Modified PINN approach integrating conservation laws for efficient multi-stage training of coupled nonlinear systems

doi:10.1088/1674-1056/ae030b

Abstract We propose an innovative conservation multi-stage algorithm (CM) based on physics-informed neural networks (PINNs) to investigate the dynamics of vector solitons governed by Zakharov equation. By ingeniously integrating conservation laws into the multi-stage training algorithm, we enhanced the method's constraint and physical consistency in solving nonlinear systems. Numerical simulations of the Zakharov and nonlinear Schrödinger (NLS) equations demonstrated that our method showed obvious improvements in approximation accuracy, convergence speed, and training efficiency, compared to traditional PINN methods. Moreover, when the parameter representing the speed of sound is sufficiently large, our method efficiently simulates the approximation from the Zakharov equation to the NLS equation. The approximative simulation not only confirms the conservation multi-stage algorithm's applicability in various physical environments but also highlights its potential in controlling sound wave propagation characteristics to simulate NLS equation behavior.

Keywords: physics-informed neural network Zakharov equation nonlinear Schrödinger equation limit behavior

Received: 21 July 2025
Revised: 25 August 2025
Accepted manuscript online: 04 September 2025

PACS:	02.60.-x	(Numerical approximation and analysis)
	52.35.-g	(Waves, oscillations, and instabilities in plasmas and intense beams)
	02.30.Jr	(Partial differential equations)
	02.60.Cb	(Numerical simulation; solution of equations)

Fund: Lijia Han is supported in part by the National Natural Science Foundation of China (Grant No. 11971166).

Corresponding Authors: Lijia Han
E-mail: hljmath@ncepu.edu.cn

Cite this article:

Jie Deng(邓婕) and Lijia Han(韩励佳) Modified PINN approach integrating conservation laws for efficient multi-stage training of coupled nonlinear systems 2026 Chin. Phys. B 35 040202

[1] Chen Z, Segev M and Christodoulides D N 2012 Rep. Prog. Phys. 75 086401
[2] Zakharov V E 1972 J. Exp. Theor. Phys. 35 1086
[3] Ablowitz M A and Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge: Cambridge Univ. Press)
[4] Hirota R 1971 Phys. Rev. Lett. 27 1192
[5] Gu C H, Zhou Z H and Tan H 1990 Soliton Theory and Its Application (Hangzhou: Zhejiang Science and Technology Press)
[6] Ern A, Guermond JL 2004 Theory and Practice of Finite Elements (New York: Springer) Appl. Math. Sci. 159
[7] Eymard R, Gallouët T and Herbin R 2000 Handb. Numer. Anal. 7 713
[8] Trefethen L N 2000 Spectral Methods in MATLAB (Philadelphia: Society for Industrial and Applied Mathematics)
[9] Canale R P and Chapra S C 2014 Numerical Methods for Engineers (Europe: McGraw-Hill Education)
[10] Washington W M, Buja L and Craig A 2008 Philos. Trans. R. Soc. A 366 2157
[11] Raissi M, Perdikaris P and Karniadakis G E 2019 J. Comput. Phys. 378 686
[12] Karniadakis G E, Kevrekidis I G, Lu L, et al. 2021 Nat. Rev. Phys. 3 422
[13] Liu Y, LiuW, Yan X, Guo S and Zhang C A 2023 J. Comput. Phys. 490 112291
[14] Lu L, Meng X, Mao Z and Karniadakis G E 2021 SIAM Rev. 63 208
[15] Cuomo S, Di Cola V S, Giampaolo F, Rozza G, Raissi M and Piccialli F 2022 Phys. Scr. 92 88
[16] Nakamura M, He H and Abdikian H 2024 Comput. Phys. Commun. 285 108460
[17] Song J, Zhong M, Karniadakis G E and Yan Z 2024 J. Comput. Phys. 505 112917
[18] Peng Y, Pu J C and Chen Y 2022 Commun. Nonlinear Sci. Numer. Simul. 105 106067
[19] Lin S and Chen Y 2023 Physica D 445 133629
[20] Pu J C and Chen Y 2022 Chaos, Solitons and Fractals 160 112182
[21] Pu J and Chen Y 2023 Physica D 454 133851
[22] Zhou Z and Yan Z 2021 Phys. Lett. A 387 127010
[23] Wu Y, Ling L and Huang Y 2024 Comput. Math. Appl. 169 132
[24] Yan Z 2007 Phys. Lett. A 361 194
[25] De Ryck T and Mishra S 2022 Adv. Comput. Math. 48 79
[26] Jagtap A D and Karniadakis G E 2020 Commun. Comput. Phys. 28 2002
[27] Wight C L and Zhao J 2021 Commun. Comput. Phys. 29 930
[28] Wang S, Teng Y and Perdikaris P 2021 SIAM J. Sci. Comput. 43 A3055
[29] Mo Y, Ling L and Zeng D 2022 Phys. Lett. A 421 127739
[30] Jagtap A D, Kharazmi E and Karniadakis G E 2020 Comput. Methods Appl. Mech. Eng. 365 113028
[31] Gurieva J, Vasiliev E and Smirnov L 2022 Procedia Comput. Sci. 212 464
[32] Baez A, ZhangW, Ma Z, Das S, Nguyen LMand Daniel L 2024 arXiv: 2410.17445
[33] Wu G Z, Fang Y, Kudryashov N A,Wang Y Y and Dai C Q 2022 Chaos Solitons Fractals 159 112143
[34] Fang Y, Wu G Z, Kudryashov N A, Wang Y Y and Dai C Q 2022 Chaos, Solitons and Fractals 158 112118
[35] Lin S and Chen Y 2022 J. Comput. Phys. 457 111053
[36] Nakamura A, Sawado N, Shimasaki K, Suzuki Y and Toda K 2024 arXiv: 2410.19014
[37] Abdikian A 2024 Phys. Plasmas 31 102105
[38] Han L, Zhang J, Gan Z and Guo B 2012 Sci. China Math. 55 509
[39] Schochet S H and Weinstein M I 1986 Commun. Math. Phys. 106 569
[40] Added H and Added S 1988 J. Funct. Anal. 79 183
[41] Ozawa T, Tsutsumi Y and Brezis H 1992 Differ. Integral Equ. 5 721
[42] Masmoudi N and Nakanishi K 2005 J. Hyperbolic Differ. Equ. 2 975
[43] He Y, Wang J L and Huang J 2023 Journal on Numerical Methods and Computer Applications 44 1
[44] Liu D C and Nocedal J 1989 Math. Program. 45 503
[45] Ling L and Zhao L C 2013 Phys. Rev. E 88 043201
[46] Zhao L C, Guo B and Ling L 2016 J. Math. Phys. 57 043508
[47] Guo B and Huo Z 2010 Commun. Partial Differ. Equ. 36 247
[48] Ling L, Zhao L C and Guo B 2015 Nonlinearity 28 3243
[49] Pei Y T, Wang J K, Guo B L and Liu W M 2023 Acta Phys. Sin. 72 100201 (in Chinese)

