IDEAS home Printed from https://ideas.repec.org/a/spr/pardea/v4y2023i5d10.1007_s42985-023-00258-8.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s42985-023-00258-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s42985-023-00258-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Laurens Van Mieghem & Antonis Papapantoleon & Jonas Papazoglou-Hennig, 2023. "Machine learning for option pricing: an empirical investigation of network architectures," Papers 2307.07657, arXiv.org.
    2. Dong, Wenfeng & Kang, Boda, 2019. "Analysis of a multiple year gas sales agreement with make-up, carry-forward and indexation," Energy Economics, Elsevier, vol. 79(C), pages 76-96.
    3. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Working Papers hal-03145949, HAL.
    4. Ben Zineb Tarik & Gobet Emmanuel, 2013. "Preliminary control variates to improve empirical regression methods," Monte Carlo Methods and Applications, De Gruyter, vol. 19(4), pages 331-354, December.
    5. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Deep Stochastic Optimization in Finance," Papers 2205.04604, arXiv.org.
    6. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.
    7. Rong Du & Duy-Minh Dang, 2023. "Fourier Neural Network Approximation of Transition Densities in Finance," Papers 2309.03966, arXiv.org.
    8. Ali Al-Aradi & Adolfo Correia & Danilo de Frietas Naiff & Gabriel Jardim & Yuri Saporito, 2019. "Extensions of the Deep Galerkin Method," Papers 1912.01455, arXiv.org, revised Apr 2022.
    9. Maximilien Germain & Huyên Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance ," Post-Print hal-03115503, HAL.
    10. Weinan E & Martin Hutzenthaler & Arnulf Jentzen & Thomas Kruse, 2021. "Multilevel Picard iterations for solving smooth semilinear parabolic heat equations," Partial Differential Equations and Applications, Springer, vol. 2(6), pages 1-31, December.
    11. Salah A. Faroughi & Ramin Soltanmohammadi & Pingki Datta & Seyed Kourosh Mahjour & Shirko Faroughi, 2023. "Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media," Mathematics, MDPI, vol. 12(1), pages 1-23, December.
    12. Jiequn Han & Ruimeng Hu & Jihao Long, 2020. "Convergence of Deep Fictitious Play for Stochastic Differential Games," Papers 2008.05519, arXiv.org, revised Mar 2021.
    13. Dehghani, Hamidreza & Zilian, Andreas, 2021. "A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 398-417.
    14. Christian Bender & Nikolaus Schweizer & Jia Zhuo, 2013. "A primal-dual algorithm for BSDEs," Papers 1310.3694, arXiv.org, revised Sep 2014.
    15. repec:dau:papers:123456789/11439 is not listed on IDEAS
    16. Polynice Oyono Ngou & Cody Hyndman, 2014. "A Fourier interpolation method for numerical solution of FBSDEs: Global convergence, stability, and higher order discretizations," Papers 1410.8595, arXiv.org, revised May 2022.
    17. Chong, Wing Fung, 2019. "Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 93-107.
    18. Jialiang Luo & Harry Zheng, 2023. "Deep Neural Network Solution for Finite State Mean Field Game with Error Estimation," Dynamic Games and Applications, Springer, vol. 13(3), pages 859-896, September.
    19. Qiang Han & Shaolin Ji, 2022. "A Multi-Step Algorithm for BSDEs Based On a Predictor-Corrector Scheme and Least-Squares Monte Carlo," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2403-2426, December.
    20. Ying Li & Longxiang Xu & Shihui Ying, 2022. "DWNN: Deep Wavelet Neural Network for Solving Partial Differential Equations," Mathematics, MDPI, vol. 10(12), pages 1-35, June.
    21. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions(Revised version of CARF-F-387)," CARF F-Series CARF-F-398, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:pardea:v:4:y:2023:i:5:d:10.1007_s42985-023-00258-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.