == 0:
                pbar.set_postfix({
                    'Loss': f'{loss.item():.4e}',
                    'PDE': f'{loss_pde.item():.4e}',
                    'BC': f'{(loss_inlet + loss_outlet + loss_wall + loss_cylinder).item():.4e}'
                })
        
    def predict(self, x, y, t):
        """予測"""
        self.model.eval()
        with torch.no_grad():
            u, v, p = self.model(x, y, t)
        return u, v, p

# サンプリング点の生成
def generate_training_data(N_domain=5000, N_boundary=500, N_initial=1000):
    """訓練データの生成"""
    
    # 計算領域の定義
    x_min, x_max = -2.0, 10.0
    y_min, y_max = -2.0, 2.0
    t_min, t_max = 0.0, 5.0
    
    # 円柱のパラメータ
    x_cyl, y_cyl = 0.0, 0.0
    r_cyl = 0.5
    
    # 領域内のランダムサンプリング(円柱内部を除く)
    x_domain = []
    y_domain = []
    t_domain = []
    
    while len(x_domain) < N_domain:
        x = np.random.uniform(x_min, x_max, N_domain * 2)
        y = np.random.uniform(y_min, y_max, N_domain * 2)
        t = np.random.uniform(t_min, t_max, N_domain * 2)
        
        # 円柱内部を除外
        mask = (x - x_cyl)**2 + (y - y_cyl)**2 > r_cyl**2
        x = x[mask]
        y = y[mask]
        t = t[mask]
        
        x_domain.extend(x[:N_domain - len(x_domain)])
        y_domain.extend(y[:N_domain - len(y_domain)])
        t_domain.extend(t[:N_domain - len(t_domain)])
    
    x_domain = torch.FloatTensor(x_domain).reshape(-1, 1).to(device)
    y_domain = torch.FloatTensor(y_domain).reshape(-1, 1).to(device)
    t_domain = torch.FloatTensor(t_domain).reshape(-1, 1).to(device)
    
    # 入口境界(x = x_min)
    x_inlet = torch.ones(N_boundary, 1).to(device) * x_min
    y_inlet = torch.FloatTensor(np.random.uniform(y_min, y_max, N_boundary)).reshape(-1, 1).to(device)
    t_inlet = torch.FloatTensor(np.random.uniform(t_min, t_max, N_boundary)).reshape(-1, 1).to(device)
    
    # 出口境界(x = x_max)
    x_outlet = torch.ones(N_boundary, 1).to(device) * x_max
    y_outlet = torch.FloatTensor(np.random.uniform(y_min, y_max, N_boundary)).reshape(-1, 1).to(device)
    t_outlet = torch.FloatTensor(np.random.uniform(t_min, t_max, N_boundary)).reshape(-1, 1).to(device)
    
    # 上壁境界(y = y_max)
    x_wall_top = torch.FloatTensor(np.random.uniform(x_min, x_max, N_boundary)).reshape(-1, 1).to(device)
    y_wall_top = torch.ones(N_boundary, 1).to(device) * y_max
    t_wall_top = torch.FloatTensor(np.random.uniform(t_min, t_max, N_boundary)).reshape(-1, 1).to(device)
    
    # 下壁境界(y = y_min)
    x_wall_bottom = torch.FloatTensor(np.random.uniform(x_min, x_max, N_boundary)).reshape(-1, 1).to(device)
    y_wall_bottom = torch.ones(N_boundary, 1).to(device) * y_min
    t_wall_bottom = torch.FloatTensor(np.random.uniform(t_min, t_max, N_boundary)).reshape(-1, 1).to(device)
    
    # 円柱表面境界
    theta = np.random.uniform(0, 2*np.pi, N_boundary)
    x_cylinder = torch.FloatTensor(x_cyl + r_cyl * np.cos(theta)).reshape(-1, 1).to(device)
    y_cylinder = torch.FloatTensor(y_cyl + r_cyl * np.sin(theta)).reshape(-1, 1).to(device)
    t_cylinder = torch.FloatTensor(np.random.uniform(t_min, t_max, N_boundary)).reshape(-1, 1).to(device)
    
    # 初期条件(t = 0)
    x_init = []
    y_init = []
    
    while len(x_init) < N_initial:
        x = np.random.uniform(x_min, x_max, N_initial * 2)
        y = np.random.uniform(y_min, y_max, N_initial * 2)
        
        mask = (x - x_cyl)**2 + (y - y_cyl)**2 > r_cyl**2
        x = x[mask]
        y = y[mask]
        
        x_init.extend(x[:N_initial - len(x_init)])
        y_init.extend(y[:N_initial - len(y_init)])
    
    x_init = torch.FloatTensor(x_init).reshape(-1, 1).to(device)
    y_init = torch.FloatTensor(y_init).reshape(-1, 1).to(device)
    t_init = torch.zeros(N_initial, 1).to(device)
    
    return (x_domain, y_domain, t_domain,
            x_inlet, y_inlet, t_inlet,
            x_outlet, y_outlet, t_outlet,
            x_wall_top, y_wall_top, t_wall_top,
            x_wall_bottom, y_wall_bottom, t_wall_bottom,
            x_cylinder, y_cylinder, t_cylinder,
            x_init, y_init, t_init)

