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SUMMARY:A theoretical analysis of machine learning and partial differentia
 l equations
DTSTART:20190409T161500
DTEND:20190409T180000
DTSTAMP:20260510T081932Z
UID:2605af6d29b329de71f444d9e608f4910665ea40a2d3d00bb5d33933
CATEGORIES:Conferences - Seminars
DESCRIPTION:Philipp Petersen (University of Oxford)\nComputational Mathema
 tics Seminar\n\nAbstract :\nNovel machine learning techniques based on dee
 p learning have achieved remarkable results in many areas such as image cl
 assification and speech recognition. As a result\, many scholars have star
 ted using them in areas which are not traditionally associated with machin
 e learning. For instance\, more and more researchers are employing deep ne
 ural networks to develop tools for the discretisation and solution of part
 ial differential equations. Two reasons can be identified to be the drivin
 g forces behind the increased interest in neural networks in the area of t
 he numerical analysis of PDEs. On the one hand\, powerful approximation th
 eoretical results have been established which demonstrate that neural netw
 orks can represent functions from the most relevant function classes with 
 a minimal number of parameters. On the other hand\, highly efficient machi
 ne learning techniques for the training of these networks are now availabl
 e and can be used as a black box. In this talk\, we will give an overview 
 of some approaches towards the numerical\ntreatment of PDEs with neural ne
 tworks and study the two aspects above. We will recall classical and some 
 novel approximation theoretical results and tie these results to PDE discr
 etisation. Additionally\, we will present theoretical results that show th
 at neural networks can very efficiently solve parametric PDEs without curs
 e of dimension if these parametric PDEs admit a sufficiently small reduced
  basis. \nProviding a counterpoint\, we analyse the structure of network 
 spaces and deduce considerable problems for the black box solver. In parti
 cular\, we will identify a number of structural properties of the set of n
 eural networks that render optimisation over this set especially challengi
 ng and sometimes impossible. \n 
LOCATION:MA A1 10 https://plan.epfl.ch/?room=MAA110
STATUS:CONFIRMED
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