Neural networks for first order HJB equations and application to front propagation with obstacle terms

Olivier Bokanowski (), Averil Prost () and Xavier Warin ()
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Olivier Bokanowski: Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL)
Averil Prost: INSA Rouen Normandie, Normandie Univ, LMI UR 3226
Xavier Warin: EDF R &D and FiME

Partial Differential Equations and Applications, 2023, vol. 4, issue 5, 1-36

Abstract: Abstract We consider a deterministic optimal control problem, focusing on a finite horizon scenario. Our proposal involves employing deep neural network approximations to capture Bellman’s dynamic programming principle. This also corresponds to solving first-order Hamilton–Jacobi–Bellman (HJB) equations. Our work builds upon the research conducted by Huré et al. (SIAM J Numer Anal 59(1):525–557, 2021), which primarily focused on stochastic contexts. However, our objective is to develop a completely novel approach specifically designed to address error propagation in the absence of diffusion in the dynamics of the system. Our analysis provides precise error estimates in terms of an average norm. Furthermore, we provide several academic numerical examples that pertain to front propagation models incorporating obstacle constraints, demonstrating the effectiveness of our approach for systems with moderate dimensions (e.g., ranging from 2 to 8) and for nonsmooth value functions.

Keywords: Neural networks; Deterministic optimal control; Dynamic programming principle; First order Hamilton–Jacobi–Bellman equation; State constraints; 35F21; 49L20; 68T07 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-023-00258-8

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