Module

$$ \begin{aligned} \Delta u + k^2(1+q)u &= f \quad \text{in}\quad \Omega\\ u &= 0\quad \text{on} \quad \partial \Omega\\ \end{aligned} $$

Given $q,f$ , to find a solution $u$ .

---

Setting

$$ \begin{aligned} \Omega &= (0,1)^2\\ q(x,y) &= \exp(-200(x-0.3)^2-100(y-0.6)^2)\\ u_{truth} &= \sin(m_1 x)\sin(m_2 y),m_1,m_2 \in \mathbb{N}^+\\ f(x,y) &= (k^2(1+q(x,y)) - m_1^2-m_2^2)\sin(m_1 x)\sin(m_2 y) \end{aligned} $$

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

---

Network

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)

Sampling

Random Sampling

Loss function

MSE LOSS

$$ \begin{aligned} l_{\text {int }}(x, y) &=\left(\Delta u_{\text {res }}+k^2(1+q) u_{\text {res }}-f\right)(x, y) \\ l_{\text {left }}(0, y) &=u(0, y) \\ l_{\text {right }}(1, y) &=u(1, y) \\ l_{\text {up }}(x, 1) &=u(x, 1) \\ l_{\text {down }}(x, 0) &=u(x, 0) \\ l_{\text {total }} &=\operatorname{MSE}\left(l_{\text {int }}\right)+\operatorname{MSE}\left(l_{\text {left }}\right)+\operatorname{MSE}\left(l_{\text {right }}\right)+\operatorname{MSE}\left(l_{\text {up }}\right)+\operatorname{MSE}\left(l_{\text {down }}\right) \end{aligned} $$

Optimizer

Adam , learning rate = 1e-4

---

Getting Started

1. 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

2. bash test.sh
3. tail -f .tmp.log to get the code progress
4. check heatmap/ for the training results and relative/ for the process of training
   filename: 'maxiter_cpu/gpu_N_n_k_m.jpg'

Examples of Results

N = 40000 n = 1000 k = 2 m = (3,4) with GPU after 10000 epochs