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

arXiv 2207.10241 ↗First posted 21 Jul 2022cs.LGcited by 0

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Abstract (arXiv)

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.

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References · 27

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15 more
  1. 2022MIONet: Learning multiple-input operators via tensor productcited by 181
  2. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  3. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 148
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  5. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
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  14. 2021Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networkscited by 5
  15. 2020Deep neural network for solving differential equations motivated by Legendre-Galerkin approximationcited by 3

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