[2603.23072] Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations

Computer Science > Machine Learning arXiv:2603.23072 (cs) [Submitted on 24 Mar 2026] Title:Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations Authors:Sebastien Andre-Sloan, Dibyakanti Kumar, Alejandro F Frangi, Anirbit Mukherjee View a PDF of the paper titled Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations, by Sebastien Andre-Sloan and 2 other authors View PDF Abstract:This work establishes rigorous first-of-its-kind upper bounds on the generalization error for the method of approximating solutions to the (d+1)-dimensional incompressible Navier-Stokes equations by training depth-2 neural networks trained via the unsupervised Physics-Informed Neural Network (PINN) framework. This is achieved by bounding the Rademacher complexity of the PINN risk. For appropriately weight bounded net classes our derived generalization bounds do not explicitly depend on the network width and our framework characterizes the generalization gap in terms of the fluid's kinematic viscosity and loss regularization parameters. In particular, the resulting sample complexity bounds are dimension-independent. Our generalization bounds suggest using novel activation functions for solving fluid dynamics. We provide empirical validation of the suggested activation functions and the corresponding bounds on a PINN setup solving the Taylor-Green vortex benchmark. Subjects: Machine Learning (cs.LG)...