Recent Advances in Numerical Solutions for Hamilton-Jacobi PDEs

Abstract

Hamilton-Jacobi partial differential equations (HJ PDEs) play a central role in many applications such as economics, physics, and engineering. These equations describe the evolution of a value function that encodes valuable information about the system, such as action, cost, or level sets of a dynamic process. Their importance lies in their ability to model diverse phenomena, ranging from the propagation of fronts in computational physics to optimal decision-making in control systems. This paper provides a review of some recent advances in numerical methods to address challenges such as high dimensionality, nonlinearity, and computational efficiency. By examining these developments, this paper sheds light on important techniques and emerging directions in the numerical solution of HJ PDEs.

Keywords

Hamilton-Jacobi partial differential equations (HJ PDEs) / Numerical methods / Optimal control / 35F21 / 49M29

References

[1] Abedian, R.: WENO schemes with adaptive order for Hamilton-Jacobi equations. Int. J. Mod. Phys. C 34(06), 2350081 (2023)

[2] Abedian, R., Dehghan, M.: RBF-ENO/WENO schemes with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations. Numer. Methods Partial Differ. Equ. 37(1), 594–613 (2021)

[3] Abgrall, R.: Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes. Commun. Pure Appl. Math. 49(12), 1339–1373 (1996)

[4] Agrawal, S., Lee, W., Fung, S.W., Nurbekyan, L.: Random features for high-dimensional nonlocal mean-field games. J. Comput. Phys. 459, 111136 (2022)

[5] Bansal, S., Chen, M., Herbert, S., Tomlin, C.J.: Hamilton-Jacobi reachability: a brief overview and recent advances. In: 2017 IEEE 56th Annual Conference on Decision and Control (CDC), pp. 2242–2253. IEEE (2017)

[6] Bansal, S., Tomlin, C.J.: Deepreach: a deep learning approach to high-dimensional reachability. In: 2021 IEEE International Conference on Robotics and Automation (ICRA), pp. 1817–1824. IEEE (2021)

[7] Bardi M, Capuzzo-Dolcetta I. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, 1997. MA, Birkhäuser Boston.

[8] Bardi, M., Faggian, S.: Hopf-type estimates and formulas for nonconvex nonconcave Hamilton-Jacobi equations. SIAM J. Math. Anal. 29(5), 1067–1086 (1998)

[9] Bażański SL. Hamilton-Jacobi formalism for geodesics and geodesic deviations. J. Math. Phys., 1989, 30(5): 1018-1029.

[10] Beck C, Hutzenthaler M, Jentzen A, Kuckuck B. An overview on deep learning-based approximation methods for partial differential equations. Discrete Contin. Dyn. Syst. Ser. B, 2023, 28(6): 3697-3746.

[11] Carmona R, Laurière M. Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games I: the ergodic case. SIAM J. Numer. Anal., 2021, 59(3): 1455-1485.

[12] Carmona R, Laurière M. Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: II—the finite horizon case. Ann. Appl. Probab., 2022, 32(6): 4065-4105.

[13] Chambolle A, Pock T. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis., 2011, 40: 120-145.

[14] Chen M, Herbert SL, Hu H, Pu Y, Fisac JF, Bansal S, Han S, Tomlin CJ. FaSTrack: a modular framework for real-time motion planning and guaranteed safe tracking. IEEE Trans. Autom. Control, 2021, 66(12): 5861-5876.

[15] Chen P, Darbon J, Meng T. Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems. Comput. Math. Appl., 2024, 161: 90-120.

[16] Chen, P., Darbon, J., Meng, T.: Lax-Oleinik-type formulas and efficient algorithms for certain high-dimensional optimal control problems. Commun. Appl. Math. Comput. 6(2), 1428–1471 (2024)

[17] Cheng, Y., Shu, C.-W.: A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations. J. Comput. Phys. 223(1), 398–415 (2007)

[18] Cheng, Y., Wang, Z.: A new discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations. J. Comput. Phys. 268, 134–153 (2014)

[19] Chow, Y.T., Darbon, J., Osher, S., Yin, W.: Algorithm for overcoming the curse of dimensionality for time-dependent non-convex Hamilton-Jacobi equations arising from optimal control and differential games problems. J. Sci. Comput. 73, 617–643 (2017)

[20] Chow, Y.T., Darbon, J., Osher, S., Yin, W.: Algorithm for overcoming the curse of dimensionality for certain non-convex Hamilton-Jacobi equations, projections and differential games. Ann. Math. Sci. Appl 3(2), 369–403 (2018)

