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SUMMARY:Statistical guarantees for neural operator surrogates - Sven Wang 
 (EPFL)
DTSTART:20251114T140000Z
DTEND:20251114T150000Z
UID:TALK237517@talks.cam.ac.uk
CONTACT:Qingyuan Zhao
DESCRIPTION:In recent years\, "operator learning" methodologies for constr
 ucting data-driven surrogates for non-linear operators are gaining widespr
 ead attention. We present statistical convergence results for the learning
  of such non-linear mappings in infinite-dimensional spaces\, e.g. arising
  from PDEs\, given noisy input-output pairs. We provide convergence result
 s for least-squares-type empirical risk minimizers over general classes\, 
 in terms of their approximation properties and metric entropy bounds. This
  generalizes classical results from finite-dimensional nonparametric regre
 ssion to an infinite-dimensional setting.\n \nAssuming $G_0$ to be holomor
 phic\, we prove algebraic (in the sample size $n$) convergence rates in th
 is setting\, thereby overcoming the curse of dimensionality. To illustrate
  the wide applicability\, as a prototypical example we discuss the learnin
 g of the non-linear solution operator to a parametric elliptic partial dif
 ferential equation\, with an encoder-decoder based neural operator archite
 cture.
LOCATION:MR12\, Centre for Mathematical Sciences
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