Zihan Wang , Ziyue Hu , Mingwei Yang , Yalin Dong , Wenlong Huang , Haijun Ren
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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.
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.