Residual Splitting Adaptive Physics-Informed Neural Networks for Solving Partial Differential Equations

FAN Kunkun; ZHANG Haoran; YUE Yucheng; YUAN Dongfang

doi:10.21656/1000-0887.460018

Volume 47 Issue 5

May 2026

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Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(5): 655-667

FAN Kunkun, ZHANG Haoran, YUE Yucheng, YUAN Dongfang. Residual Splitting Adaptive Physics-Informed Neural Networks for Solving Partial Differential Equations[J]. Applied Mathematics and Mechanics, 2026, 47(5): 655-667. doi: 10.21656/1000-0887.460018

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Residual Splitting Adaptive Physics-Informed Neural Networks for Solving Partial Differential Equations

doi: 10.21656/1000-0887.460018

1. School of Automation and Electrical Engineering, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia 014010, P.R. China;
2. School of science, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia 014010, P.R. China

Received Date: 2025-01-24
Rev Recd Date: 2025-03-24

Available Online: 2026-06-04

Publish Date: 2026-05-01

Abstract

Abstract

The magnitude difference between the loss functions of physical-informed neural networks (PINN) leads to a slow convergence of the training process and sometimes even training failure in some regions. To address this challenge, a physical-informed neural network model incorporating residual splitting and weight self-adaptation was proposed. The method improves the convergence of PINN by splitting the residual terms of PDE dominating the training process of PINN, into multiple independent components according to the domain decomposition, and adopts a self-adaptive weighting strategy to automatically adjust the weights among the components, thus promoting the convergence of PINN. This method makes up for the defects of the global residual strategy ignoring and smoothing out the local features, and increases the attention to the local features by splitting the subterms, which improves the efficiency of the optimization process, and thus enhances the solution accuracy. Through numerical experiments, the results show that, the proposed method not only surpasses the existing models in terms of accuracy, but also achieves an improvement of 2-3 orders of magnitude with superior computational efficiency.
- physics-informed neural network,
- adaptive weighting,
- residual splitting,
- domain decomposition

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References

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