Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method
Abstract
Physics-informed machine learning has become a major direction in predictive modeling of fluid dynamics, combining governing physics with learned representations. Yet prevailing formulations neither guarantee that small PDE-residual losses imply small solution errors nor scale effectively with increasing problem size. Here we present the neural basis method (NBM) as a new approach for Navier--Stokes dynamics. NBM is built on a ``representation-then-projection'' principle: fixed neural bases define the representation space, while projection of the governing equations determines the expansion coefficients. The same bases define common coordinates for the parametric solution manifold, recasting operator learning as a parameter-to-coefficient mapping and enabling scalability by removing explicit spatial dependence. For incompressible flow, NBM yields a velocity-only formulation whose residual provides an a posteriori velocity-error estimate, establishing a certified, data-free training objective for coefficient learning. Theoretical analysis and numerical experiments demonstrate the accuracy, efficiency, generalization, and scalability of NBM, with substantial improvements over baselines.
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BibTeX
@misc{wang2026solving,
title = {{Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method}},
author = {Yuhe Wang and Zhu Wang and Min Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32167},
eprint = {2610.32167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32167},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method
AU - Yuhe Wang
AU - Zhu Wang
AU - Min Wang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32167
UR - https://arxiv.org/abs/2610.32167
AB - Physics-informed machine learning has become a major direction in predictive modeling of fluid dynamics, combining governing physics with learned representations. Yet prevailing formulations neither guarantee that small PDE-residual losses imply small solution errors nor scale effectively with increasing problem size. Here we present the neural basis method (NBM) as a new approach for Navier--Stokes dynamics. NBM is built on a ``representation-then-projection'' principle: fixed neural bases define the representation space, while projection of the governing equations determines the expansion coefficients. The same bases define common coordinates for the parametric solution manifold, recasting operator learning as a parameter-to-coefficient mapping and enabling scalability by removing explicit spatial dependence. For incompressible flow, NBM yields a velocity-only formulation whose residual provides an a posteriori velocity-error estimate, establishing a certified, data-free training objective for coefficient learning. Theoretical analysis and numerical experiments demonstrate the accuracy, efficiency, generalization, and scalability of NBM, with substantial improvements over baselines.
ER -