# 可視化
def visualize_flow(pinn, t_value=2.0, save_path='flow_field.png'):
    """流れ場の可視化"""
    
    x = np.linspace(-2, 10, 150)
    y = np.linspace(-2, 2, 80)
    X, Y = np.meshgrid(x, y)
    
    x_flat = torch.FloatTensor(X.flatten()).reshape(-1, 1).to(device)
    y_flat = torch.FloatTensor(Y.flatten()).reshape(-1, 1).to(device)
    t_flat = torch.ones_like(x_flat) * t_value
    
    u, v, p = pinn.predict(x_flat, y_flat, t_flat)
    
    u = u.cpu().numpy().reshape(X.shape)
    v = v.cpu().numpy().reshape(X.shape)
    p = p.cpu().numpy().reshape(X.shape)
    
    # 円柱内部をマスク
    mask = (X - 0.0)**2 + (Y - 0.0)**2 < 0.5**2
    u[mask] = np.nan
    v[mask] = np.nan
    p[mask] = np.nan
    
    # 速度の大きさ
    velocity_magnitude = np.sqrt(u**2 + v**2)
    
    fig, axes = plt.subplots(2, 2, figsize=(16, 8))
    
    # u速度
    im1 = axes[0, 0].contourf(X, Y, u, levels=20, cmap='RdBu_r')
    axes[0, 0].set_title(f'u-velocity at t={t_value}')
    axes[0, 0].set_xlabel('x')
    axes[0, 0].set_ylabel('y')
    axes[0, 0].add_patch(plt.Circle((0, 0), 0.5, color='black'))
    plt.colorbar(im1, ax=axes[0, 0])
    
    # v速度
    im2 = axes[0, 1].contourf(X, Y, v, levels=20, cmap='RdBu_r')
    axes[0, 1].set_title(f'v-velocity at t={t_value}')
    axes[0, 1].set_xlabel('x')
    axes[0, 1].set_ylabel('y')
    axes[0, 1].add_patch(plt.Circle((0, 0), 0.5, color='black'))
    plt.colorbar(im2, ax=axes[0, 1])
    
    # 圧力
    im3 = axes[1, 0].contourf(X, Y, p, levels=20, cmap='RdBu_r')
    axes[1, 0].set_title(f'Pressure at t={t_value}')
    axes[1, 0].set_xlabel('x')
    axes[1, 0].set_ylabel('y')
    axes[1, 0].add_patch(plt.Circle((0, 0), 0.5, color='black'))
    plt.colorbar(im3, ax=axes[1, 0])
    
    # 速度の大きさとストリームライン
    im4 = axes[1, 1].contourf(X, Y, velocity_magnitude, levels=20, cmap='viridis')
    axes[1, 1].streamplot(X, Y, u, v, color='white', linewidth=0.5, density=1.5)
    axes[1, 1].set_title(f'Velocity magnitude at t={t_value}')
    axes[1, 1].set_xlabel('x')
    axes[1, 1].set_ylabel('y')
    axes[1, 1].add_patch(plt.Circle((0, 0), 0.5, color='black'))
    plt.colorbar(im4, ax=axes[1, 1])
    
    plt.tight_layout()
    plt.savefig(save_path, dpi=150, bbox_inches='tight')
    plt.show()
    
    print(f"Flow field visualization saved to {save_path}")

def plot_loss_history(pinn, save_path='loss_history.png'):
    """損失関数の履歴をプロット"""
    
    plt.figure(figsize=(10, 6))
    plt.plot(pinn.loss_history)
    plt.yscale('log')
    plt.xlabel('Iteration')
    plt.ylabel('Loss')
    plt.title('Training Loss History')
    plt.grid(True)
    plt.savefig(save_path, dpi=150, bbox_inches='tight')
    plt.show()
    
    print(f"Loss history plot saved to {save_path}")

# メイン実行
if __name__ == "__main__":
    # ハイパーパラメータ
    layers = [3, 64, 64, 64, 64, 3]  # [入力層, 隠れ層..., 出力層]
    Re = 100  # レイノルズ数
    epochs = 10000  # 訓練エポック数
    
    print("=" * 60)
    print("2D Cylinder Flow Simulation using PINN")
    print("=" * 60)
    print(f"Reynolds number: {Re}")
    print(f"Network architecture: {layers}")
    print(f"Training epochs: {epochs}")
    print("=" * 60)
    
    # PINNモデルの初期化
    pinn = NavierStokesPINN(layers, Re=Re)
    
    # 訓練データの生成
    print("\nGenerating training data...")
    training_data = generate_training_data(
        N_domain=5000,
        N_boundary=500,
        N_initial=1000
    )
    
    # モデルの訓練
    print("\nStarting training...")
    pinn.train(epochs, *training_data)
    
    # 損失関数の履歴をプロット
    print("\nPlotting loss history...")
    plot_loss_history(pinn, save_path='/workspace/outputs/loss_history.png')
    
    # 流れ場の可視化(複数の時刻)
    print("\nVisualizing flow fields...")
    for t in [0.5, 1.0, 2.0, 3.0]:
        visualize_flow(pinn, t_value=t, 
                      save_path=f'/workspace/outputs/flow_field_t{t:.1f}.png')
    
    # モデルの保存
    torch.save(pinn.model.state_dict(), '/workspace/outputs/pinn_cylinder_model.pth')
    print("\nModel saved to /workspace/outputs/pinn_cylinder_model.pth")
    
    print("\n" + "=" * 60)
    print("Simulation completed!")
    print("=" * 60)