[21] Chow, Y.T., Darbon, J., Osher, S., Yin, W.: Algorithm for overcoming the curse of dimensionality for state-dependent Hamilton-Jacobi equations. J. Comput. Phys. 387, 376–409 (2019)

[22] Chow, Y.T., Fung, S.W., Liu, S., Nurbekyan, L., Osher, S.: A numerical algorithm for inverse problem from partial boundary measurement arising from mean field game problem. Inverse Probl. 39(1), 014001 (2022)

[23] Chow, Y.T., Li, W., Osher, S., Yin, W.: Algorithm for Hamilton-Jacobi equations in density space via a generalized Hopf formula. J. Sci. Comput. 80, 1195–1239 (2019)

[24] Cockburn, B., Shu, C.-W.: The Runge-Kutta local projection-discontinuous-Galerkin finite element method for scalar conservation laws. ESAIM Math. Model. Numer. Anal. 25(3), 337–361 (1991)

[25] Crandal, M., Lions, P.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43(167), 1–19 (1984)

[26] Crandall, M.G., Evans, L.C., Lions, P.-L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282(2), 487–502 (1984)

[27] Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)

[28] Darbon, J., Dower, P.M., Meng, T.: Neural network architectures using min-plus algebra for solving certain high-dimensional optimal control problems and Hamilton-Jacobi PDEs. Math. Control Signals Syst. 35(1), 1–44 (2023)

[29] Darbon, J., Langlois, G.P.: On Bayesian posterior mean estimators in imaging sciences and Hamilton-Jacobi partial differential equations. J. Math. Imaging Vis. 63(7), 821–854 (2021)

[30] Darbon, J., Langlois, G.P., Meng, T.: Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures. Res. Math. Sci. 7(3), 20 (2020)

[31] Darbon, J., Langlois, G.P., Meng, T.: Connecting Hamilton-Jacobi partial differential equations with maximum a posteriori and posterior mean estimators for some non-convex priors. In: Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging: Mathematical Imaging and Vision, pp. 1–25 (2021)

[32] Darbon, J., Meng, T.: On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton-Jacobi partial differential equations. J. Comput. Phys. 425, 109907 (2021)

[33] Darbon J, Osher S. Algorithms for overcoming the curse of dimensionality for certain Hamilton-Jacobi equations arising in control theory and elsewhere. Res. Math. Sci., 2016, 3(1): 19.

[34] Di, N., Chi, E.C., Fung, S.W.: A Monte Carlo approach to nonsmooth convex optimization via proximal splitting algorithms. arXiv:2509.07914 (2025)

[35] Dower, P.M., McEneaney, W.M., Zhang, H.: Max-plus fundamental solution semigroups for optimal control problems. In: 2015 Proceedings of the Conference on Control and Its Applications, pp. 368–375. SIAM (2015)

[36] Esteve-Yagüe, C., Tsai, R., Massucco, A.: Finite-difference least square methods for solving Hamilton-Jacobi equations using neural networks. J. Comput. Phys. 524, 113721 (2025)

[37] Evans, L.C.: Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)

[38] Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, Philadelphia (2013)

[39] Falcone, M., Ferretti, R.: Numerical methods for Hamilton-Jacobi type equations. Handb. Numer. Anal. 17, 603–626 (2016)

[40] Fleming, W.H., McEneaney, W.M.: A max-plus-based algorithm for a Hamilton-Jacobi-Bellman equation of nonlinear filtering. SIAM J. Control. Optim. 38(3), 683–710 (2000)

[41] Fleming WH, Soner HM. Controlled Markov Processes and Viscosity Solutions, 2006. New York, Springer

[42] Gaubert, S., McEneaney, W., Qu, Z.: Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 1054–1061. IEEE (2011)

[43] Gelphman, E., Verma, D., Yang, N., Osher, S., Fung, S.W.: End-to-end training of high-dimensional optimal control with implicit Hamiltonians via Jacobian-free backpropagation. arXiv:2510.00359 (2025)

[44] Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial networks. Commun. ACM 63(11), 139–144 (2020)

[45] Guo W, Huang J, Tao Z, Cheng Y. An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions. J. Comput. Phys., 2021, 436. ArticleID: 110294

[46] Heaton H, Fung SW, Osher S. Global solutions to nonconvex problems by evolution of Hamilton-Jacobi PDEs. Commun. Appl. Math. Comput., 2024, 6(2): 790-810.

