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A Theoretical Analysis of Deep Neural Networks and Parametric PDEs

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arxiv 1904.00377 v3 pith:XSQUNK7M submitted 2019-03-31 math.NA cs.LGcs.NAmath.FAstat.ML

Gitta Kutyniok , Philipp Petersen , Mones Raslan , Reinhold Schneider This is my paper · ORCID
classification math.NAcs.LGcs.NAmath.FAstat.ML
keywords networksneuralparametricsolutionapproximationbasisdifferentialequations
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We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Deep neural network approximations for Monte Carlo algorithms

    math.NA 2019-08 conditional novelty 6.0 of 10

    A general theorem shows that neural networks inherit the absence of the curse of dimensionality from any discrete Monte Carlo scheme they can emulate, with applications to Kolmogorov PDEs.

  2. Space-time error estimates for deep neural network approximations for differential equations

    math.NA 2019-08 accept novelty 6.0 of 10

    The paper proves the first space-time error estimates for deep ReLU network approximations of Euler approximations of perturbed differential equations.

  3. Reduced Order Models and Conditional Expectation -- Analysing Parametric Low-Order Approximations

    cs.LG 2024-12 conditional novelty 4.0 of 10

    Parametric reduced-order models built by least-squares projection, including POD, reduced basis methods, and Gaussian process emulation, can be viewed as conditional expectations in a Bayesian updating framework.

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