Kronecker-Factored Approximate Curvature for Physics-Informed Neural Networks

Felix Dangel, Johannes Müller, Marius Zeinhofer

Keywords: KFAC, PINNs, Gauss-Newton, PDEs, Taylor mode automatic differentiation, Forward Laplacian, Second-order optimization, Higher-order derivatives

TL;DR: We derive a KFAC approximation for PINN losses which scales to high-dimensional NNs and PDEs and consistently outperforms first-order methods for training PINNs.

Abstract:

Physics-Informed Neural Networks (PINNs) are infamous for being hard to train.
Recently, second-order methods based on natural gradient and Gauss-Newton methods have shown promising performance, improving the accuracy achieved by first-order methods by several orders of magnitude.
While promising, the proposed methods only scale to networks with a few thousand parameters due to the high computational cost to evaluate, store, and invert the curvature matrix.
We propose Kronecker-factored approximate curvature (KFAC) for PINN losses that greatly reduces the computational cost and allows scaling to much larger networks.
Our approach goes beyond the popular KFAC for traditional deep learning problems as it captures contributions from a PDE's differential operator that are crucial for optimization.
To establish KFAC for such losses, we use Taylor-mode automatic differentiation to describe the differential operator's computation graph as a forward network with shared weights which allows us to apply a variant of KFAC for networks with weight-sharing.
Empirically, we find that our KFAC-based optimizers are competitive with expensive second-order methods on small problems, scale more favorably to higher-dimensional neural networks and PDEs, and consistently outperform first-order methods.

Primary Area: Machine learning for physical sciences (for example: climate, physics)

Submission Number: 9354