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

• URL: http://arxiv.org/abs/2005.05409v1
• Date: Mon, 11 May 2020 20:14:02 GMT
• Title: Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space
• Authors: Nikolas N\"usken, Lorenz Richter
• Abstract summary: Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate the potential of iterative diffusion techniques. We develop a principled framework based on divergences between path measures, encompassing various existing methods. The promise of the developed approach is exemplified by a range of high-dimensional and metastable numerical examples.
• Score: 3.1219977244201056
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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