Unsupervised Legendre-Galerkin Neural Network for Solving Partial Differential Equationsopen access
- Authors
- Choi, Junho; Kim, Namjung; Hong, Youngjoon
- Issue Date
- Feb-2023
- Publisher
- IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
- Keywords
- Unsupervised learning; deep neural network; Legendre-Galerkin approximation; spectral bias; boundary layer; singular perturbation
- Citation
- IEEE ACCESS, v.11, pp.23433 - 23446
- Journal Title
- IEEE ACCESS
- Volume
- 11
- Start Page
- 23433
- End Page
- 23446
- URI
- https://scholarworks.bwise.kr/gachon/handle/2020.sw.gachon/87657
- DOI
- 10.1109/ACCESS.2023.3244681
- ISSN
- 2169-3536
- Abstract
- In recent years, machine learning methods have been used to solve partial differential equations (PDEs) and dynamical systems, leading to the development of a new research field called scientific machine learning, which combines techniques such as deep neural networks and statistical learning with classical problems in applied mathematics. In this paper, we present a novel numerical algorithm that uses machine learning and artificial intelligence to solve PDEs. Based on the Legendre-Galerkin framework, we propose an unsupervised machine learning algorithm that learns multiple instances of the solutions for different types of PDEs. Our approach addresses the limitations of both data-driven and physics-based methods. We apply the proposed neural network to general 1D and 2D PDEs with various boundary conditions, as well as convection-dominated singularly perturbed PDEs that exhibit strong boundary layer behavior.
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