Hybridized Data Driven Flux-Conservative Solvers: Towards Foundation Models for PDE-Solving

Dirk Hartmann

Keywords: PDE solving, Hybridized Methods, Domain Decomposition, Flux Conservation

TL;DR: Work-in-progress report on a scalable flux conservation focused PDE solver leveraging differentiability for effective solving architectures.

Abstract: Numerical solution of Partial Differential Equations (PDE) is an indispensable tool in science and engineering. While Scientific Machine Learning offers potentially unique opportunities, current methods face challenges in scaling to real world applications. This work in progress report introduces Hybridized Data-Driven Flux-Conservative Solvers (H-DD-FCS). These combine three core principles: scalable domain decomposition, explicit flux conservation enforcement, and Newton-like iterative solvers leveraging modern differentiable ML frameworks. We demonstrate the feasibility of this approach along the 2D Poisson problem. Compared to state-of-the-art approaches, H-DD-FCS explicitly considers flux conservation and thus allows for better robustness, scalability, and accessibility to mathematical analysis. It offers a promising direction towards Foundation Models for PDE-solving.

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Submission Number: 54