[47] Hu, C., Shu, C.-W.: A discontinuous Galerkin finite element method for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21(2), 666–690 (1999)

[48] Kaipio, J., Somersalo, E.: Statistical and computational inverse problems. Springer Science & Business Media (2006)

[49] Kim CH, Ha Y, Yang H, Yoon J. A third-order WENO scheme based on exponential polynomials for Hamilton-Jacobi equations. Appl. Numer. Math., 2021, 165: 167-183.

[50] Kim, J., Shin, J., Yang, I.: Hamilton-Jacobi deep Q-learning for deterministic continuous-time systems with Lipschitz continuous controls. J. Mach. Learn. Res. 22(206), 1–34 (2021)

[51] Kirchner, M.R., Hewer, G., Darbon, J., Osher, S.: A primal-dual method for optimal control and trajectory generation in high-dimensional systems. In: 2018 IEEE Conference on Control Technology and Applications (CCTA), pp. 1583–1590. IEEE (2018)

[52] Kirchner, M.R., Mar, R., Hewer, G., Darbon, J., Osher, S., Chow, Y.T.: Time-optimal collaborative guidance using the generalized Hopf formula. IEEE Control Syst. Lett. 2(2), 201–206 (2017)

[53] Klingenberg C, Schnücke G, Xia Y. An arbitrary Lagrangian-Eulerian local discontinuous Galerkin method for Hamilton-Jacobi equations. J. Sci. Comput., 2017, 73: 906-942.

[54] Laplace, P.S.: Mémoire sur la probabilité de causes par les événements. Mémoire de l’académie royale des sciences (1774)

[55] Lee, B., Darbon, J., Osher, S., Kang, M.: Revisiting the redistancing problem using the Hopf-Lax formula. J. Comput. Phys. 330, 268–281 (2017)

[56] Li F, Shu C-W. Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations. Appl. Math. Lett., 2005, 18(11): 1204-1209.

[57] Li, F., Yakovlev, S.: A central discontinuous Galerkin method for Hamilton-Jacobi equations. J. Sci. Comput. 45(1), 404–428 (2010)

[58] Lin AT, Fung SW, Li W, Nurbekyan L, Osher SJ. Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games. Proc. Natl. Acad. Sci., 2021, 118(31. ArticleID: e2024713118

[59] Liu, H., Pollack, M.: Alternating evolution discontinuous Galerkin methods for Hamilton-Jacobi equations. J. Comput. Phys. 258, 31–46 (2014)

[60] Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

[61] Margellos, K., Lygeros, J.: Hamilton-Jacobi formulation for reach-avoid differential games. IEEE Trans. Autom. Control 56(8), 1849–1861 (2011)

[62] McEneaney WM. Max-Plus Methods for Nonlinear Control and Estimation, 2006. Boston, Birkhäuser

[63] Meng, T., Hao, W., Liu, S., Osher, S.J., Li, W.: Primal-dual hybrid gradient algorithms for computing time-implicit Hamilton-Jacobi equations. arXiv:2310.01605 (2023)

[64] Meng, T., Liu, S., Li, W., Osher, S.: A primal-dual hybrid gradient method for solving optimal control problems and the corresponding Hamilton-Jacobi PDEs. arXiv:2403.02468 (2024)

[65] Mitchell, I.M., Bayen, A.M., Tomlin, C.J.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Autom. Control 50(7), 947–957 (2005)

[66] Onken, D., Fung, S.W., Li, X., Ruthotto, L.: OT-flow: fast and accurate continuous normalizing flows via optimal transport. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, pp. 9223–9232 (2021)

[67] Onken, D., Nurbekyan, L., Li, X., Fung, S.W., Osher, S., Ruthotto, L.: A neural network approach applied to multi-agent optimal control. In: 2021 European Control Conference (ECC), pp. 1036–1041. IEEE (2021)

[68] Onken, D., Nurbekyan, L., Li, X., Fung, S.W., Osher, S., Ruthotto, L.: A neural network approach for high-dimensional optimal control applied to multiagent path finding. IEEE Trans. Control Syst. Technol. 31(1), 235–251 (2022)

[69] Osher, S., Heaton, H., Fung, S.W.: A Hamilton-Jacobi-based proximal operator. Proc. Natl. Acad. Sci. 120(14), e2220469120 (2023)

[70] Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

[71] Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991)

