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

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

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.

Keywords

finite difference method / loss function / Fourier neural operator / physics-informed neural networks

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] J. De Luca, Chemical principle and PDE of variational electrodynamics, J. Differ. Equations, 268 (2019), 272-300. https://doi.org/10.1016/j.jde.2019.08.020

[2] J. Lv, J. Li, The delayed reaction-diffusion Filippov system with threshold control: Modeling and analysis of plateau pika-vegetation dynamics, Commun. Nonlinear Sci. Numer. Simul., 152 (2026), 109366. https://doi.org/10.1016/j.cnsns.2025.109366

[3] D. A. Nield, A. V. Kuznetsov, The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer, 52 (2009), 5792-5795. https://doi.org/10.1016/j.ijheatmasstransfer.2009.07.024

[4] P. Suchde, J. Kuhnert, A meshfree generalized finite difference method for surface PDEs, Comput. Math. Appl., 78 (2019), 2789-2805. https://doi.org/10.1016/j.camwa.2019.04.030

[5] M. Li, C. S. Chen, Y. C. Hon, P. H. Wen, Finite integration method for solving multi-dimensional partial differential equations, Appl. Math. Modell., 39 (2015), 4979-4994. https://doi.org/10.1016/j.apm.2015.03.049

[6] Y. Liu, Y. Cheng, S. Chen, Y. Zhang, Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations, J. Comput. Phys., 388 (2019), 90-102. https://doi.org/10.1016/j.jcp.2019.03.021

[7] A. Townsend, S. Olver, The automatic solution of partial differential equations using a global spectral method, J. Comput. Phys., 299 (2015), 106-123. https://doi.org/10.1016/j.jcp.2015.06.031

[8] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686-707. https://doi.org/10.1016/j.jcp.2018.10.045

[9] Y. Ren, H. Feng, S. Zhao, A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains, J. Comput. Phys., 448 (2022), 110762. https://doi.org/10.1016/j.jcp.2021.110762

[10] J. Berg, K. Nyström, A unified deep artificial neural network approach to partial differential equations in complex geometries, Neurocomputing, 317 (2018), 28-41. https://doi.org/10.1016/j.neucom.2018.06.056

[11] C. Beck, S. Becker, P. Grohs, N. Jaafari, A. Jentzen, Solving the Kolmogorov PDE by means of deep learning, J. Sci. Comput., 88 (2021), 73. https://doi.org/10.1007/s10915-021-01590-0

[12] L. Yuan, Y. Ni, X. Deng, S. Hao, A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations, J. Comput. Phys., 462 (2022), 111260. https://doi.org/10.1016/j.jcp.2022.111260

[13] Y. Chen, L. Lu, G. E. Karniadakis, L. Dal Negro, Physics-informed neural networks for inverse problems in nano-optics and metamaterials, Opt. Express, 28 (2020), 11618-11633. https://doi.org/10.1364/OE.384875

[14] G. Pang, L. Lu, G. E. Karniadakis, fPINNs: Fractional physics-informed neural networks, SIAM J. Sci. Comput., 41 (2019), 2603-2626. https://doi.org/10.1137/18M1229845

[15] S. Wang, H. Wang, P. Perdikaris, On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks, Comput. Methods Appl. Mech. Eng., 384 (2021), 113938. https://doi.org/10.1016/j.cma.2021.113938

[16] S. Wang, Y. Teng, P. Perdikaris, Understanding and mitigating gradient flow pathologies in physics-informed neural networks, preprint, arXiv:2001.04536.

[17] J. Li, J. Chen, B. Li, Gradient-optimized physics-informed neural networks (GOPINNs): A deep learning method for solving the complex modified KDV equation, Nonlinear Dyn., 107 (2022), 781-792. https://doi.org/10.1007/s11071-021-06996-x

[18] A. D. Jagtap, G. E. Karniadakis, Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations, Commun. Comput. Phys., 28 (2020), 2002-2041. https://doi.org/10.4208/cicp.OA-2020-0164

[19] L. Guo, H. Wu, X. Yu, T. Zhou, Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations, Comput. Methods Appl. Mech. Eng., 400 (2022), 115523. https://doi.org/10.1016/j.cma.2022.115523

[20] Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, et al., Fourier neural operator for parametric partial differential equations, preprint, arXiv:2010.08895.

[21] Z. K. Lawal, H. Yassin, D. T. C. Lai, A. C. Idris, Modeling the complex spatio-temporal dynamics of ocean wave parameters: A hybrid PINN-LSTM approach for accurate wave forecasting, Measurement, 252 (2025), 117383. https://doi.org/10.1016/j.measurement.2025.117383

[22] S. Cai, Z. Wang, S. Wang, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks for heat transfer problems, ASME. J. Heat Mass Transfer., 143 (2021), 060801. https://doi.org/10.1115/1.4050542

[23] B. Bahtiri, B. Arash, S. Scheffler, M. Jux, R. Rolfes, A thermodynamically consistent physics-informed deep learning material model for short fiber/polymer nanocomposites, Comput. Methods Appl. Mech. Eng., 427 (2024), 117038. https://doi.org/10.1016/j.cma.2024.117038

[24] J. Zhao, Highly accurate compact mixed methods for two point boundary value problems, Appl. Math. Comput., 188 (2007), 1402-1418. https://doi.org/10.1016/j.amc.2006.11.006

[25] C. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439-471. https://doi.org/10.1016/0021-9991(88)90177-5