Unsupervised Legendre-Galerkin Neural Network for Singularly Perturbed Partial Differential Equations

TL;DR

This paper introduces an unsupervised neural network method based on Legendre-Galerkin techniques to solve various PDEs, including challenging singularly perturbed problems with boundary layers, advancing scientific machine learning.

Contribution

It develops a novel unsupervised neural network algorithm within the Legendre-Galerkin framework for solving multiple PDE instances, including singularly perturbed equations.

Findings

Successfully applied to 1D and 2D PDEs with various boundary conditions.

Effectively handles convection-dominated singularly perturbed PDEs.

Overcomes limitations of existing data-driven and physics-based methods.

Abstract

Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques such as deep neural networks and statistical learning are applied to classical problems of applied mathematics. In this paper, we develop a novel numerical algorithm that incorporates machine learning and artificial intelligence to solve PDEs. Based on the Legendre-Galerkin framework, we propose the {\it unsupervised machine learning} algorithm to learn {\it multiple instances} of the solutions for different types of PDEs. Our approach overcomes the limitations of data-driven and physics-based methods. The proposed neural network is applied to general 1D and 2D PDEs with various boundary conditions as well as convection-dominated {\it singularly perturbed PDEs} that exhibit strong boundary layer behavior.