[72] Parkinson, C., Arnold, D., Bertozzi, A.L., Chow, Y.T., Osher, S.: Optimal human navigation in steep terrain: a Hamilton-Jacobi-Bellman approach. arXiv:1805.04973 (2018)

[73] Parkinson, C., Bertozzi, A.L., Osher, S.J.: A Hamilton-Jacobi formulation for time-optimal paths of rectangular nonholonomic vehicles. In: 2020 59th IEEE Conference on Decision and Control (CDC), pp. 4073–4078. IEEE (2020)

[74] Parkinson, C., Ceccia, M.: Time-optimal paths for simple cars with moving obstacles in the Hamilton-Jacobi formulation. In: 2022 American Control Conference (ACC), pp. 2944–2949. IEEE (2022)

[75] Peng D, Merriman B, Osher S, Zhao H, Kang M. A PDE-based fast local level set method. J. Comput. Phys., 1999, 155(2): 410-438.

[76] Qiu J, Shu C-W. Hermite WENO schemes for Hamilton-Jacobi equations. J. Comput. Phys., 2005, 204(1): 82-99.

[77] Ruthotto L, Osher SJ, Li W, Nurbekyan L, Fung SW. A machine learning framework for solving high-dimensional mean field game and mean field control problems. Proc. Natl. Acad. Sci., 2020, 117(17): 9183-9193.

[78] Shu, C.-W.: High order numerical methods for time dependent Hamilton-Jacobi equations. In: Mathematics and Computation in Imaging Science and Information Processing, pp. 47–91 (2007)

[79] Shu C-W, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 1988, 77(2): 439-471.

[80] Tibshirani, R.J., Fung, S.W., Heaton, H., Osher, S.: Laplace meets Moreau: smooth approximation to infimal convolutions using Laplace’s method. J. Mach. Learn. Res. 26(72), 1–36 (2025)

[81] Vidal, A., Fung, S.W., Osher, S., Tenorio, L., Nurbekyan, L.: Kernel expansions for high-dimensional mean-field control with non-local interactions. In: 2025 American Control Conference (ACC), pp. 4164–4171. IEEE (2025)

[82] Vidal A, Fung SW, Tenorio L, Osher S, Nurbekyan L. Taming hyperparameter tuning in continuous normalizing flows using the JKO scheme. Sci. Rep., 2023, 13(1): 4501.

[83] Wang, X., Fung, S.W., Nurbekyan, L.: A primal-dual price-optimization method for computing equilibrium prices in mean-field games models. arXiv:2506.04169 (2025)

[84] Witzany V. Hamilton-Jacobi equation for spinning particles near black holes. Phys. Rev. D, 2019, 100(10. ArticleID: 104030

[85] Xu C, Cheng X, Xie Y. Normalizing flow neural networks by JKO scheme. Adv. Neural. Inf. Process. Syst., 2023, 36: 47379-47405

[86] Yan J, Osher S. A local discontinuous Galerkin method for directly solving Hamilton-Jacobi equations. J. Comput. Phys., 2011, 230(1): 232-244.

[87] Zhang, M., Han, F., Chow, Y.T., Osher, S., Schaeffer, H.: Inexact proximal point algorithms for zeroth-order global optimization. arXiv:2412.11485 (2024)

[88] Zhang Y, Zhu J. A new type of increasingly higher order of finite difference ghost multi-resolution WENO schemes for Hamilton-Jacobi equations. J. Sci. Comput., 2025, 102(2): 54.

[89] Zheng F, Qiu J. Directly solving the Hamilton-Jacobi equations by Hermite WENO schemes. J. Comput. Phys., 2016, 307: 423-445.

[90] Zhou M, Han J, Lu J. Actor-critic method for high dimensional static Hamilton-Jacobi-Bellman partial differential equations based on neural networks. SIAM J. Sci. Comput., 2021, 43(6): A4043-A4066.

[91] Zhou, M., Lu, J.: A policy gradient framework for stochastic optimal control problems with global convergence guarantee. arXiv:2302.05816 (2023)

[92] Zhou, M., Lu, J.: Solving time-continuous stochastic optimal control problems: algorithm design and convergence analysis of actor-critic flow. arXiv:2402.17208 (2024)

Funding

National Science Foundation(DMS-2110745)

Office of Naval Research(N00014-20-1-2787)

National Science Foundation(NSF 2345256)

RIGHTS & PERMISSIONS

Shanghai University