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Volume 37, Issue 4
Convergence Analysis of PINNs with Over-Parameterization

Mo Chen, Zhao Ding, Yuling Jiao, Xiliang Lu, Peiying Wu & Jerry Zhijian Yang

DOI: 10.4208/cicp.OA-2023-0101

Commun. Comput. Phys., 37 (2025), pp. 942-974.

Published online: 2025-04

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  • Abstract

Recently, physics-informed neural networks (PINNs) have been shown to be a simple and efficient method for solving PDEs empirically. However, the numerical analysis of PINNs is still incomplete, especially why over-parameterized PINNs work remains unknown. This paper presents the first convergence analysis of the over-parameterized PINNs for the Laplace equations with Dirichlet boundary conditions. We demonstrate that the convergence rate can be controlled by the weight norm, regardless of the number of parameters in the network.

  • Keywords

PINNs, over-parameterization, convergence rate, deep approximation with norm control.

  • AMS Subject Headings

65M15, 65N15, 65Y20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-37-942, author = {Chen , MoDing , ZhaoJiao , YulingLu , XiliangWu , Peiying and Yang , Jerry Zhijian}, title = {Convergence Analysis of PINNs with Over-Parameterization}, journal = {Communications in Computational Physics}, year = {2025}, volume = {37}, number = {4}, pages = {942--974}, abstract = {

Recently, physics-informed neural networks (PINNs) have been shown to be a simple and efficient method for solving PDEs empirically. However, the numerical analysis of PINNs is still incomplete, especially why over-parameterized PINNs work remains unknown. This paper presents the first convergence analysis of the over-parameterized PINNs for the Laplace equations with Dirichlet boundary conditions. We demonstrate that the convergence rate can be controlled by the weight norm, regardless of the number of parameters in the network.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0101}, url = {http://global-sci.org/intro/article_detail/cicp/24028.html} }
TY - JOUR T1 - Convergence Analysis of PINNs with Over-Parameterization AU - Chen , Mo AU - Ding , Zhao AU - Jiao , Yuling AU - Lu , Xiliang AU - Wu , Peiying AU - Yang , Jerry Zhijian JO - Communications in Computational Physics VL - 4 SP - 942 EP - 974 PY - 2025 DA - 2025/04 SN - 37 DO - http://doi.org/10.4208/cicp.OA-2023-0101 UR - https://global-sci.org/intro/article_detail/cicp/24028.html KW - PINNs, over-parameterization, convergence rate, deep approximation with norm control. AB -

Recently, physics-informed neural networks (PINNs) have been shown to be a simple and efficient method for solving PDEs empirically. However, the numerical analysis of PINNs is still incomplete, especially why over-parameterized PINNs work remains unknown. This paper presents the first convergence analysis of the over-parameterized PINNs for the Laplace equations with Dirichlet boundary conditions. We demonstrate that the convergence rate can be controlled by the weight norm, regardless of the number of parameters in the network.

Chen , MoDing , ZhaoJiao , YulingLu , XiliangWu , Peiying and Yang , Jerry Zhijian. (2025). Convergence Analysis of PINNs with Over-Parameterization. Communications in Computational Physics. 37 (4). 942-974. doi:10.4208/cicp.OA-2023-0101
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