Repository: Freie Universität Berlin, Math Department

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

Nüsken, N. and Richter, L. (2020) Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space. SFB 1114 Preprint in arXiv . pp. 1-40. (Submitted)

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Official URL: https://arxiv.org/abs/2005.05409

Abstract

Optimal control of diffusion processes is intimately connected to the problem of solving certain Hamilton- Jacobi-Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate the potential of iterative diffusion optimisation techniques, in particular considering applications in importance sampling and rare event simulation. The choice of an appropriate loss function being a central element in the algorithmic design, we develop a principled framework based on divergences between path measures, encompassing various existing methods. Motivated by connections to forward-backward SDEs, we propose and study the novel log-variance divergence, showing favourable properties of corresponding Monte Carlo estimators. The promise of the developed approach is exemplified by a range of high-dimensional and metastable numerical examples.

Item Type:Article
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics
ID Code:2435
Deposited By: Monika Drueck
Deposited On:25 May 2020 09:05
Last Modified:25 May 2020 09:05

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