Invited talk "An exponential integrating perspective on training neural networks for linear PDEs"

by Dr. Martin Schreiber, "Simulation großer Systeme" Seminar at the University of Stuttgart

Time integration of PDEs to simulate the atmosphere is a non-trivial task once considering wallclock time as well as accuracy constraints. There are claims about machine learning with neural networks being able to approximate high-dimensional spaces as well as non-linearities due to the activation functions used. This motivates an investigation of their suitability to overcome time step restrictions in the context of dynamical cores.

This work targets exploiting neural networks for temporal integration of PDEs using large time step sizes beyond the CFL condition. We design the underlying neural network with an inspiration from exponential integrators: Instead of using arbitrary off-the-shelf neural networks in a blackbox fashion, we will design neural networks to reflect as a first step the underlying linear terms. First, purely linear one- dimensional problems will be discussed and problems of high errors when using state-of-the-art gradient-based optimizers, which interestingly already exists for linear optimization of neural networks. As a first step towards a robust training, this will be overcome with a reformulation to a linear optimization problem and using a preconditioned conjugate gradient solvers as an optimizer. As a second part, possible extensions to non-linear problems will be discussed.