@article {Bar-Sinai15344,
author = {Bar-Sinai, Yohai and Hoyer, Stephan and Hickey, Jason and Brenner, Michael P.},
title = {Learning data-driven discretizations for partial differential equations},
volume = {116},
number = {31},
pages = {15344--15349},
year = {2019},
doi = {10.1073/pnas.1814058116},
publisher = {National Academy of Sciences},
abstract = {In many physical systems, the governing equations are known with high confidence, but direct numerical solution is prohibitively expensive. Often this situation is alleviated by writing effective equations to approximate dynamics below the grid scale. This process is often impossible to perform analytically and is often ad hoc. Here we propose data-driven discretization, a method that uses machine learning to systematically derive discretizations for continuous physical systems. On a series of model problems, data-driven discretization gives accurate solutions with a dramatic drop in required resolution.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 1 spatial dimension at resolutions 4{\texttimes} to 8{\texttimes} coarser than is possible with standard finite-difference methods.},
issn = {0027-8424},
URL = {https://www.pnas.org/content/116/31/15344},
eprint = {https://www.pnas.org/content/116/31/15344.full.pdf},
journal = {Proceedings of the National Academy of Sciences}
}