Improving Energy Natural Gradient Descent through Woodbury, Momentum, and Randomization
This work addresses efficiency bottlenecks in training PINNs, which is incremental for computational physics and engineering applications.
The authors tackled the high computational cost of energy natural gradient descent (ENGD) for Physics-Informed Neural Networks (PINNs) by introducing techniques like the Woodbury formula and randomization, achieving the same L^2 error up to 75 times faster than original ENGD.
Natural gradient methods significantly accelerate the training of Physics-Informed Neural Networks (PINNs), but are often prohibitively costly. We introduce a suite of techniques to improve the accuracy and efficiency of energy natural gradient descent (ENGD) for PINNs. First, we leverage the Woodbury formula to dramatically reduce the computational complexity of ENGD. Second, we adapt the Subsampled Projected-Increment Natural Gradient Descent algorithm from the variational Monte Carlo literature to accelerate the convergence. Third, we explore the use of randomized algorithms to further reduce the computational cost in the case of large batch sizes. We find that randomization accelerates progress in the early stages of training for low-dimensional problems, and we identify key barriers to attaining acceleration in other scenarios. Our numerical experiments demonstrate that our methods outperform previous approaches, achieving the same $L^2$ error as the original ENGD up to $75\times$ faster.