Learned numerical methods for solving partial differential equations

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc.

This talk will give an overview of how we've used machine learning to develop better numerical methods for solving PDEs on coarse grids. Our approach is based upon the new paradigm of "differentiable programming", which allows for end-to-end optimization of simulations built upon the combination of neural networks and traditional simulation methods. We demonstrate results for Burgers' equation and the 2D Navier-Stokes equation, which we are able to solve on 4-12x coarser grids and up to two orders of magnitude than standard numerical methods.