2018/08/31 by Yohai Bar-Sinai, Stephan Hoyer, Jason Hickey +1 · 4 citations
Physics and Astronomy · #cond-mat.dis-nn #physics.comp-ph
paper · pdf · doi:10.1073/pnas.1814058116
published as PNAS July 30, 2019 116 (31) 15344-15349 · YBS and SH contributed equally to this work. 7 pages, 4 figures (+ Appendix: 9 pages, 10 figures)
arxiv created 2019/06/26 · arxiv updated 2019/08/22
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. Here we introduce data driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end-to-end to best satisfy the equations on a low resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in one spatial dimension at resolutions 4-8x coarser than is possible with standard finite difference methods.