AI and the World's Hard Problems · Physics · arXiv · 2026-10-26
Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method
Physics-informed machine learning for fluid dynamics often does not guarantee that a small PDE residual means a small solution error, and it scales poorly with problem size. The authors propose the neural basis method (NBM), which uses fixed neural bases to represent the solution and projects the governing equations to obtain the expansion coefficients. For incompressible flow, its residual gives an a posteriori velocity-error estimate, providing a certified, data-free training objective, and theory and experiments report substantial gains over baselines in accuracy, efficiency, generalization and scalability.