Neural basis method improves fluid flow prediction accuracy and scale
Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method
Machine Learning
Summary
Predicting how fluids like air and water move is very important but hard because the math equations are complex and computers struggle as problems get bigger. This paper presents a new approach called the neural basis method that breaks down the problem using fixed building blocks (bases) learned by a neural network, then figures out the best combination to satisfy the physics rules. The authors show that this way of representing fluid flow is more accurate, efficient, and can handle larger problems than previous methods. They also introduce a way to measure how good the solution is without needing extra data.
What this means in practice
- •For computational fluid dynamics engineers: Improve simulation accuracy and efficiency for large-scale fluid flow problems using neural basis representation with certified error estimates.
- •For industrial simulation software developers: Integrate scalable neural basis methods to provide reliable fluid dynamics predictions without requiring extensive labeled data.$Commercial implications: Enables advanced fluid simulation modules that reduce training data dependency, improving commercial CFD software products.
Authors
Yuhe Wang, Zhu Wang, Min Wang
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.