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

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  • Olivier Bokanowski

    (Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL)
    Sorbonne Université, CNRS, LJLL)

  • Averil Prost

    (INSA Rouen Normandie, Normandie Univ, LMI UR 3226)

  • Xavier Warin

    (EDF R &D and FiME)

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.

Suggested Citation

  • Olivier Bokanowski & Averil Prost & Xavier Warin, 2023. "Neural networks for first order HJB equations and application to front propagation with obstacle terms," Partial Differential Equations and Applications, Springer, vol. 4(5), pages 1-36, October.
    • Handle: RePEc:spr:pardea:v:4:y:2023:i:5:d:10.1007_s42985-023-00258-8
    DOI: 10.1007/s42985-023-00258-8
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    References listed on IDEAS

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    1. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1793-1812, December.
    2. Christophe Barrera-Esteve & Florent Bergeret & Charles Dossal & Emmanuel Gobet & Asma Meziou & Rémi Munos & Damien Reboul-Salze, 2006. "Numerical Methods for the Pricing of Swing Options: A Stochastic Control Approach," Methodology and Computing in Applied Probability, Springer, vol. 8(4), pages 517-540, December.
    3. Achref Bachouch & Côme Huré & Nicolas Langrené & Huyên Pham, 2022. "Deep Neural Networks Algorithms for Stochastic Control Problems on Finite Horizon: Numerical Applications," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 143-178, March.
    4. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
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