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dc.contributor.author
Lanthaler, Samuel
dc.contributor.author
Mishra, Siddhartha
dc.contributor.author
Karniadakis, George
dc.date.accessioned
2021-06-24T09:43:41Z
dc.date.available
2021-06-24T08:20:34Z
dc.date.available
2021-06-24T09:43:41Z
dc.date.issued
2021-02
dc.identifier.uri
http://hdl.handle.net/20.500.11850/491123
dc.description.abstract
DeepOnets have recently been proposed as a framework for learning nonlinear operators mapping between infinite dimensional Banach spaces. We analyze DeepOnets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepOnets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ODE, an elliptic PDE with variable coefficients and nonlinear parabolic and hyperbolic PDEs. In all these examples, we prove that DeepOnets \emph{break the curse of dimensionality}, thus demonstrating the efficient approximation of infinite-dimensional operators with this machine learning framework.
en_US
dc.language.iso
en
en_US
dc.publisher
Seminar for Applied Mathematics, ETH Zurich
en_US
dc.subject
Deep learning
en_US
dc.subject
Operator learning
en_US
dc.title
Error estimates for DeepOnets: A deep learning framework in infinite dimensions
en_US
dc.type
Report
ethz.journal.title
SAM Research Report
ethz.journal.volume
2021-07
en_US
ethz.size
113 p.
en_US
ethz.publication.place
Zurich
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03851 - Mishra, Siddhartha / Mishra, Siddhartha
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03851 - Mishra, Siddhartha / Mishra, Siddhartha
en_US
ethz.identifier.url
https://math.ethz.ch/sam/research/reports.html?id=949
ethz.relation.isPreviousVersionOf
10.3929/ethz-b-000558811
ethz.date.deposited
2021-06-24T08:20:41Z
ethz.source
FORM
ethz.eth
yes
en_US
ethz.identifier.internal
https://math.ethz.ch/sam/research/reports.html?id=949
en_US
ethz.availability
Metadata only
en_US
ethz.rosetta.installDate
2021-06-24T09:43:47Z
ethz.rosetta.lastUpdated
2021-06-24T09:43:47Z
ethz.rosetta.versionExported
true
ethz.COinS
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