Parallel Physics-Informed Neural Networks Method with Regularization Strategies for the Forward-Inverse Problems of the Variable Coefficient Modified KdV Equation

IF 2.6 3区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Journal of Systems Science & Complexity Pub Date : 2024-01-26 DOI:10.1007/s11424-024-3467-7

Huijuan Zhou

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Abstract

This paper mainly introduces the parallel physics-informed neural networks (PPINNs) method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-de Vries (VC-MKdV) equation. For the forward problem of the VC-MKdV equation, the authors use the traditional PINN method to obtain satisfactory data-driven soliton solutions and provide a detailed analysis of the impact of network width and depth on solving accuracy and speed. Furthermore, the author finds that the traditional PINN method outperforms the one with locally adaptive activation functions in solving the data-driven forward problems of the VC-MKdV equation. As for the data-driven inverse problem of the VC-MKdV equation, the author introduces a parallel neural networks to separately train the solution function and coefficient function, successfully addressing the function discovery problem of the VC-MKdV equation. To further enhance the network’s generalization ability and noise robustness, the author incorporates two regularization strategies into the PPINNs. An amount of numerical experimental data in this paper demonstrates that the PPINNs method can effectively address the function discovery problem of the VC-MKdV equation, and the inclusion of appropriate regularization strategies in the PPINNs can improves its performance.

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针对变系数修正 KdV 方程正反问题的并行物理信息神经网络方法与正则化策略

摘要本文主要介绍了采用正则化策略的并行物理信息神经网络（PPINNs）方法求解变系数修正Korteweg-de Vries（VC-MKdV）方程的数据驱动正演反演问题。对于 VC-MKdV 方程的正演问题，作者使用传统 PINN 方法获得了令人满意的数据驱动孤子解，并详细分析了网络宽度和深度对求解精度和速度的影响。此外，作者还发现，在求解 VC-MKdV 方程的数据驱动正向问题时，传统 PINN 方法优于具有局部自适应激活函数的方法。对于数据驱动的 VC-MKdV 方程反演问题，作者引入并行神经网络分别训练解函数和系数函数，成功解决了 VC-MKdV 方程的函数发现问题。为了进一步提高网络的泛化能力和噪声鲁棒性，作者在 PPINN 中加入了两种正则化策略。本文中的大量数值实验数据表明，PPINNs 方法可以有效解决 VC-MKdV 方程的函数发现问题，而在 PPINNs 中加入适当的正则化策略可以提高其性能。

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来源期刊

Journal of Systems Science & Complexity 数学-数学跨学科应用

CiteScore

3.80

自引率

9.50%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Systems Science and Complexity is dedicated to publishing high quality papers on mathematical theories, methodologies, and applications of systems science and complexity science. It encourages fundamental research into complex systems and complexity and fosters cross-disciplinary approaches to elucidate the common mathematical methods that arise in natural, artificial, and social systems. Topics covered are: complex systems, systems control, operations research for complex systems, economic and financial systems analysis, statistics and data science, computer mathematics, systems security, coding theory and crypto-systems, other topics related to systems science.