基于预处理-预训练物理信息神经网络的偏微分方程求解方法

IF 7.2 2区 物理与天体物理 Q1 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Zihan Wang , Ziyue Hu , Mingwei Yang , Yalin Dong , Wenlong Huang , Haijun Ren
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引用次数: 0

摘要

物理信息神经网络(PINN)是一种深度学习框架,已被广泛应用于求解各个领域的时空偏微分方程(PDEs)。然而,最近的数值实验表明,香草- pinn经常与具有高频解或强非线性的偏微分方程作斗争。为了提高PINN的性能,我们提出了一种新的策略,称为预调节-预训练物理信息神经网络(PP-PINN)。该方法将原有的任务转化为一个具有低频和弱非线性特征的新系统。然后使用预训练方法求解变换后的偏微分方程。此外,我们还引入了一个新的约束,称为“不动点”,这有利于极高频率或强非线性的场景。为了证明我们方法的有效性,我们将新开发的策略应用于三个不同的方程,与之前结合预训练技术的方法相比,实现了更高的精度和更低的计算成本。我们还讨论了PP-PINN的有效性和可解释性,强调了其在处理高频解和强非线性方面的优势,从而为其在复杂数学建模中的更广泛适用性提供了见解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Solving partial differential equations based on preconditioning-pretraining physics-informed neural network
Physics-Informed Neural Network (PINN) is a deep learning framework that has been widely employed to solve spatial-temporal partial differential equations (PDEs) across various fields. However, recent numerical experiments indicate that the vanilla-PINN often struggles with PDEs featuring high-frequency solutions or strong nonlinearity. To enhance PINN's performance, we propose a novel strategy called the Preconditioning-Pretraining Physics-Informed Neural Network (PP-PINN). This approach involves transforming the original task into a new system characterized by low frequency and weak nonlinearity over an extended time scale. The transformed PDEs are then solved using a pretraining approach. Additionally, we introduce a new constraint termed “fixed point”, which is beneficial for scenarios with extremely high frequency or strong nonlinearity. To demonstrate the efficacy of our method, we apply the newly developed strategy to three different equations, achieving improved accuracy and reduced computational costs compared to previous approaches which incorporate the pretraining technique. The effectiveness and interpretability of our PP-PINN are also discussed, emphasizing its advantages in tackling high-frequency solutions and strong nonlinearity, thereby offering insights into its broader applicability in complex mathematical modeling.
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来源期刊
Computer Physics Communications
Computer Physics Communications 物理-计算机:跨学科应用
CiteScore
12.10
自引率
3.20%
发文量
287
审稿时长
5.3 months
期刊介绍: The focus of CPC is on contemporary computational methods and techniques and their implementation, the effectiveness of which will normally be evidenced by the author(s) within the context of a substantive problem in physics. Within this setting CPC publishes two types of paper. Computer Programs in Physics (CPiP) These papers describe significant computer programs to be archived in the CPC Program Library which is held in the Mendeley Data repository. The submitted software must be covered by an approved open source licence. Papers and associated computer programs that address a problem of contemporary interest in physics that cannot be solved by current software are particularly encouraged. Computational Physics Papers (CP) These are research papers in, but are not limited to, the following themes across computational physics and related disciplines. mathematical and numerical methods and algorithms; computational models including those associated with the design, control and analysis of experiments; and algebraic computation. Each will normally include software implementation and performance details. The software implementation should, ideally, be available via GitHub, Zenodo or an institutional repository.In addition, research papers on the impact of advanced computer architecture and special purpose computers on computing in the physical sciences and software topics related to, and of importance in, the physical sciences may be considered.
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