Operator compression with deep neural networks View Full Text


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Article Info

DATE

2022-04-09

AUTHORS

Fabian Kröpfl, Roland Maier, Daniel Peterseim

ABSTRACT

This paper studies the compression of partial differential operators using neural networks. We consider a family of operators, parameterized by a potentially high-dimensional space of coefficients that may vary on a large range of scales. Based on the existing methods that compress such a multiscale operator to a finite-dimensional sparse surrogate model on a given target scale, we propose to directly approximate the coefficient-to-surrogate map with a neural network. We emulate local assembly structures of the surrogates and thus only require a moderately sized network that can be trained efficiently in an offline phase. This enables large compression ratios and the online computation of a surrogate based on simple forward passes through the network is substantially accelerated compared to classical numerical upscaling approaches. We apply the abstract framework to a family of prototypical second-order elliptic heterogeneous diffusion operators as a demonstrating example. More... »

PAGES

29

References to SciGraph publications

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  • 2020-10-29. Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized Equations in JOM
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  • 2017-11-10. Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations in COMMUNICATIONS IN MATHEMATICS AND STATISTICS
  • 2018-10-17. Explicit computational wave propagation in micro-heterogeneous media in BIT NUMERICAL MATHEMATICS
  • 2021-06-02. A Theoretical Analysis of Deep Neural Networks and Parametric PDEs in CONSTRUCTIVE APPROXIMATION
  • 2014-09-20. Computation of eigenvalues by numerical upscaling in NUMERISCHE MATHEMATIK
  • 2020-04-06. A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations in PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS
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  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1186/s13662-022-03702-y

    DOI

    http://dx.doi.org/10.1186/s13662-022-03702-y

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1146967700

    PUBMED

    https://www.ncbi.nlm.nih.gov/pubmed/35531267


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