The improved PINNs for solving partial differential equations with nonlinear couple systems

Jie Xu; Jinhua Ran; Jiao She; Jie Xu; Jinhua Ran; Jiao She

doi:10.3934/era.2026007

Electronic Research Archive

2026, Volume 34, Issue 1: 113-159. doi: 10.3934/era.2026007

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Research article

The improved PINNs for solving partial differential equations with nonlinear couple systems

1.
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
2.
Ningxia Key Laboratory of Intelligent Information and Big Data Processing, Yinchuan 750021, China

Received: 10 October 2025 Revised: 17 December 2025 Accepted: 22 December 2025 Published: 30 December 2025

For partial differential equations, traditional methods are characterized by high accuracy, yet they face difficulties in handling complex high-dimensional. Although physics-informed neural networks (PINNs) can effectively solve high-dimensional problems, their accuracy still needs to be improved. The Fourier neural operator (FNO) has an advantage in capturing the global characteristics of physical systems, but it relies on a large amount of training data. To address these limitations, this paper proposes a hybrid method, FNO-PINN, which integrates the FNO with a PINN. This method devises a hybrid architecture consisting of an enhanced FNO, high precision difference calculation, multi-objective physical constraints, and post processing mechanisms. Meanwhile, a dynamic weight adjustment strategy and an iterative auxiliary data loss for different time-scale variables are proposed to effectively alleviate the imbalance problem during the training process. The numerical experimental results of three examples show that compared with the traditional PINNs method, the proposed FNO-PINN method has a faster convergence speed and smaller error, and it demonstrates superior performance in handling coupled systems with multiple time scales.
- finite difference method,
- loss function,
- Fourier neural operator,
- physics-informed neural networks
Citation: Jie Xu, Jinhua Ran, Jiao She. The improved PINNs for solving partial differential equations with nonlinear couple systems[J]. Electronic Research Archive, 2026, 34(1): 113-159. doi: 10.3934/era.2026007

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Abstract

For partial differential equations, traditional methods are characterized by high accuracy, yet they face difficulties in handling complex high-dimensional. Although physics-informed neural networks (PINNs) can effectively solve high-dimensional problems, their accuracy still needs to be improved. The Fourier neural operator (FNO) has an advantage in capturing the global characteristics of physical systems, but it relies on a large amount of training data. To address these limitations, this paper proposes a hybrid method, FNO-PINN, which integrates the FNO with a PINN. This method devises a hybrid architecture consisting of an enhanced FNO, high precision difference calculation, multi-objective physical constraints, and post processing mechanisms. Meanwhile, a dynamic weight adjustment strategy and an iterative auxiliary data loss for different time-scale variables are proposed to effectively alleviate the imbalance problem during the training process. The numerical experimental results of three examples show that compared with the traditional PINNs method, the proposed FNO-PINN method has a faster convergence speed and smaller error, and it demonstrates superior performance in handling coupled systems with multiple time scales.

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