Unsupervised Legendre-Galerkin Neural Network for Singularly Perturbed Partial Differential Equations
This work addresses the challenge of solving convection-dominated PDEs with boundary layers for applied mathematics and scientific computing, presenting a novel hybrid method.
The authors tackled the problem of solving singularly perturbed PDEs with strong boundary layers by developing an unsupervised Legendre-Galerkin neural network, achieving solutions for multiple instances across 1D and 2D cases with various boundary conditions.
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.