Bridging Traditional and Machine Learning-Based Algorithms for Solving PDEs: The Random Feature Method
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.
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 Email
Xurong Chi Email
Weinan E Email
Zhouwang Yang Email
-
Applied AI Techniques in the Process Industry
Reverse Design of Heat Exchange Systems Using Physics‐Informed Machine Learning
He, Chang | Chen, Yunquan2025
https://doi.org/10.1002/9783527845491.ch6 [Citations: 0] -
Solving a class of multi-scale elliptic PDEs by Fourier-based mixed physics informed neural networks
Li, Xi'an | Wu, Jinran | Tai, Xin | Xu, Jianhua | Wang, You-GanJournal of Computational Physics, Vol. 508 (2024), Iss. P.113012
https://doi.org/10.1016/j.jcp.2024.113012 [Citations: 6] -
Randomized radial basis function neural network for solving multiscale elliptic equations
Wu, Yuhang | Liu, Ziyuan | Sun, Wenjun | Qian, XuMachine Learning: Science and Technology, Vol. 6 (2025), Iss. 1 P.015033
https://doi.org/10.1088/2632-2153/ad979c [Citations: 0] -
Local randomized neural networks with finite difference methods for interface problems
Li, Yunlong | Wang, FeiJournal of Computational Physics, Vol. 529 (2025), Iss. P.113847
https://doi.org/10.1016/j.jcp.2025.113847 [Citations: 0] -
Deciphering and integrating invariants for neural operator learning with various physical mechanisms
Zhang, Rui | Meng, Qi | Ma, Zhi-MingNational Science Review, Vol. 11 (2024), Iss. 4
https://doi.org/10.1093/nsr/nwad336 [Citations: 3] -
Transferable Neural Networks for Partial Differential Equations
Zhang, Zezhong | Bao, Feng | Ju, Lili | Zhang, GuannanJournal of Scientific Computing, Vol. 99 (2024), Iss. 1
https://doi.org/10.1007/s10915-024-02463-y [Citations: 4] -
The Application of Physics-Informed Machine Learning in Multiphysics Modeling in Chemical Engineering
Wu, Zhiyong | Wang, Huan | He, Chang | Zhang, Bingjian | Xu, Tao | Chen, QinglinIndustrial & Engineering Chemistry Research, Vol. 62 (2023), Iss. 44 P.18178
https://doi.org/10.1021/acs.iecr.3c02383 [Citations: 21] -
A multiple transferable neural network method with domain decomposition for elliptic interface problems
Lu, Tianzheng | Ju, Lili | Zhu, LiyongJournal of Computational Physics, Vol. 530 (2025), Iss. P.113902
https://doi.org/10.1016/j.jcp.2025.113902 [Citations: 0] -
The neural network basis method for nonlinear partial differential equations and its Gauss–Newton optimizer
Huang, Jianguo | Wu, HaohaoCommunications in Nonlinear Science and Numerical Simulation, Vol. 143 (2025), Iss. P.108608
https://doi.org/10.1016/j.cnsns.2025.108608 [Citations: 0] -
Darboux transformation-based LPNN generating novel localized wave solutions
Pu, Juncai | Chen, YongPhysica D: Nonlinear Phenomena, Vol. 467 (2024), Iss. P.134262
https://doi.org/10.1016/j.physd.2024.134262 [Citations: 5] -
Operator Learning Using Random Features: A Tool for Scientific Computing
Nelsen, Nicholas H. | Stuart, Andrew M.SIAM Review, Vol. 66 (2024), Iss. 3 P.535
https://doi.org/10.1137/24M1648703 [Citations: 3] -
Adaptive trajectories sampling for solving PDEs with deep learning methods
Chen, Xingyu | Cen, Jianhuan | Zou, QingsongApplied Mathematics and Computation, Vol. 481 (2024), Iss. P.128928
https://doi.org/10.1016/j.amc.2024.128928 [Citations: 0] -
Runge–Kutta random feature method for solving multiphase flow problems of cells
Deng, Yangtao | He, QiaolinPhysics of Fluids, Vol. 37 (2025), Iss. 2
https://doi.org/10.1063/5.0252273 [Citations: 0] -
Optimization of Random Feature Method in the High-Precision Regime
Chen, Jingrun | E, Weinan | Sun, YifeiCommunications on Applied Mathematics and Computation, Vol. 6 (2024), Iss. 2 P.1490
https://doi.org/10.1007/s42967-024-00389-8 [Citations: 0] -
The random feature method for solving interface problems
Chi, Xurong | Chen, Jingrun | Yang, ZhouwangComputer Methods in Applied Mechanics and Engineering, Vol. 420 (2024), Iss. P.116719
https://doi.org/10.1016/j.cma.2023.116719 [Citations: 1] -
The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach
Song, Yongcun | Yuan, Xiaoming | Yue, HangruiSIAM Journal on Scientific Computing, Vol. 46 (2024), Iss. 6 P.C659
https://doi.org/10.1137/23M1566935 [Citations: 0] -
Lax pairs informed neural networks solving integrable systems
Pu, Juncai | Chen, YongJournal of Computational Physics, Vol. 510 (2024), Iss. P.113090
https://doi.org/10.1016/j.jcp.2024.113090 [Citations: 10] -
A Hybrid Iterative Method for Elliptic Variational Inequalities of the Second Kind
Cao, Yujian | Huang, Jianguo | Wang, HaoqinCommunications on Applied Mathematics and Computation, Vol. (2024), Iss.
https://doi.org/10.1007/s42967-024-00423-9 [Citations: 0] -
Accelerating heat exchanger design by combining physics-informed deep learning and transfer learning
Wu, Zhiyong | Zhang, Bingjian | Yu, Haoshui | Ren, Jingzheng | Pan, Ming | He, Chang | Chen, QinglinChemical Engineering Science, Vol. 282 (2023), Iss. P.119285
https://doi.org/10.1016/j.ces.2023.119285 [Citations: 15] -
Digital Twins - a golden age for industrial mathematics
Hartmann, Dirk | Van der Auweraer, HermanJournal of Mathematics in Industry, Vol. 15 (2025), Iss. 1
https://doi.org/10.1186/s13362-025-00170-3 [Citations: 0] -
Deep Learning without Global Optimization by Random Fourier Neural Networks
Davis, Owen | Geraci, Gianluca | Motamed, MohammadSIAM Journal on Scientific Computing, Vol. 47 (2025), Iss. 2 P.C265
https://doi.org/10.1137/24M1661777 [Citations: 0]