Solving and learning advective multiscale Darcian dynamics with the Neural Basis Method

TL;DR

The paper introduces the Neural Basis Method, a projection-based approach that improves physics-informed neural models by providing a stable, interpretable residual metric, demonstrated on advective multiscale Darcian dynamics.

Contribution

It proposes a new neural basis formulation that couples a physics-conforming basis with an operator residual metric for stable, interpretable solutions and efficient parametric inference.

Findings

Accurate solutions for advective multiscale Darcian dynamics.

Stable and reliable enforcement of physics constraints.

Fast parametric inference enabled by the method.

Abstract

Physics-governed models are increasingly paired with machine learning for accelerated predictions, yet most "physics--informed" formulations treat the governing equations as a penalty loss whose scale and meaning are set by heuristic balancing. This blurs operator structure, thereby confounding solution approximation error with governing-equation enforcement error and making the solving and learning progress hard to interpret and control. Here we introduce the Neural Basis Method, a projection-based formulation that couples a predefined, physics-conforming neural basis space with an operator-induced residual metric to obtain a well-conditioned deterministic minimization. Stability and reliability then hinge on this metric: the residual is not merely an optimization objective but a computable certificate tied to approximation and enforcement, remaining stable under basis enrichment and yielding reduced coordinates that are learnable across parametric instances. We use advective multiscale Darcian dynamics as a concrete demonstration of this broader point. Our method produce accurate and robust solutions in single solves and enable fast and effective parametric inference with operator learning.