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SEQUENCE:1
X-APPLE-TRAVEL-ADVISORY-BEHAVIOR:AUTOMATIC
224601
20250929T091755Z
DTSTART;TZID=America/New_York:20251015T093000
DTEND;TZID=America/New_York:2
 0251015T103000
URL;TYPE=URI:/news/calendar/events/depar
 tment-mathematical-sciences-taouri-wang-dissertation-proposal
Department of Mathematical Sciences: Taouri Wang, Dissertation Proposal
\n\n\n      \n      \n\n\n\nZoom link:\nhttps://wpi.zoom.us/j/4833920729?om
 n=98284733385\nMeeting ID: 483 392 0729\n\nTitle: Physics-Informed Neural 
 Networks for High Dimensional Partial Differential Equations Arising from 
 Stochastic Dynamics\n\nAbstract: Partial differential equations connected 
 with stochastic dynamics are widely used in modeling, control, and modern 
 optimization. In this work, we focus on two such equations: Fokker-Planck 
 (FP) equations for densities of stochas-tic differential equations(SDEs) a
 nd Hamilton-Jacobi-Bellman (HJB) equations for Langevin diffusions with co
 ntrols. Classical grid-based numerical solvers face difficulties from unbo
 unded domains and density normalization for FP, and fully nonlinear operat
 ors for HJB. These issues exacerbated at high dimension. Ex-isting neural 
 solvers for high dimensional problems often require heavy Monte-Carlo data
 , struggle with normalization for FP, or rely on policy-iteration loops fo
 r HJB that are costly in computation.\nBased on Physics-informed neural ne
 tworks(PINNs), I address three tasks in high dimension (dimension greater 
 than 3): (i) steady-state FP; (ii) ex-ploratory HJB and its use to approxi
 mate the classical HJB; and (iii) Langevin-based non-convex optimization w
 ith state-dependent temperature. To efficiently solve FP equations, we app
 ly (a) tensor neural networks with efficient auto-differentiation(AD); (b)
  SDE-guided numerical-support; and (c) accurate nor-malization for probabi
 lity density. For HJB, we solve the exploratory PDE di-rectly (no policy i
 teration) by embedding the log-integral operator in the residual with a nu
 merically stable scheme for small exploration weight 位; the result ap-pro
 ximates the classical HJB at small 位. We then deploy the resulting solver
  to design state-dependent temperature schedules for Langevin-based non-co
 nvex optimization.\nFor FP, our method in 6-10 dimensions achieves less th
 an 10% relative er-ror. For HJB, current experiments show that exploratory
  solutions with our methods accurately approximate their classical counter
 parts at small 位; Prelim-inary experiments with Langevin diffusion indica
 te promising performance for non-convex optimization with learned temperat
 ure schedules.\nI request committee鈥檚 feedback on two issues: (1) my wor
 k on Fokker-Planck equation solvers; and (2) my ongoing development of the
  numerical solver for exploratory HJB and its application to Langevin diff
 usion with state-dependent temperature for non-convex optimization. For (2
 ), I will add experiments on (a) high-dimensional exploratory HJB with con
 vergence tests toward the classical HJB as exploration weight 位 goes to 0
 , and (b) different algorithmic design based on the obtained state-depende
 nt temperature and additional non-convex optimization benchmarks.\n\nDisse
 rtation Committee:\nProf. Zhongqiang Zhang (advisor), Worcester Polytechni
 c InstituteProf. Gu Wang, Worcester Polytechnic InstituteProf. Xun Li, ext
 ernal member, Hong Kong Polytechnic UniversityProf. Marcus Sarkis-Martins,
  Worcester Polytechnic InstituteProf. Jacob Whitehill, Worcester Polytechn
 ic Institute\n
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