$$
\begin{aligned}
\Delta u + k^2(1+q)u &= f \quad \text{in}\quad \Omega\\
u &= 0\quad \text{on} \quad \partial \Omega\\
\end{aligned}
$$
Given $q,f$ , to find a solution $u$ .
$$
\begin{aligned}
\Omega &= (0,1)^2\\
q(x,y) &= \exp(-200(x-0.3)^2-100(y-0.6)^2)\\
u_{truth} &= \sin(m_1 x)\sin(m_2 y),m_1,m_2 \in \mathbb{N}^+\\
f(x,y) &= (k^2(1+q(x,y)) - m_1^2-m_2^2)\sin(m_1 x)\sin(m_2 y)
\end{aligned}
$$
We input the $q,f$ to the network, aftre some training, we get the output $u_{res}$ .
We test the $L_2$ relative error on the resolution = 256 $\times$ 256
Unsupervised Learning
nn.Linear(2,64)nn.Tanh()nn.Linear(64,64)nn.Tanh()nn.Linear(64,64)nn.Tanh()nn.Linear(64,64)nn.Tanh()nn.Linear(64,1)
Random Sampling
MSE LOSS
$$
\begin{aligned}
l_{\text {int }}(x, y) &=\left(\Delta u_{\text {res }}+k^2(1+q) u_{\text {res }}-f\right)(x, y) \\
l_{\text {left }}(0, y) &=u(0, y) \\
l_{\text {right }}(1, y) &=u(1, y) \\
l_{\text {up }}(x, 1) &=u(x, 1) \\
l_{\text {down }}(x, 0) &=u(x, 0) \\
l_{\text {total }} &=\operatorname{MSE}\left(l_{\text {int }}\right)+\operatorname{MSE}\left(l_{\text {left }}\right)+\operatorname{MSE}\left(l_{\text {right }}\right)+\operatorname{MSE}\left(l_{\text {up }}\right)+\operatorname{MSE}\left(l_{\text {down }}\right)
\end{aligned}
$$
Adam , learning rate = 1e-4
- Modifying the following arguments in
test.sh:
-
maxiter : total epochs of training
-
N : number of points sampled in $\Omega$
-
n : number of points sampled on each boundary of $\partial \Omega$
-
grids : resolution of testing
-
k : frequency of the equation
-
m : frequency of $u_{truth}$
-
gpu : 'yes' or 'no' to compute with gpu or not
bash test.shtail -f .tmp.log to get the code progress- check
heatmap/ for the training results and relative/ for the process of training
filename: 'maxiter_cpu/gpu_N_n_k_m.jpg'
N = 40000 n = 1000 k = 2 m = (3,4) with GPU after 10000 epochs
