2025/02/26 by Zhenyi Zhu, Yuchen Huang, Zhu, Zhenyi +3 · 3 citations
Materials Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Machine Learning in Materials Science #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2502.19290
openalex publication_date 2025/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Time-dependent partial differential equations are a significant class of equations that describe the evolution of various physical phenomena over time. One of the open problems in scientific computing is predicting the behaviour of the solution outside the given temporal region. Most traditional numerical methods are applied to a given time-space region and can only accurately approximate the solution of the given region. To address this problem, many deep learning-based methods, basically data-driven and data-free approaches, have been developed to solve these problems. However, most data-driven methods require a large amount of data, which consumes significant computational resources and fails to utilize all the necessary information embedded underlying the partial differential equations (PDEs). Moreover, data-free approaches such as Physics-Informed Neural Networks (PINNs) may not be that ideal in practice, as traditional PINNs, which primarily rely on multilayer perceptrons (MLPs) and convolutional neural networks (CNNs), tend to overlook the crucial temporal dependencies inherent in real-world physical systems. We propose a method denoted as PhysicsSolver that merges the strengths of two approaches: data-free methods that can learn the intrinsic properties of physical systems without using data, and data-driven methods, which are effective at making predictions. Extensive numerical experiments have demonstrated the efficiency and robustness of our proposed method. We provide the code at \hrefhttps://github.com/PhysicsSolver/PhysicsSolverhttps://github.com/PhysicsSolver.