Gauss-Newton Natural Gradient Descent for Physics-Informed Computational Fluid Dynamics

TL;DR

This paper introduces a Gauss-Newton natural gradient method for physics-informed neural networks to solve Navier-Stokes equations with high accuracy, demonstrating scalability and efficiency in computational fluid dynamics.

Contribution

It is the first to apply Gauss-Newton's method in function space for PINNs solving Navier-Stokes equations with near single-precision accuracy.

Findings

Achieved near single-precision accuracy on benchmark problems.

Demonstrated the method's scalability to large neural networks.

Established equivalence with Gauss-Newton's method in parameter space.

Abstract

We propose Gauss-Newton's method in function space for the solution of the Navier-Stokes equations in the physics-informed neural network (PINN) framework. Upon discretization, this yields a natural gradient method that provably mimics the function space dynamics. Our computational results demonstrate close to single-precision accuracy measured in relative $L^2$ norm on a number of benchmark problems. To the best of our knowledge, this constitutes the first contribution in the PINN literature that solves the Navier-Stokes equations to this degree of accuracy. Finally, we show that given a suitable integral discretization, the proposed optimization algorithm agrees with Gauss-Newton's method in parameter space. This allows a matrix-free formulation enabling efficient scalability to large network sizes.