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.
翻译:计算机代数可以解答关于使用象征性算法进行部分差异方程式的各种问题。 但是,在计算机代数中,将数据纳入等式的情况很少。 因此,最近计算机代数模型已经与机器学习中的回归模型Gaussian进程相结合,以描述数据下某些差异方程式的行为。 虽然可以在此背景下描述多元边界条件,但我们将这些模型扩大到分析边界条件。 此外,我们用某些分析系数描述Weyl代数的Gr\'obner和Janet基点的必要算法。我们利用这些算法,提供了受分析功能约束的领域中无差异流动的实例,并适应了观测结果。