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在本研究中，我们提出了扩展混合训练物理信息神经网络（X-MTPINNs），该模型通过整合扩展物理信息神经网络（X-PINNs）的域分解技术与混合训练物理信息神经网络（MTPINNs）框架，有效提升了非线性波动问题的求解能力。相较于经典PINNs模型，新模型具有双重优势：其一，混合训练框架通过优化初边值条件的处理机制，显著改善了模型收敛特性，在提升非线性波解拟合精度的同时，将计算时间降低约40%；其二，X-PINNs的域分解技术增强了模型对复杂动力学行为的表征能力。基于非线性薛定谔方程（NLSE）的数值实验表明，X-MTPINNs在亮双孤子解及三阶怪波求解以及参数反演等任务中均表现优异，其预测精度较传统PINNs提升一至两个数量级。对于逆问题，X-MTPINNs算法在有噪声和无噪声条件下都能准确识别NLSE中的未知参数，解决了经典PINNs在本研究条件下NLSE参数识别中完全失效的问题，表现出很强的鲁棒性。In recent years, Physics-Informed Neural Networks (PINNs) have provided effcient data-driven methods for solving both forward and inverse problems of partial differential equations (PDEs). However, when addressing complex PDEs, PINNs face significant challenges in computational effciency and accuracy. In this study, we propose the Extended Mixed-Training Physics-Informed Neural Networks (X-MTPINNs), as illustrated in fig. 1., which effectively enhance the capability for solving nonlinear wave problems by integrating the domain decomposition technique of extended Physics-Informed Neural Networks (X-PINNs) with the mixed-training Physics-Informed Neural Networks (MTPINNs) framework. Compared to the classical PINNs model, the new model exhibits dual advantages: First, the mixed-training framework significantly improves convergence properties by optimizing the handling mechanism of initial and boundary conditions, achieving higher fitting accuracy for nonlinear wave solutions while reducing the computation time by approximately 40%. Second, the domain decomposition technique from X-PINNs strengthens the model’s representation capability for complex dynamical behaviors. Numerical experiments based on the Nonlinear Schrödinger Equation (NLSE) demonstrate that X-MTPINNs excel in solving two bright solitons, third-order rogue waves, and parameter inversion tasks, with prediction accuracy improved by one to two orders of magnitude over traditional PINNs. For inverse problems, the X-MTPINNs algorithm accurately identifies unknown parameters in the NLSE under noise-free, 2%, and 5% noisy conditions, addressing the complete failure of classical PINNs in parameter identification for NLSE under the studied scenarios, thereby exhibiting strong robustness.
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Keywords:
- PINNs /
- NLSE /
- X-MTPINNs /
- domain decomposition technique /
- parameters discovery
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[1] Dissanayake M, PHAN-THIEN N 1994 Commun. Numer. Meth. Eng. 10 195
[2] Lagaris I E, Likas A 1998 IEEE Trans. Neural Netw. 9 987
[3] Eymard R, Gallouët T, Herbin R 2000 Handb. Numer. Anal. 7 713
[4] Zhang Y 2009 Appl. Math. Comput. 215 524
[5] Taylor C A, Hughes T J R, Zarins C K 1998 Comput. Methods Appl. Mech. Eng. 158 155
[6] He K M, Zhang X Y,Ren S Q, Sun J 2016 CPRV. 2016-December 770
[7] Ding Y, Li J T, Zhang J X, Li P P, Bai H, Fang B, Fang H X, Huang K, Wang G Y, Nowell C J, Voelcker N H, Peng B, Li L, Huang W 2025 Nat. Commun. 16 743
[8] Liu J, Chen Y B, Liu K,Zhao J 2018 AAAI. 32 4865
[9] Raissi M, Perdikaris P, Karniadakis G E 2019 J. Comput. Phys. 378 686
[10] Wang F J, Zhai Z, Zhao Z B, Di Y, Chen X F 2024 Nat. Commun. 15 4332
[11] James D, Alireza D, Soroush A 2019 Sci. Total Environ. 912 168814
[12] Xu S F, Sun Z X, Huang R F, Guo D L, Yang G W, Ju S J 2023 AMS. 39 322302
[13] Chiu P H, Wong J C, Ooi C C, Dao H 2022 Comput. Methods Appl. Mech. Eng. 395 114909
[14] Yang L, Meng X H, Karniadakis G E 2021 J. Comput. Phys. 425 109913
[15] Mojgani R, Balajewicz M, Hassanzadeh P 2023 Comput. Methods Appl. Mech. Eng. 404 115810
[16] Jagtap A D, Karniadakis G E 2021 CEUR Workshop Proc. 2964
[17] Che W J, Chen S C, Liu C, Zhao L C, Akhmediev N 2022 Phys. Rev. A 105 043526
[18] Liu C, Chen S C, Yao X K, Akhmediev N 2022 Physica D 433 133192
[19] Fang Y, Bo W B, Wang R R, Wang Y Y, Dai C Q 2022 Chaos Solit. Fractals 165 112908
[20] Li W T, Chen Y P, Zhang Z, Yang X Y, Li B 2025 Chin. Phys. Lett. 42 010202
[21] Chen M, Wang Z 2023 Chin. Phys. B 32 090504
[22] Li J, Chen Y 2020 Commun. Theor. Phys. 72 105005
[23] Li J, Chen Y 2020 Commun. Theor. Phys. 72 115003
[24] Song J, Zhong M, Karniadakis G E, Yan Z Y 2024 J. Comput. Phys. 505 112917
[25] Wang X L, Wu Z K, Song J, Han W J, Yan Z Y 2024 Chaos Solit. Fractals 180 114509
[26] Qiu T W, Wei G M, Song Y X, Wang Z 2025 Appl. Math. Mech. 46 105
[27] Tian S F, Li B 2023 Acta Phys. Sin. 72 100202
[28] Li J H, Li B 2022 Chaos Solit. Fractals 164 112712
[29] Tian S F, Li B, Zhang Z 2024 Chin. Phys. Lett. 41 030201
[30] Zhou H J, Pu J C, Chen Y 2023 Nonlinear Dyn. 111 14667
[31] Qiu W X, Si Z Z, Mou D S, Dai C Q, Li J T, Liu W 2025 Nonlinear Dyn. 113 4063
[32] Liu X M, Zhang Z Y, Liu W J 2023 Chin. Phys. Lett. 40 070501
[33] Liu H Y, Zhang Y B, Wang L 2024 J. Syst. Sci. 37 494
[34] Zhang W X, Chen Y 2024 Nonlinear Dyn.
[35] Pu J C, Chen Y 2024 J. Comput. Phys. 510 113090
[36] Zhang J F, Dai C Q, Wang Y Y 2016 Rogue Wave Theory and Its Applications Based on the Nonlinear Schrödinger Equation (Beijing： Science Press) pp44-63 (in Chinese) [张解放, 戴朝卿, 王悦悦 2016 基于非线性薛定谔方程的畸形波理论及其应用（北京：科学出版社）第 44-63 页]
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