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Volume 38, Issue 3
Two-Scale Neural Networks for Partial Differential Equations with Small Parameters

Qiao Zhuang, Chris Ziyi Yao, Zhongqiang Zhang & George Em Karniadakis

DOI: 10.4208/cicp.OA-2024-0040

Commun. Comput. Phys., 38 (2025), pp. 603-629.

Published online: 2025-08

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  • Abstract

We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.

  • Keywords

Two-scale neural networks, partial differential equations, small parameters, successive training.

  • AMS Subject Headings

65N35, 35B25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-38-603, author = {Zhuang , QiaoYao , Chris ZiyiZhang , Zhongqiang and Karniadakis , George Em}, title = {Two-Scale Neural Networks for Partial Differential Equations with Small Parameters}, journal = {Communications in Computational Physics}, year = {2025}, volume = {38}, number = {3}, pages = {603--629}, abstract = {

We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0040}, url = {http://global-sci.org/intro/article_detail/cicp/24309.html} }
TY - JOUR T1 - Two-Scale Neural Networks for Partial Differential Equations with Small Parameters AU - Zhuang , Qiao AU - Yao , Chris Ziyi AU - Zhang , Zhongqiang AU - Karniadakis , George Em JO - Communications in Computational Physics VL - 3 SP - 603 EP - 629 PY - 2025 DA - 2025/08 SN - 38 DO - http://doi.org/10.4208/cicp.OA-2024-0040 UR - https://global-sci.org/intro/article_detail/cicp/24309.html KW - Two-scale neural networks, partial differential equations, small parameters, successive training. AB -

We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.

Zhuang , QiaoYao , Chris ZiyiZhang , Zhongqiang and Karniadakis , George Em. (2025). Two-Scale Neural Networks for Partial Differential Equations with Small Parameters. Communications in Computational Physics. 38 (3). 603-629. doi:10.4208/cicp.OA-2024-0040
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