Near-optimal Sketchy Natural Gradients for Physics-Informed Neural Networks

Natural gradient methods for PINNs have achieved state-of-the-art performance with errors several orders of magnitude smaller than those achieved by standard optimizers such as ADAM or L-BFGS. However, computing natural gradients for PINNs is prohibitively computationally costly and memory-intensive for all but small neural network architectures. We develop a randomized algorithm for natural gradient descent for PINNs that uses sketching to approximate the natural gradient descent direction. We prove that the change of coordinate Gram matrix used in a natural gradient descent update has rapidly-decaying eigenvalues for a one-layer, one-dimensional neural network and empirically demonstrate that this structure holds for four different example problems. Under this structure, our sketching algorithm is guaranteed to provide a near-optimal lowrank approximation of the Gramian. Our algorithm dramatically speeds up computation time and reduces memory overhead. Additionally, in our experiments, the sketched natural gradient outperforms the original natural gradient in terms of accuracy, often achieving an error that is an order of magnitude smaller. Training time for a network with around 5,000 parameters is reduced from several hours to under two minutes. Training can be practically scaled to large network sizes; we optimize a PINN for a network with over a million parameters within a few minutes, a task for which the full Gram matrix does not fit in memory.