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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.17776 |
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| _version_ | 1866914464442351616 |
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| author | Wang, Yuhe Wang, Min |
| author_facet | Wang, Yuhe Wang, Min |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17776 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solving and learning advective multiscale Darcian dynamics with the Neural Basis Method Wang, Yuhe Wang, Min Numerical Analysis Machine Learning 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. |
| title | Solving and learning advective multiscale Darcian dynamics with the Neural Basis Method |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2602.17776 |