[1]	LT-PINNs: Physics-informed neural networks based on Laplace transform for solving Caputo-type fractional partial differential equations Ruibo Zhang(张瑞波), Fengjun Li(李风军), and Jianqiang Liu(刘建强). Chin. Phys. B, 2026, 35(3): 030201.
[2]	Study and application of solitary wave propagation at fractional order of time based on SPH method Luyang Ma(马璐阳), Rahmatjan Imin(热合买提江·依明), and Azhar Halik(艾孜海尔·哈力克). Chin. Phys. B, 2025, 34(5): 050202.
[3]	Dynamical analysis and localized waves of the n-component nonlinear Schrödinger equation with higher-order effects Yu Lou(娄瑜) and Guoan Xu(许国安). Chin. Phys. B, 2025, 34(3): 030201.
[4]	Periodic lump, soliton, and some mixed solutions of the (2+1)-dimensional generalized coupled nonlinear Schrödinger equations Xiao-Min Wang(王晓敏), Ji Li(李吉), and Xiao-Xiao Hu(胡霄骁). Chin. Phys. B, 2025, 34(11): 110502.
[5]	Dynamics of fundamental and double-pole breathers and solitons for a nonlinear Schrödinger equation with sextic operator under non-zero boundary conditions Luyao Zhang(张路瑶) and Xiyang Xie(解西阳). Chin. Phys. B, 2024, 33(9): 090207.
[6]	TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method Jia Guo(郭嘉), Haifeng Wang(王海峰), Shilin Gu(古仕林), and Chenping Hou(侯臣平). Chin. Phys. B, 2024, 33(5): 050701.
[7]	MetaPINNs: Predicting soliton and rogue wave of nonlinear PDEs via the improved physics-informed neural networks based on meta-learned optimization Yanan Guo(郭亚楠), Xiaoqun Cao(曹小群), Junqiang Song(宋君强), and Hongze Leng(冷洪泽). Chin. Phys. B, 2024, 33(2): 020203.
[8]	Bright soliton dynamics for resonant nonlinear Schrödinger equation with generalized cubic-quintic nonlinearity Keyu Bao(鲍柯宇), Xiaogang Tang(唐晓刚), and Ying Wang(王颖). Chin. Phys. B, 2024, 33(12): 124203.
[9]	Effective regulation of the interaction process among three optical solitons Houhui Yi(伊厚会), Xiaofeng Li(李晓凤), Junling Zhang(张俊玲), Xin Zhang(张鑫), and Guoli Ma(马国利). Chin. Phys. B, 2024, 33(10): 100502.
[10]	Breather and its interaction with rogue wave of the coupled modified nonlinear Schrödinger equation Ming Wang(王明), Tao Xu(徐涛), Guoliang He(何国亮), and Yu Tian(田雨). Chin. Phys. B, 2023, 32(5): 050503.
[11]	Meshfree-based physics-informed neural networks for the unsteady Oseen equations Keyi Peng(彭珂依), Jing Yue(岳靖), Wen Zhang(张文), and Jian Li(李剑). Chin. Phys. B, 2023, 32(4): 040207.
[12]	All-optical switches based on three-soliton inelastic interaction and its application in optical communication systems Shubin Wang(王树斌), Xin Zhang(张鑫), Guoli Ma(马国利), and Daiyin Zhu(朱岱寅). Chin. Phys. B, 2023, 32(3): 030506.
[13]	From breather solutions to lump solutions: A construction method for the Zakharov equation Feng Yuan(袁丰), Behzad Ghanbari, Yongshuai Zhang(张永帅), and Abdul Majid Wazwaz. Chin. Phys. B, 2023, 32(12): 120201.
[14]	Nondegenerate solitons of the (2+1)-dimensional coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optical fibers Wei Yang(杨薇), Xueping Cheng(程雪苹), Guiming Jin(金桂鸣), and Jianan Wang(王佳楠). Chin. Phys. B, 2023, 32(12): 120202.
[15]	Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schrödinger equation Xuefeng Zhang(张雪峰), Tao Xu(许韬), Min Li(李敏), and Yue Meng(孟悦). Chin. Phys. B, 2023, 32(1): 010505.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Modified PINN approach integrating conservation laws for efficient multi-stage training of coupled nonlinear systems

Cite this article:

Online attention