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DTSTAMP:20260512T065748Z
SUMMARY:Control and Optimisation seminar – Yun Zhao (Imperial College Lon
 don)
DESCRIPTION:Title\nDeep Neural Network Convergence for Variational Inequali
 ties\nAbstract\nMotivated by recent progresses in applying deep neural net
 works to high-dimensional PDEs\, we propose an approach to apply them on l
 inear parabolic variational inequalities. We begin by developing a deep ne
 ural network framework that approximates the solution in a bounded domain\
 , using loss functions that directly incorporate the variational inequalit
 y on the whole domain\, so there is no need to determine the stopping regi
 on in advance. Crucially\, we design the loss function by the inverse trac
 e theorem\, and prove the existence of neural networks whose losses conver
 ge to zero. Moreover\, we prove the functional convergence in the Sobolev 
 space $H^{0\,1}(\\Omega_T)$. To align with most optimal stopping problems\
 , we extend these results to unbounded domains\, ensuring that our neural 
 network surrogates satisfy boundary conditions while maintaining convergen
 ce guarantees.\nWe then apply our approach to a specific mixed optimal sto
 pping and control problem in finance. By leveraging duality\, the nonlinea
 r HJB-type operator of the primal problem is converted into a linear parab
 olic operator in the dual formulation. A key step in this process is to pr
 ove the convergence of the primal value function from the dual neural netw
 ork solution — an outcome made possible by our Sobolev norm analysis. Af
 ter the theoretical convergence analysis\, we illustrate the versatility a
 nd accuracy of our method with numerical experiments for both power and no
 n-HARA utilities\, and discuss practical considerations such as domain tru
 ncation and sampling strategies. Our results underscore the potential of d
 eep neural networks as a reliable and efficient tool for variational inequ
 alities in optimization and control problems.\nBio\n\nYun Zhao is a doctor
 al student in the Department of Mathematics at Imperial College London.\n
URL:https://www.imperial.ac.uk/events/196501/control-and-optimisation-semin
 ar-yun-zhao-imperial-college-london/
DTSTART;TZID=Europe/London:20250806T144500
DTEND;TZID=Europe/London:20250806T153000
LOCATION:503\, Huxley Building\, South Kensington Campus\, Imperial College
  London\, London\, SW7 2AZ\, United Kingdom
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