An Intelligent Validated, Energy-Aware and Decision-Oriented Numerical Framework for Solving Higher-Order Partial Differential Equations

Abstract

The development of higher-order PDEs that model many of the complex processes in materials science including phase separation, interfacial dynamics, and plate bending have made it increasingly challenging to develop numerically reliable methods to solve these problems, particularly when developing validation for decision-making systems. In this paper we describe an automated system for evaluating the performance of solvers for higher-order PDEs. The proposed framework is a series of chained tools designed to ensure reliability and reproducibility. HOG-AR is a graph auto-reduction method that transforms higher-order operator graphs into lower-order forms with preservation of continuity and satisfaction of all boundary conditions; EAS-FDG is a stencil design method based on a graph-regularized discretization scheme that ensures both the stability of the discretization and its consistency with respect to the physical energy of the problem; DRX-Res is a diagnostic tool that uses strong form residual analysis to identify localized errors in a solution by examining each element individually; CARMA is a reinforcement-based mesh adaptation technique that provides a systematic means of mesh refinement; DSS-NO is a surrogate assessment technique based on neural operators that can provide rapid, uncertainty-aware estimates of material behaviour without the need to perform the full simulation. Overall, the proposed system has the potential to improve the accuracy, efficiency, stability, and interpretability of models used to support decision making.