济南学生网站建设求职,域名备案需要有网站吗,视频建设网站,wordpress无法在线安装插件PINN解偏微分方程实例11. PINN简介2. 偏微分方程实例3. 基于pytorch实现代码4. 数值解参考资料1. PINN简介 PINN是一种利用神经网络求解偏微分方程的方法#xff0c;其计算流程图如下图所示#xff0c;这里以偏微分方程(1)为例。 ∂u∂tu∂u∂xv∂2u∂x2\begin{align} \frac{…
PINN解偏微分方程实例11. PINN简介2. 偏微分方程实例3. 基于pytorch实现代码4. 数值解参考资料1. PINN简介 PINN是一种利用神经网络求解偏微分方程的方法其计算流程图如下图所示这里以偏微分方程(1)为例。 ∂u∂tu∂u∂xv∂2u∂x2\begin{align} \frac{\partial u}{\partial t}u \frac{\partial u}{\partial x}v\frac{\partial^2 u}{\partial x^2} \end{align} ∂t∂uu∂x∂uv∂x2∂2u 神经网络输入位置x,y,z和时间t的值预测偏微分方程解u在这个时空条件下的数值解。 上图中可以看出PINN的损失函数包含两部分内容一部分是来源于训练数据误差另一部分来源于偏微分方程误差可以记作(2)式。 lwdataldatawPDElPDE\begin{align} \mathcal{l} w_{data}\mathcal{l}_{data}w_{PDE}\mathcal{l}_{PDE} \end{align} lwdataldatawPDElPDE 其中 ldata1Ndata∑i1Ndata(u(xi,ti)−ui)2lPDE1Ndata∑j1NPDE(∂u∂tu∂u∂x−v∂2u∂x2)2∣(xj,tj)\begin{align} \begin{aligned} \mathcal{l}_{data} \frac{1}{N_{data}}\sum_{i1}^{N_{data}} (u(x_i,t_i)-u_i)^2 \\ \mathcal{l}_{PDE} \frac{1}{N_{data}}\sum_{j1}^{N_{PDE}} \left( \frac{\partial u}{\partial t}u \frac{\partial u}{\partial x}-v\frac{\partial^2 u}{\partial x^2} \right)^2|_{(x_j,t_j)} \end{aligned} \end{align} ldatalPDENdata1i1∑Ndata(u(xi,ti)−ui)2Ndata1j1∑NPDE(∂t∂uu∂x∂u−v∂x2∂2u)2∣(xj,tj)
2. 偏微分方程实例 考虑偏微分方程如下 ∂2u∂x2−∂4u∂y4(2−x2)e−y\begin{align} \begin{aligned} \frac{\partial^2 u}{\partial x^2} - \frac{\partial^4 u}{\partial y^4} (2-x^2)e^{-y} \end{aligned} \end{align} ∂x2∂2u−∂y4∂4u(2−x2)e−y 考虑以下边界条件 uyy(x,0)x2uyy(x,1)x2eu(x,0)x2u(x,1)x2eu(0,y)0u(1,y)e−y\begin{align} \begin{aligned} u_{yy}(x,0) x^2 \\ u_{yy}(x,1) \frac{x^2}{e} \\ u(x,0) x^2 \\ u(x,1) \frac{x^2}{e} \\ u(0,y) 0 \\ u(1,y) e^{-y} \\ \end{aligned} \end{align} uyy(x,0)uyy(x,1)u(x,0)u(x,1)u(0,y)u(1,y)x2ex2x2ex20e−y 以上偏微分方程真解为u(x,y)x2e−yu(x,y)x^2 e^{-y}u(x,y)x2e−y,在区域[0,1]×[0,1][0,1]\times[0,1][0,1]×[0,1]上随机采样配置点和数据点其中配置点用来构造PDE损失函数l1,l2,⋯,l7\mathcal{l}_1,\mathcal{l}_2,\cdots,\mathcal{l}_7l1,l2,⋯,l7数据点用来构造数据损失函数l8\mathcal{l}_8l8. l11N1∑(xi,yi)∈Ω(u^xx(xi,yi;θ)−u^yyyy(xi,yi;θ)−(2−xi2)e−yi)2l21N2∑(xi,yi)∈[0,1]×{0}(u^yy(xi,yi;θ)−xi2)2l31N3∑(xi,yi)∈[0,1]×{1}(u^yy(xi,yi;θ)−xi2e)2l41N4∑(xi,yi)∈[0,1]×{0}(u^(xi,yi;θ)−xi2)2l51N5∑(xi,yi)∈[0,1]×{1}(u^(xi,yi;θ)−xi2e)2l61N6∑(xi,yi)∈{0}×[0,1](u^(xi,yi;θ)−0)2l71N7∑(xi,yi)∈{1}×[0,1](u^(xi,yi;θ)−e−yi)2l81N8∑i1N8(u^(xi,yi;θ)−ui)2\begin{align} \begin{aligned} \mathcal{l}_1 \frac{1}{N_1}\sum_{(x_i,y_i)\in\Omega} (\hat{u}_{xx}(x_i,y_i;\theta) - \hat{u}_{yyyy}(x_i,y_i;\theta) - (2-x_i^2)e^{-y_i})^2 \\ \mathcal{l}_2 \frac{1}{N_2}\sum_{(x_i,y_i)\in[0,1]\times\{0\}} (\hat{u}_{yy}(x_i,y_i;\theta) - x_i^2)^2 \\ \mathcal{l}_3 \frac{1}{N_3}\sum_{(x_i,y_i)\in[0,1]\times\{1\}} (\hat{u}_{yy}(x_i,y_i;\theta) - \frac{x_i^2}{e})^2 \\ \mathcal{l}_4 \frac{1}{N_4}\sum_{(x_i,y_i)\in[0,1]\times\{0\}} (\hat{u}(x_i,y_i;\theta) - x_i^2)^2 \\ \mathcal{l}_5 \frac{1}{N_5}\sum_{(x_i,y_i)\in[0,1]\times\{1\}} (\hat{u}(x_i,y_i;\theta) - \frac{x_i^2}{e})^2 \\ \mathcal{l}_6 \frac{1}{N_6}\sum_{(x_i,y_i)\in\{0\}\times [0,1]}(\hat{u}(x_i,y_i;\theta) - 0)^2 \\ \mathcal{l}_7 \frac{1}{N_7}\sum_{(x_i,y_i)\in\{1\}\times [0,1]}(\hat{u}(x_i,y_i;\theta) - e^{-y_i})^2 \\ \mathcal{l}_8 \frac{1}{N_{8}}\sum_{i1}^{N_{8}} (\hat{u}(x_i,y_i;\theta)-u_i)^2 \end{aligned} \end{align} l1l2l3l4l5l6l7l8N11(xi,yi)∈Ω∑(u^xx(xi,yi;θ)−u^yyyy(xi,yi;θ)−(2−xi2)e−yi)2N21(xi,yi)∈[0,1]×{0}∑(u^yy(xi,yi;θ)−xi2)2N31(xi,yi)∈[0,1]×{1}∑(u^yy(xi,yi;θ)−exi2)2N41(xi,yi)∈[0,1]×{0}∑(u^(xi,yi;θ)−xi2)2N51(xi,yi)∈[0,1]×{1}∑(u^(xi,yi;θ)−exi2)2N61(xi,yi)∈{0}×[0,1]∑(u^(xi,yi;θ)−0)2N71(xi,yi)∈{1}×[0,1]∑(u^(xi,yi;θ)−e−yi)2N81i1∑N8(u^(xi,yi;θ)−ui)2
3. 基于pytorch实现代码 A scratch for PINN solving the following PDE
u_xx-u_yyyy(2-x^2)*exp(-y)
Author: suntao
Date: 2023/2/26import torch
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3Depochs 10000 # 训练代数
h 100 # 画图网格密度
N 1000 # 内点配置点数
N1 100 # 边界点配置点数
N2 1000 # PDE数据点def setup_seed(seed):torch.manual_seed(seed)torch.cuda.manual_seed_all(seed)torch.backends.cudnn.deterministic True# 设置随机数种子
setup_seed(888888)# Domain and Sampling
def interior(nN):# 内点x torch.rand(n, 1)y torch.rand(n, 1)cond (2 - x ** 2) * torch.exp(-y)return x.requires_grad_(True), y.requires_grad_(True), conddef down_yy(nN1):# 边界 u_yy(x,0)x^2x torch.rand(n, 1)y torch.zeros_like(x)cond x ** 2return x.requires_grad_(True), y.requires_grad_(True), conddef up_yy(nN1):# 边界 u_yy(x,1)x^2/ex torch.rand(n, 1)y torch.ones_like(x)cond x ** 2 / torch.ereturn x.requires_grad_(True), y.requires_grad_(True), conddef down(nN1):# 边界 u(x,0)x^2x torch.rand(n, 1)y torch.zeros_like(x)cond x ** 2return x.requires_grad_(True), y.requires_grad_(True), conddef up(nN1):# 边界 u(x,1)x^2/ex torch.rand(n, 1)y torch.ones_like(x)cond x ** 2 / torch.ereturn x.requires_grad_(True), y.requires_grad_(True), conddef left(nN1):# 边界 u(0,y)0y torch.rand(n, 1)x torch.zeros_like(y)cond torch.zeros_like(x)return x.requires_grad_(True), y.requires_grad_(True), conddef right(nN1):# 边界 u(1,y)e^(-y)y torch.rand(n, 1)x torch.ones_like(y)cond torch.exp(-y)return x.requires_grad_(True), y.requires_grad_(True), conddef data_interior(nN2):# 内点x torch.rand(n, 1)y torch.rand(n, 1)cond (x ** 2) * torch.exp(-y)return x.requires_grad_(True), y.requires_grad_(True), cond# Neural Network
class MLP(torch.nn.Module):def __init__(self):super(MLP, self).__init__()self.net torch.nn.Sequential(torch.nn.Linear(2, 32),torch.nn.Tanh(),torch.nn.Linear(32, 32),torch.nn.Tanh(),torch.nn.Linear(32, 32),torch.nn.Tanh(),torch.nn.Linear(32, 32),torch.nn.Tanh(),torch.nn.Linear(32, 1))def forward(self, x):return self.net(x)# Loss
