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A multi-level procedure for enhancing accuracy of machine learning algorithms

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  14 July 2020

KJETIL O. LYE ,
SIDDHARTHA MISHRA  and
ROBERTO MOLINARO Open the ORCID record for ROBERTO MOLINARO [Opens in a new window]
KJETIL O. LYE
Affiliation:
SINTEF Digital, Oslo, Norway, email: kjetil.olsen.lye@sintef.no
SIDDHARTHA MISHRA
Affiliation:
Seminar for Applied Mathematics (SAM), D-Math, ETH Zürich, Rämistrasse 101, Zürich-8092, Switzerland, emails: smishra@sam.math.ethz.ch; roberto.molinaro@sam.math.ethz.ch
ROBERTO MOLINARO
Affiliation:
Seminar for Applied Mathematics (SAM), D-Math, ETH Zürich, Rämistrasse 101, Zürich-8092, Switzerland, emails: smishra@sam.math.ethz.ch; roberto.molinaro@sam.math.ethz.ch
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Abstract

We propose a multi-level method to increase the accuracy of machine learning algorithms for approximating observables in scientific computing, particularly those that arise in systems modelled by differential equations. The algorithm relies on judiciously combining a large number of computationally cheap training data on coarse resolutions with a few expensive training samples on fine grid resolutions. Theoretical arguments for lowering the generalisation error, based on reducing the variance of the underlying maps, are provided and numerical evidence, indicating significant gains over underlying single-level machine learning algorithms, are presented. Moreover, we also apply the multi-level algorithm in the context of forward uncertainty quantification and observe a considerable speedup over competing algorithms.

Keywords

Variance ReductionMulti-level AlgorithmsMachine LearningUncertainity Quantification

MSC classification

Primary: 65M06: Finite difference methods
Secondary: 65M08: Finite volume methods 65M75: Probabilistic methods, particle methods, etc.
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 3: Connections between Deep learning and Partial Differential Equations , June 2021 , pp. 436 - 469
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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