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Bridging Traditional and Machine Learning-Based Algorithms for Solving PDEs: The Random Feature Method

Bridging Traditional and Machine Learning-Based Algorithms for Solving PDEs: The Random Feature Method
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Year:    2022

Author:    Jingrun Chen, Xurong Chi, Weinan E, Zhouwang Yang

Journal of Machine Learning, Vol. 1 (2022), Iss. 3 : pp. 268–298

Abstract

One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A long list of numerical methods have been proposed and successfully used for various applications. In recent years, deep learning methods have shown their superiority for high-dimensional PDEs where traditional methods fail. However, for low dimensional problems, it remains unclear whether these methods have a real advantage over traditional algorithms as a direct solver. In this work, we propose the random feature method (RFM) for solving PDEs, a natural bridge between traditional and machine learning-based algorithms. RFM is based on a combination of well-known ideas: 1. representation of the approximate solution using random feature functions; 2. collocation method to take care of the PDE; 3. penalty method to treat the boundary conditions, which allows us to treat the boundary condition and the PDE in the same footing. We find it crucial to add several additional components including multi-scale representation and adaptive weight rescaling in the loss function. We demonstrate that the method exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and efficiency. In addition, we find that RFM is particularly suited for problems with complex geometry, where both traditional and machine learning-based algorithms encounter difficulties.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jml.220726

Journal of Machine Learning, Vol. 1 (2022), Iss. 3 : pp. 268–298

Published online:    2022-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    31

Keywords:    Partial differential equation Machine learning Random feature method Rescaling.

Author Details

Jingrun Chen

Xurong Chi

Weinan E

Zhouwang Yang

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