loss torch.nn.MSELoss()def gradients(u, x, order1):if order 1:return torch.autograd.grad(u, x, grad_outputstorch.ones_like(u),create_graphTrue,only_inputsTrue, )[0]else:return gradients(gradients(u, x), x, orderorder - 1)# 以下7个损失是PDE损失
def l_interior(u):# 损失函数L1x, y, cond interior()uxy u(torch.cat([x, y], dim1))return loss(gradients(uxy, x, 2) - gradients(uxy, y, 4), cond)def l_down_yy(u):# 损失函数L2x, y, cond down_yy()uxy u(torch.cat([x, y], dim1))return loss(gradients(uxy, y, 2), cond)def l_up_yy(u):# 损失函数L3x, y, cond up_yy()uxy u(torch.cat([x, y], dim1))return loss(gradients(uxy, y, 2), cond)def l_down(u):# 损失函数L4x, y, cond down()uxy u(torch.cat([x, y], dim1))return loss(uxy, cond)def l_up(u):# 损失函数L5x, y, cond up()uxy u(torch.cat([x, y], dim1))return loss(uxy, cond)def l_left(u):# 损失函数L6x, y, cond left()uxy u(torch.cat([x, y], dim1))return loss(uxy, cond)def l_right(u):# 损失函数L7x, y, cond right()uxy u(torch.cat([x, y], dim1))return loss(uxy, cond)# 构造数据损失
def l_data(u):# 损失函数L8x, y, cond data_interior()uxy u(torch.cat([x, y], dim1))return loss(uxy, cond)# Trainingu MLP()
opt torch.optim.Adam(paramsu.parameters())for i in range(epochs):opt.zero_grad()l l_interior(u) \ l_up_yy(u) \ l_down_yy(u) \ l_up(u) \ l_down(u) \ l_left(u) \ l_right(u) \ l_data(u)l.backward()opt.step()if i % 100 0:print(i)# Inference
xc torch.linspace(0, 1, h)
xm, ym torch.meshgrid(xc, xc)
xx xm.reshape(-1, 1)
yy ym.reshape(-1, 1)
xy torch.cat([xx, yy], dim1)
u_pred u(xy)
u_real xx * xx * torch.exp(-yy)
u_error torch.abs(u_pred-u_real)
u_pred_fig u_pred.reshape(h,h)
u_real_fig u_real.reshape(h,h)
u_error_fig u_error.reshape(h,h)
print(Max abs error is: , float(torch.max(torch.abs(u_pred - xx * xx * torch.exp(-yy)))))
# 仅有PDE损失 Max abs error: 0.004852950572967529
# 带有数据点损失 Max abs error: 0.0018916130065917969# 作PINN数值解图
fig plt.figure()
ax Axes3D(fig)
ax.plot_surface(xm.detach().numpy(), ym.detach().numpy(), u_pred_fig.detach().numpy())
ax.text2D(0.5, 0.9, PINN, transformax.transAxes)
plt.show()
fig.savefig(PINN solve.png)# 作真解图
fig plt.figure()
ax Axes3D(fig)
ax.plot_surface(xm.detach().numpy(), ym.detach().numpy(), u_real_fig.detach().numpy())
ax.text2D(0.5, 0.9, real solve, transformax.transAxes)
plt.show()
fig.savefig(real solve.png)# 误差图
fig plt.figure()
ax Axes3D(fig)
ax.plot_surface(xm.detach().numpy(), ym.detach().numpy(), u_error_fig.detach().numpy())
ax.text2D(0.5, 0.9, abs error, transformax.transAxes)
plt.show()
fig.savefig(abs error.png)4. 数值解 参考资料
[1]. Physics-informed machine learning [2]. 知乎-PaperWeekly
