On linear models for discrete operator inference in time dependent problems

Abstract

Numerical methods such as finite elements or finite differences to solve ordinary (ODEs) or partial differential equations (PDEs) are essential to computing in science and engineering. These discrete methods approximate the differential operators as matrices with non-zero coefficients for close-by degrees of freedom — the so-called stencil. For the computation of the stencil coefficients, the symbolic differential equation is generally required beforehand.In a recent work, we have demonstrated the usage of linear regression models for the accurate inference of stencil coefficients from spatial field data of linear, one- and two-dimensional PDEs with constant coefficients.In this paper, we introduce the propagation error, which may be used to evaluate regressed stencil coefficients with respect to the numerical stability of the induced system, which is a critical aspect for applications. Based on the propagation error, we continue by comparing several optimization techniques for variable selection that allow us to extend the applicability of linear models in stencil regression to non-linear PDEs. Finally, we highlight how a sliding-window approach may be utilized on spatio-temporal data to consider PDEs with non-constant coefficients.In summary, the analyses provided in this paper extend the use of linear regression models for the inference of stencil coefficients to a much broader class of applications.All source code associated with this study is publicly available athttps://github.com/HSU-HPC/ESCO_OperatorInference_TimeDependentProblems.