On boundary conditions parametrized by analytic functions

TL;DR

This paper extends computer algebra models combined with Gaussian processes to handle analytic boundary conditions in differential equations, providing algorithms for Weyl algebras and demonstrating applications to divergence-free flows.

Contribution

It introduces methods to incorporate analytic boundary conditions into algebraic models and develops algorithms for Weyl algebras with analytic coefficients.

Findings

Algorithms for Gr"obner and Janet bases in Weyl algebras with analytic coefficients

Examples of divergence-free flow in domains bounded by analytic functions

Enhanced modeling of differential equations with data using algebraic methods

Abstract

Computer algebra can answer various questions about partial differential equations using symbolic algorithms. However, the inclusion of data into equations is rare in computer algebra. Therefore, recently, computer algebra models have been combined with Gaussian processes, a regression model in machine learning, to describe the behavior of certain differential equations under data. While it was possible to describe polynomial boundary conditions in this context, we extend these models to analytic boundary conditions. Additionally, we describe the necessary algorithms for Gr\"obner and Janet bases of Weyl algebras with certain analytic coefficients. Using these algorithms, we provide examples of divergence-free flow in domains bounded by analytic functions and adapted to observations.