Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space

Abstract Optimal control of diffusion processes is intimately connected to the problem solving certain Hamilton–Jacobi–Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate potential iterative optimisation techniques, in particular considering applications importance sampling and rare event simulation, focusing problems without control, with linearly controlled drift running costs that depend quadratically control. More generally, our methods apply nonlinear parabolic PDEs a shift invariance. The choice an appropriate loss function being central element algorithmic design, develop principled framework based divergences between path measures, encompassing various existing methods. Motivated by connections forward-backward SDEs, propose study novel log-variance divergence, showing favourable properties corresponding Monte Carlo estimators. promise developed approach exemplified range metastable numerical examples.