Learning data-driven discretizations for partial differential equations

Yohai Bar-Sinai*, Stephan Hoyer, Jason Hickey, Michael P. Brenner

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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× to 8× coarser than is possible with standard finite-difference methods.

Original languageEnglish
Pages (from-to)15344-15349
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Volume116
Issue number31
DOIs
StatePublished - 30 Jul 2019
Externally publishedYes

Keywords

  • Coarse graining
  • Computational physics
  • Machine learning

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