An AI-aided algorithm for multivariate polynomial reconstruction on Cartesian grids and the PLG finite difference method

Abstract: Polynomial reconstruction on Cartesian grids is fundamental in many scientific and engineering applications, yet it is still an open problem how to construct for a finite subset $K$ of $\mathbb{Z}^{\textsf{D}}$ a lattice $\mathcal{T}\subset K$ so that multivariate polynomial interpolation on this lattice is unisolvent. In this work, we solve this open problem of poised lattice generation (PLG) via an interdisciplinary research of approximation theory, abstract algebra, and AI. Specifically, we focus on the triangular lattices in approximation theory, study group actions of permutations upon triangular lattices, prove an isomorphism between the group of permutations and that of triangular lattices, and dynamically organize the AI state space of permutations so that a depth-first search of poised lattices has optimal efficiency. Based on this algorithm, we further develop the PLG finite difference method that retains the simplicity of Cartesian grids yet overcomes the disadvantage of legacy finite difference methods in handling irregular geometries. Results of various numerical tests demonstrate the effectiveness of our algorithm and the simplicity, efficiency, and fourth-order accuracy of the PLG finite difference method.