Trying to solve NS equation, but got RuntimeError

[baichen99]

Hi Lu Lu/ All,

Sorry I made a mistake and reversed the order of u and x. Please help me delete this disscussion

Thanks for your great work !
But I have some problems with deepxde and would like to ask for your help

NS equation

NS equation is as follows:

$$ \frac{\partial \mathbf{U}}{\partial t}+\nabla \cdot(\mathbf{U U})=-\nabla \frac{p}{\rho}+\nabla \cdot(\nabla \mathbf{U}) $$

In which:

$U$ is a 2d vector

$$ \frac{\partial \mathbf{U}}{\partial t}=\left[\begin{array}{c} \frac{\partial u_{1}}{\partial t} \\ \frac{\partial u_{2}}{\partial t} \\ \end{array}\right] $$

$$ \mathbf{U U}=\mathbf{U} \cdot \mathbf{U}^{T}=\left[\begin{array}{l} u_{1} \\ u_{2} \\ \end{array}\right]\left[u_{1}, u_{2}\right]=\left[\begin{array}{lll} u_{1} u_{1} & u_{1} u_{2} \\ u_{2} u_{1} & u_{2} u_{2} \\ \end{array}\right] $$

$$ \nabla \cdot(\mathbf{U U})=\nabla \cdot\left[\begin{array}{lll} u_{1} u_{1} & u_{1} u_{2} \\ u_{2} u_{1} & u_{2} u_{2} \\ \end{array}\right]=\left[\begin{array}{l} \frac{\partial u_{1} u_{1}}{\partial x}+\frac{\partial u_{2} u_{1}}{\partial y} \\ \frac{\partial u_{1} u_{2}}{\partial x}+\frac{\partial u_{2} u_{2}}{\partial y} \\ \end{array}\right] $$

$$ \nabla p=\left[\begin{array}{l} \frac{\partial p}{\partial x} \\ \frac{\partial p}{\partial y} \\ \end{array}\right] $$

$$ \nabla \cdot(\nabla \mathbf{U})=\nabla \cdot\left[\begin{array}{lll} \frac{\partial u_{1}}{\partial x} & \frac{\partial u_{2}}{\partial x} \\ \frac{\partial u_{1}}{\partial y} & \frac{\partial u_{2}}{\partial y} \\ \end{array}\right]=\left[\begin{array}{l} \frac{\partial}{\partial x}\left(\frac{\partial u_{1}}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_{1}}{\partial y}\right)\\ \frac{\partial}{\partial x}\left(\frac{\partial u_{2}}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_{2}}{\partial y}\right) \\ \end{array}\right] $$

code

define pde

#pde.py
import deepxde as dde
import torch


def pde(x, u):
    # x[:, 0:1] x
    # x[:, 1:2] y
    # x[:, 2:]  t
    ux = u[:, 0:1]
    uy = u[:, 1:2] 
    # p = u[:, 2:]
    print('u.shape', u.shape)
    print('x.shape', x.shape)
    rho = 5550.0
    
    ux_t = dde.grad.jacobian(u, x, i=0, j=2)
    uy_t = dde.grad.jacobian(u, x, i=1, j=2)
    guu_1 = dde.grad.jacobian(ux*ux, x, i=0, j=0) + dde.grad.jacobian(uy*ux, x, i=0, j=1)
    guu_2 = dde.grad.jacobian(ux*uy, x, i=0, j=1) + dde.grad.jacobian(uy*uy, x, i=0, j=1)
    p_x = dde.grad.jacobian(u, x, i=2, j=0)
    p_y = dde.grad.jacobian(u, x, i=2, j=1)
    

    ggu_1 = dde.grad.hessian(u, x, i=0, j=0, component=0) +  dde.grad.hessian(u, x, i=1, j=1, component=0)
    ggu_2= dde.grad.hessian(u, x, i=0, j=0, component=1) +  dde.grad.hessian(u, x, i=1, j=1, component=1)
    
    eq_1 = ux_t + guu_1 + p_x / rho - ggu_1
    eq_2 = uy_t + guu_2 + p_y / rho - ggu_2
    
    return [eq_1, eq2]

train network

#train.py
import pandas as pd
import numpy as np
import torch
from glob import glob
import deepxde as dde

def get_train_data():
    ...
    t = np.reshape(t, (-1, 1))
    x = np.reshape(x, (-1, 1))
    y = np.reshape(y, (-1, 1))
    u_x = np.reshape(u_x, (-1, 1))
    u_y = np.reshape(u_y, (-1, 1))
    p = np.reshape(p, (-1, 1))
    
    X = np.hstack((x, y, t))
    return X, u_x, u_y, p

observe_y1 = dde.icbc.PointSetBC(X, u_x, component=0)
observe_y2 = dde.icbc.PointSetBC(X, u_y, component=1)
net = dde.nn.FNN([3] + [20] * 3 + [3], "tanh", "Glorot uniform")

from pde import pde

geom = dde.geometry.Rectangle([0, -0.3], [0.6, 0.3])
timedomain = dde.geometry.TimeDomain(0, 8000)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

data = dde.data.TimePDE(
    geomtime,
    pde,
    [observe_y1, observe_y2],
    num_domain=2000,
    num_test=50000,
)

model = dde.Model(data, net)
model.compile("adam", lr=0.001)

losshistory, train_state = model.train(iterations=80000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)