Learning data-driven discretizations for partial differential equations. Bar-Sinai, Y., Hoyer, S., Hickey, J., & Brenner, M. P Proc. Natl. Acad. Sci. U. S. A., 116(31):15344–15349, July, 2019.
abstract   bibtex   
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.
@ARTICLE{Bar-Sinai2019-mn,
  title    = "Learning data-driven discretizations for partial differential
              equations",
  author   = "Bar-Sinai, Yohai and Hoyer, Stephan and Hickey, Jason and
              Brenner, Michael P",
  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$\times$ to 8$\times$ coarser
              than is possible with standard finite-difference methods.",
  journal  = "Proc. Natl. Acad. Sci. U. S. A.",
  volume   =  116,
  number   =  31,
  pages    = "15344--15349",
  month    =  jul,
  year     =  2019,
  keywords = "coarse graining; computational physics; machine learning",
  language = "en"
}

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