Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra
For researchers using neural network-based PDE solvers, this work offers a more efficient optimization method, though it is an incremental improvement over existing NGD approaches.
The paper tackles the high computational cost of Natural Gradient Descent for Physics-Informed Neural Networks by proposing randomized preconditioning for the inner conjugate gradient solver, achieving substantial performance improvements over existing methods.
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 Numerical Linear Algebra (RandNLA) techniques for efficient preconditioning 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.