Subregion–identified physics-informed neural networks for the Navier–Stokes/Darcy interface model

Mulin Wang; Jinyi Luo; Xiangpeng Xin; Changxin Qiu; Mulin Wang; Jinyi Luo; Xiangpeng Xin; Changxin Qiu

doi:10.3934/era.2025223

Electronic Research Archive

2025, Volume 33, Issue 8: 4964-4983. doi: 10.3934/era.2025223

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Subregion–identified physics-informed neural networks for the Navier–Stokes/Darcy interface model

1.
School of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China
2.
School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, China
3.
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China

Received: 19 June 2025 Revised: 01 August 2025 Accepted: 06 August 2025 Published: 25 August 2025

Physics-informed neural networks (PINNs), built upon general neural networks, integrate physical information into the loss function, achieving remarkable results in solving partial differential equations (PDEs). Meanwhile, PINNs face challenges in solving complex interface problems, such as the Navier–Stokes/Darcy interface model considered in this paper. To address these challenges, we extend the general PINNs to propose subregion–identified physics-informed neural networks (SI-PINNs), which consist of three different neural networks for different subregions (Navier–Stokes, Darcy, and interface) with separated datasets. By identifying and classifying the regional features of the neural networks and datasets, the complexity of the physical information that SI-PINNs need to process is effectively split, while the networks also take all physical information into account during the training. Additionally, this splitting feature substantially reduces the training costs while enhancing or maintaining the accuracy of the general PINNs for the target model. Numerical experiments are performed to validate the effectiveness of the SI-PINN method.
- SI-PINNs,
- subregion–identification,
- physics-informed neural networks,
- Navier–Stokes/Darcy interface model,
- Beavers–Joseph interface condition
Citation: Mulin Wang, Jinyi Luo, Xiangpeng Xin, Changxin Qiu. Subregion–identified physics-informed neural networks for the Navier–Stokes/Darcy interface model[J]. Electronic Research Archive, 2025, 33(8): 4964-4983. doi: 10.3934/era.2025223

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Abstract

Physics-informed neural networks (PINNs), built upon general neural networks, integrate physical information into the loss function, achieving remarkable results in solving partial differential equations (PDEs). Meanwhile, PINNs face challenges in solving complex interface problems, such as the Navier–Stokes/Darcy interface model considered in this paper. To address these challenges, we extend the general PINNs to propose subregion–identified physics-informed neural networks (SI-PINNs), which consist of three different neural networks for different subregions (Navier–Stokes, Darcy, and interface) with separated datasets. By identifying and classifying the regional features of the neural networks and datasets, the complexity of the physical information that SI-PINNs need to process is effectively split, while the networks also take all physical information into account during the training. Additionally, this splitting feature substantially reduces the training costs while enhancing or maintaining the accuracy of the general PINNs for the target model. Numerical experiments are performed to validate the effectiveness of the SI-PINN method.

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