论文解读:DeLISA: Deep learning based iteration scheme approximation for solving PDEs

2. Methodology

2.1. 问题设置

In this work, we consider a family of PDEs in a bounded domain $\Omega \in {\mathbb{R}}^d$:

$u_t(t,x)+\mathcal{L}u(t,x)=0,(t,x)\in [0,T]\times \Omega$

where $u(t,x)$ is the latent solution to be solved, $\mathcal{L}$ is differential operator, and $u_t$ denotes first-order derivative of $u$ with respect to $t$. Initial and boundary conditions are given by practical problem.

2.2. DeLISA 架构

We consider one-step implicit multistep method [36], namely trapezoidal rule, on one dimension:

$u^{n+1}=u^n-\frac{1}{2}\Delta t\left(\mathcal{L}{u}^{n+1}+\mathcal{L}{u}^n\right),n=0,1,2,3,....,N-1$$(2)$

where $u^n$, that is $u(t_n,x)$, corresponds to solution $u$ at time $n\Delta t$, $\Delta t$ is time step. Then the approximation $f(t,x;\theta )$ of $u$ is applied in Eq. (2) to construct a new iteration:

[36] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2009