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Title: Learning data-driven discretizations for 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. 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.  more » « less
Award ID(s):
1715477
PAR ID:
10108709
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
Proceedings of the National Academy of Sciences
Volume:
116
Issue:
31
ISSN:
0027-8424
Page Range / eLocation ID:
15344 to 15349
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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