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Representation Results and Error Estimates for Differential Games with Applications Using Neural Networks

Author

Listed:
  • Olivier Bokanowski

    (Université Paris Cité
    Sorbonne Université)

  • Xavier Warin

    (EDF R&D & FiME)

Abstract

We study deterministic optimal control problems for differential games with finite horizon. We propose new approximations of the strategies in feedback form and show error estimates and a convergence result of the value in some weak sense for one of the formulations. This result applies in particular to neural network approximations. This work follows some ideas introduced in Bokanowski, Prost and Warin (PDEA, 2023) for deterministic optimal control problems, yet with a simplified approach for the error estimates, which allows to consider a global optimization scheme instead of a time-marching scheme. We also give a new approximation result between the continuous and the semi-discrete optimal control value in the game setting, improving the classical convergence order $$O({\Delta t}^{1/2})$$ O ( Δ t 1 / 2 ) to $$O({\Delta t})$$ O ( Δ t ) , under some assumptions on the dynamical system. Numerical examples are performed on elementary academic problems related to backward reachability, with exact analytic solutions given, as well as a two-player game in the presence of state constraints, using stochastic gradient-type algorithms to deal with the minimax problem.

Suggested Citation

  • Olivier Bokanowski & Xavier Warin, 2025. "Representation Results and Error Estimates for Differential Games with Applications Using Neural Networks," Dynamic Games and Applications, Springer, vol. 15(2), pages 417-453, May.
    • Handle: RePEc:spr:dyngam:v:15:y:2025:i:2:d:10.1007_s13235-024-00597-0
    DOI: 10.1007/s13235-024-00597-0
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    References listed on IDEAS

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    1. 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.
    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. Piernicola Bettiol & Pierre Cardaliaguet & Marc Quincampoix, 2006. "Zero-sum state constrained differential games: existence of value for Bolza problem," International Journal of Game Theory, Springer;Game Theory Society, vol. 34(4), pages 495-527, November.
    4. R. Buckdahn & P. Cardaliaguet & M. Quincampoix, 2011. "Some Recent Aspects of Differential Game Theory," Dynamic Games and Applications, Springer, vol. 1(1), pages 74-114, March.
    5. 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.
    6. repec:dau:papers:123456789/6046 is not listed on IDEAS
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