Zoom link:
https://wpi.zoom.us/j/4833920729?omn=98284733385
Meeting ID: 483 392 0729
Title: Physics-Informed Neural Networks for High Dimensional Partial Differential Equations Arising from Stochastic Dynamics
Abstract: 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) and Hamilton-Jacobi-Bellman (HJB) equations for Langevin diffusions with controls. Classical grid-based numerical solvers face difficulties from unbounded domains and density normalization for FP, and fully nonlinear operators 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 for HJB that are costly in computation.
Based on Physics-informed neural networks(PINNs), I address three tasks in high dimension (dimension greater than 3): (i) steady-state FP; (ii) ex-ploratory HJB and its use to approximate the classical HJB; and (iii) Langevin-based non-convex optimization with state-dependent temperature. To efficiently solve FP equations, we apply (a) tensor neural networks with efficient auto-differentiation(AD); (b) SDE-guided numerical-support; and (c) accurate nor-malization for probability density. For HJB, we solve the exploratory PDE di-rectly (no policy iteration) by embedding the log-integral operator in the residual with a numerically stable scheme for small exploration weight λ; the result ap-proximates the classical HJB at small λ. We then deploy the resulting solver to design state-dependent temperature schedules for Langevin-based non-convex optimization.
For FP, our method in 6-10 dimensions achieves less than 10% relative er-ror. For HJB, current experiments show that exploratory solutions with our methods accurately approximate their classical counterparts at small λ; Prelim-inary experiments with Langevin diffusion indicate promising performance for non-convex optimization with learned temperature schedules.
I request committee’s feedback on two issues: (1) my work on Fokker-Planck equation solvers; and (2) my ongoing development of the numerical solver for exploratory HJB and its application to Langevin diffusion with state-dependent temperature for non-convex optimization. For (2), I will add experiments on (a) high-dimensional exploratory HJB with convergence tests toward the classical HJB as exploration weight λ goes to 0, and (b) different algorithmic design based on the obtained state-dependent temperature and additional non-convex optimization benchmarks.
Dissertation Committee:
Prof. Zhongqiang Zhang (advisor), Worcester Polytechnic Institute
Prof. Gu Wang, Worcester Polytechnic Institute
Prof. Xun Li, external member, Hong Kong Polytechnic University
Prof. Marcus Sarkis-Martins, Worcester Polytechnic Institute
Prof. Jacob Whitehill, Worcester Polytechnic Institute