Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2212.00798

Pre-training strategy for solving evolution equations based on physics-informed neural networks

Jiawei Guo·Yanzhong Yao·Han Wang·Tongxiang Gu
View original source ↗Read source document ↗Export metadata ↓

Abstract

The physics informed neural network (PINN) is a promising method for solving time-evolution partial differential equations (PDEs). However, the standard PINN method may fail to solve the PDEs with strongly nonlinear characteristics or those with high-frequency solutions. The physics informed neural network (PINN) is a promising method for solving time-evolution partial differential equations (PDEs). However, the standard PINN method may fail to solve the PDEs with strongly nonlinear characteristics or those with high-frequency solutions. The PT-PINN method transforms the difficult problem on the entire time domain to relatively simple problems defined on small subdomains. The neural network trained on small subdomains provides the neural network initialization and extra supervised learning data for the problems on larger subdomains or on the entire time-domain. By numerical experiments, we demonstrate that the PT-PINN succeeds in solving the evolution PDEs with strong non-linearity and/or high frequency solutions, including the strongly nonlinear heat equation, the Allen-Cahn equation, the convection equation with high-frequency solutions and so on, and that the convergence and accuracy of the PT-PINN is superior to the standard PINN method. The PT-PINN method is a competitive method for solving the time-evolution PDEs.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jiawei Guo, Yanzhong Yao, Han Wang, Tongxiang Gu. 2022-11-12. Pre-training strategy for solving evolution equations based on physics-informed neural networks. https://doi.org/10.1016/j.jcp.2023.112258

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