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
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

[1]	ZHANG S, GU W, ZHANG X P, et al. Dynamic modeling and simulation of integrated electricity and gas systems[J]. IEEE Transactions on Smart Grid, 2023, 14 (2): 1011-1026. doi: 10.1109/TSG.2022.3203485
[2]	AVALOS G, LASIECKA I, TRIGGIANI R. Heat-wave interaction in 2/3 dimensions: optimal rational decay rate[J]. Journal of Mathematical Analysis and Applications, 2016, 437 (2): 782-815. doi: 10.1016/j.jmaa.2015.12.051
[3]	ARQUB O A. Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm[J]. International Journal of Numerical Methods for Heat & Fluid Flow, 2018, 28 (4): 828-856.
[4]	ZHANG Y. A finite difference method for fractional partial differential equation[J]. Applied Mathematics and Computation, 2009, 215 (2): 524-529. doi: 10.1016/j.amc.2009.05.018
[5]	QIN X Q, MA Y C, ZHANG Y. Two-grid method for characteristics finite-element solution of 2D nonlinear convection-dominated diffusion problem[J]. Applied Mathematics and Mechanics, 2005, 26 (11): 1506-1514. doi: 10.1007/BF03246258
[6]	EYMARD R, GALLOUËT T, HERBIN R. Handbook of Numerical Analysis: Finite Volume Methods[M]. Elsevier, 2000, 7 : 713-1018.
[7]	NOCHETTO R H, SIEBERT K G, VEESER A. Theory of adaptive finite element methods: an introduction[C]//Multiscale, Nonlinear and Adaptive Approximation. Berlin, Heidelberg: Springer, 2009: 409-542.
[8]	RASP S, PRITCHARD M S, GENTINE P. Deep learning to represent subgrid processes in climate models[J]. Proceedings of the National Academy of Sciences, 2018, 115 (39): 9684-9689. doi: 10.1073/pnas.1810286115
[9]	LENG K, NISSEN-MEYER T, VAN DRIEL M, et al. AxiSEM3D: broad-band seismic wavefields in 3-D global earth models with undulating discontinuities[J]. Geophysical Journal International, 2019, 217 (3): 2125-2146. doi: 10.1093/gji/ggz092
[10]	RAISSI M, PERDIKARIS P, KARNIADAKIS G E. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational Physics, 2019, 378 : 686-707. doi: 10.1016/j.jcp.2018.10.045
[11]	LU L, MENG X, MAO Z, et al. DeepXDE: a deep learning library for solving differential equations[J]. SIAM Review, 2021, 63 (1): 208-228. doi: 10.1137/19M1274067
[12]	SUN L, GAO H, PAN S, et al. Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 361 : 112732. doi: 10.1016/j.cma.2019.112732
[13]	ARZANI A, WANG J X, D'SOUZA R M. Uncovering near-wall blood flow from sparse data with physics-informed neural networks[J]. Physics of Fluids, 2021, 33 (7): 071905. doi: 10.1063/5.0055600
[14]	WANG Y, BAI J, LIN Z, et al. Artificial intelligence for partial differential equations in computational mechanics: a review[PP/OL]. arXiv(2024-11-23)[2025-03-24]. https://arxiv.org/abs/2410.19843v2.
[15]	WU H, LUO H, MA Y, et al. RoPINN: region optimized physics-informed neural networks[PP/OL]. arXiv(2024-10-23)[2025-03-24]. https://arxiv.org/abs/2405.14369v3.
[16]	JAGTAP A D, KARNIADAKIS G E. Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations[J]. Communications in Computational Physics, 2020, 28 (5): 2002-2041. doi: 10.4208/cicp.OA-2020-0164
[17]	WIGHT C L, ZHAO J. Solving Allen-cahn and cahn-Hilliard equations using the adaptive physics informed neural networks[J]. Communications in Computational Physics, 2021, 29 (3).
[18]	WANG Y, YAO Y, GUO J, et al. A practical PINN framework for multi-scale problems with multi-magnitude loss terms[J]. Journal of Computational Physics, 2024, 510 : 113112. doi: 10.1016/j.jcp.2024.113112
[19]	WANG S, YU X, PERDIKARIS P. When and why PINNs fail to train: a neural tangent kernel perspective[J]. Journal of Computational Physics, 2022, 449 : 110768. doi: 10.1016/j.jcp.2021.110768
[20]	ELHAMOD M, BU J, SINGH C, et al. CoPhy-PGNN: learning physics-guided neural networks with competing loss functions for solving eigenvalue problems[J]. ACM Transactions on Intelligent Systems and Technology, 2022, 13 (6): 1-23.
[21]	WANG S, TENG Y, PERDIKARIS P. Understanding and mitigating gradient flow pathologies in physics-informed neural networks[J]. SIAM Journal on Scientific Computing, 2021, 43 (5): A3055-A3081. doi: 10.1137/20M1318043
[22]	SONG Y, WANG H, YANG H, et al. Loss-attentional physics-informed neural networks[J]. Journal of Computational Physics, 2024, 501 : 112781. doi: 10.1016/j.jcp.2024.112781
[23]	QIU L, WANG Y, GU Y, et al. Adaptive physics-informed neural networks for dynamic coupled thermo-mechanical problems in large-size-ratio functionally graded materials[J]. Applied Mathematical Modelling, 2025, 140 : 115906. doi: 10.1016/j.apm.2024.115906
[24]	YANG J, LIU X, DIAO Y, et al. Adaptive task decomposition physics-informed neural networks[J]. Computer Methods in Applied Mechanics and Engineering, 2024, 418 : 116561.
[25]	XIANG Z, PENG W, LIU X, et al. Self-adaptive loss balanced physics-informed neural networks[J]. Neurocomputing, 2022, 496 : 11-34.
[26]	MCCLENNY L, BRAGA-NETO U. Self-adaptive physics-informed neural networks using a soft attention mechanism[PP/OL]. arXiv(2024-06-18)[2025-03-24]. https://arxiv.org/abs/2009.04544v5.
[27]	KINGMA D P, BA J. Adam: a method for stochastic optimization[PP/OL]. arXiv(2017-01-30)[2025-03-24]. https://arxiv.org/abs/1412.6980v9.
[28]	BYRD R H, LU P, NOCEDAL J, et al. A limited memory algorithm for bound constrained optimization[J]. SIAM Journal on Scientific Computing, 1995, 16 (5): 1190-1208.
[29]	WANG Y, LAI C Y. Multi-stage neural networks: function approximator of machine precision[J]. Journal of Computational Physics, 2024, 504 : 112865.
[30]	闵建, 傅卓佳, 郭远. 课程-迁移学习物理信息神经网络用于曲面长时间对流扩散行为模拟[J]. 应用数学和力学, 2024, 45 (9): 1212-1223. doi: 10.21656/1000-0887.440320 MIN Jian, FU Zhuojia, GUO Yuan. Curriculum-transfer-learning-based physics-informed neural networks for simulating long-term-evolution convection-diffusion behaviors on curved surfaces[J]. Applied Mathematics and Mechanics, 2024, 45 (9): 1212-1223. (in Chinese) doi: 10.21656/1000-0887.440320