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SUMMARY:Yulong Lu (University of Massachusetts)
DTSTART:20210629T140000Z
DTEND:20210629T150000Z
DTSTAMP:20260423T034447Z
UID:DSCSS/22
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/DSCSS/22/">A
  priori generalization error analysis of neural network methods for solvin
 g high dimensional elliptic PDEs</a>\nby Yulong Lu (University of Massachu
 setts) as part of Data Science and Computational Statistics Seminar\n\n\nA
 bstract\nNeural network-based machine learning methods\, including the mos
 t notably deep learning have achieved extraordinary successes in numerous 
 fields. Despite the rapid development of learning algorithms based on neur
 al networks\, their mathematical analysis is far from understood. In parti
 cular\, it has been a big mystery that neural network-based machine learni
 ng methods work extremely well for solving high dimensional problems.\n\nI
 n this talk\, we will demonstrate the power of neural network methods for 
 solving high dimensional elliptic PDEs. Specifically\, we will discuss an 
 a priori generalization error analysis of the Deep Ritz Method for solving
  two classes of high dimensional Schrödinger problems: the stationary Sch
 rödinger equation and the ground state of Schrödinger operator.  Assumin
 g the exact solution or the ground state lies in a low-complexity function
  space called spectral Barron space\, we show that the convergence rate of
  the generalization error is independent of dimension. We also develop a n
 ew regularity theory for the PDEs of consideration on the spectral Barron 
 space. This can be viewed as an analog of the classical Sobolev regularity
  theory for PDEs.\n
LOCATION:https://researchseminars.org/talk/DSCSS/22/
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