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

• URL: http://arxiv.org/abs/2602.17776v1
• Date: Thu, 19 Feb 2026 19:17:55 GMT
• Title: Solving and learning advective multiscale Darcian dynamics with the Neural Basis Method
• Authors: Yuhe Wang, Min Wang,
• Abstract summary: We introduce the Neural Basis Method, a projection-based formulation that couples a physics-conforming neural basis space with an operator-induced residual metric.<n>Our method produce accurate and robust solutions in single solves and enable fast and effective parametric inference with operator learning.
• Score: 4.331539387944184
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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