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A Tailored Physics-informed Neural Network Method for Solving Singularly Perturbed Differential Equations

Published: 14 March 2023 Publication History

Abstract

Physics-informed neural networks (PINNs) have recently been demonstrated to be effective for the numerical solution of differential equations, with the advantage of small real labelled data needed. However, the performance of PINN greatly depends on the differential equation. The solution of singularly perturbed differential equations (SPDEs) usually contains a boundary layer, which makes it difficult for PINN to approximate the solution of SPDEs. In this paper, we analyse the reasons for the failure of PINN in solving SPDE and provide a feasible solution by adding prior knowledge of the boundary layer to the neural network. The new method is called the tailored physics-informed neural network (TPINN) since the network is tailored to some particular properties of the problem. Numerical experiments show that our method can effectively improve both the training speed and accuracy of neural networks.

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Cited By

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  • (2024)Multistep asymptotic pre-training strategy based on PINNs for solving steep boundary singular perturbation problemsComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2024.117222431(117222)Online publication date: Nov-2024

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  1. A Tailored Physics-informed Neural Network Method for Solving Singularly Perturbed Differential Equations

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    cover image ACM Other conferences
    ACAI '22: Proceedings of the 2022 5th International Conference on Algorithms, Computing and Artificial Intelligence
    December 2022
    770 pages
    ISBN:9781450398336
    DOI:10.1145/3579654
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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    Publication History

    Published: 14 March 2023

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    Author Tags

    1. boundary layer
    2. physics-informed neural network
    3. prior knowledge
    4. singularly perturbed differential equation

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    View all
    • (2024)Multistep asymptotic pre-training strategy based on PINNs for solving steep boundary singular perturbation problemsComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2024.117222431(117222)Online publication date: Nov-2024

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