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

TL;DR

This paper derives new theoretical bounds on the generalization error of physics-informed neural networks applied to the incompressible Navier-Stokes equations, with empirical validation on fluid dynamics benchmarks.

Contribution

It provides the first rigorous generalization bounds for PINNs solving Navier-Stokes equations, linking error to physical parameters and regularization, independent of network width.

Findings

Generalization bounds do not depend on network width.

Bounds relate to fluid viscosity and regularization parameters.

Empirical validation on Taylor-Green vortex benchmark.

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.