基本信息

题目：Deep learning-based method coupled with small sample learning for solving partial differential equations

作者：Ying Li1 · Fangjun Mei1 Received:

期刊：Multimedia Tools and Applications

发表日期：Received: 3 February 2020 / Revised: 24 April 2020 / Accepted: 27 May 2020

内容

摘要

我们develop了一种基于深度学习的通用数值方法，并结合小样本学习(SSL)来求解偏微分方程。更具体地说，通过一个深度前馈神经网络来逼近解，该神经网络经过训练来满足具有初值和有界条件的偏微分方程。然后将所提出的方法建模，通过最小化设计的成本函数来解决一个优化问题，其中包括微分方程的残差、初始/边界条件和少量观测值的残差。将develop的方法应用于classical Burgers equations, Schr¨odinger equations, Buckley-Leverett equation, Navier-Stokes equation, and Carburiz- ing diffusion equations，证明了该方法是effective, flexible and robust without relying on trial solutions

Introduction

动机：

It is widely known that PDEs are existing widely in the field of mathematics, physics and engineering. However, most of the PDEs encountered in practical applications cannot be solved analytically. Thus, various approximation methods is needed. Traditional methods, such as finite element method (FEM) [34], finite difference method (FDM) [39], or finite volume method (FVM)
Traditional methods, such as finite element method (FEM) [34], finite difference method (FDM) [39], or finite volume method (FVM).These methods are almost mature in solving PDEs, but there are some important limitations. For instance, mesh-based methods are easy to cause curse of dimensionality when the dimension grows. Also, the solution is only computed at the mesh points. Moreover, the traditional numerical methods are usually iterative in nature, where we fix the step size before the start of the computation. After the solution is obtained, if we want to know the solution in between steps, then again the procedure is to be repeated from initial stage.
In recent decades, deep learning has achieved remarkable success in solving PDE. Raissi developed data-driven algorithms for general linear equations using Gaussian process priors¹ ² ³ ⁴. Although Gaussian process regression has a major advantage in prediction prob- lems, the high nonlinearity of the problem limits its application. The Gaussian process is developed on the basis of Bayesian learning theory, and the characteristics of Bayesian prior information may affect the expressive ability of the model. Deep learning is a data-hungry learning manner. However, the cost of data acquisition is prohibitive, especially in some expert domains. Thus, a new learning paradigm, called Small Sample Learning (SSL) [8], has been popular in the recent years.

Algorithm

One-dimensional Schr ¨odinger equation

{iψt+0.5ψxx+∣ψ∣2ψ=0,x∈[−5,5],t∈[0,π2]ψ(0,x)=2sech⁡(x)ψ(t,−5)=ψ(t,5)ψx(t,−5)=ψx(t,5)\left\{\begin{array}{l} i \psi_{t}+0.5 \psi_{x x}+|\psi|^{2} \psi=0, x \in[-5,5], t \in\left[0, \frac{\pi}{2}\right] \\ \psi(0, x)=2 \operatorname{sech}(x) \\ \psi(t,-5)=\psi(t, 5) \\ \psi_{x}(t,-5)=\psi_{x}(t, 5) \end{array}\right.⎩⎪⎪⎨⎪⎪⎧iψt+0.5ψxx+∣ψ∣2ψ=0,x∈[−5,5],t∈[0,2π]ψ(0,x)=2sech(x)ψ(t,−5)=ψ(t,5)ψx(t,−5)=ψx(t,5)
ψ(t,x)\psi(t, x)ψ(t,x)is a complex-valued solution，ψ(t,x)=[u(t,x),v(t,x)]\psi(t, x)=[u(t, x), v(t, x)]ψ(t,x)=[u(t,x),v(t,x)]
{ut=−0.5vxx−(u2+v2)vvt=0.5uxx+(u2+v2)u\left\{\begin{array}{l}u_{t}=-0.5 v_{x x}-\left(u^{2}+v^{2}\right) v \\ v_{t}=0.5 u_{x x}+\left(u^{2}+v^{2}\right) u\end{array}\right.{ut=−0.5vxx−(u2+v2)vvt=0.5uxx+(u2+v2)u

Reference

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Deep learning-based method coupled with small sample learning for solving partial differential equat