Accelerating Natural Gradient Descent for PINNs with Randomized Nystr\"om Preconditioning

TL;DR

This paper introduces a randomized Nyström preconditioning technique to significantly accelerate Natural Gradient Descent for training Physics-Informed Neural Networks, reducing computational costs and improving convergence on PDE problems.

Contribution

The paper proposes a novel randomized Nyström preconditioning method to enhance the efficiency of matrix-free NGD for PDE solvers, extending its applicability and performance.

Findings

Substantial performance improvements over existing NGD methods.

Faster convergence of the conjugate gradient solver.

Effective acceleration across various PDE discretizations.

Abstract

Natural Gradient Descent (NGD) has emerged as a promising optimization algorithm for training neural network-based solvers for partial differential equations (PDEs), such as Physics-Informed Neural Networks (PINNs). However, its practical use is often limited by the high computational cost of solving linear systems involving the Gramian matrix. While matrix-free NGD methods based on the conjugate gradient (CG) method avoid explicit matrix inversion, the ill-conditioning of the Gramian significantly slows the convergence of CG. In this work, we extend matrix-free NGD to broader classes of problems than previously considered and propose the use of Randomized Nystr\"om preconditioning to accelerate convergence of the inner CG solver. The resulting algorithm demonstrates substantial performance improvements over existing NGD-based methods and other state-of-the-art optimizers on a range of PDE problems discretized using neural networks.