SympOCnet: solving optimal control problems with applications to high-dimensional multiagent path planning problems. (English) Zbl 1504.49052
Summary: Solving high-dimensional optimal control problems in real-time is an important but challenging problem, with applications to multiagent path planning problems, which have drawn increased attention given the growing popularity of drones in recent years. In this paper, we propose a novel neural network method called SympOCnet that applies the symplectic network to solve high-dimensional optimal control problems with state constraints. We present several numerical results on path planning problems in two- and three-dimensional spaces. Specifically, we demonstrate that our SympOCnet can solve a problem with more than 500 dimensions in 1.5 hours on a single GPU, which shows the effectiveness and efficiency of SympOCnet. The proposed method is scalable and has the potential to solve truly high-dimensional path planning problems in real-time.
MSC:
| 49M37 | Numerical methods based on nonlinear programming |
| 49M20 | Numerical methods of relaxation type |
| 49M29 | Numerical methods involving duality |
| 68T07 | Artificial neural networks and deep learning |
References:
| [1] | M. Akian, R. Bapat, and S. Gaubert, Max-plus algebra, in Handbook of Linear Algebra, L. Hogben, ed., Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2007, 25. · Zbl 0922.15001 |
| [2] | M. Akian, S. Gaubert, and A. Lakhoua, The max-plus finite element method for solving deterministic optimal control problems: Basic properties and convergence analysis, SIAM J. Control Optim., 47 (2008), pp. 817-848, https://doi.org/10.1137/060655286. · Zbl 1157.49034 |
| [3] | A. Alla, M. Falcone, and L. Saluzzi, An efficient DP algorithm on a tree-structure for finite horizon optimal control problems, SIAM J. Sci. Comput., 41 (2019), pp. A2384-A2406, https://doi.org/10.1137/18M1203900. · Zbl 1423.49024 |
| [4] | A. Alla, M. Falcone, and S. Volkwein, Error analysis for POD approximations of infinite horizon problems via the dynamic programming approach, SIAM J. Control Optim., 55 (2017), pp. 3091-3115, https://doi.org/10.1137/15M1039596. · Zbl 1378.49025 |
| [5] | A. Bachouch, C. Huré, N. Langrené, and H. Pham, Deep Neural Networks Algorithms for Stochastic Control Problems on Finite Horizon: Numerical Applications, preprint, https://arxiv.org/abs/1812.05916, 2018. · Zbl 1496.93112 |
| [6] | S. Bansal and C. J. Tomlin, DeepReach: A deep learning approach to high-dimensional reachability, in Proceedings of the 2021 IEEE International Conference on Robotics and Automation (ICRA), 2021, pp. 1817-1824. |
| [7] | M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, with appendices by Maurizio Falcone and Pierpaolo Soravia, Systems Control Found. Appl., Birkhäuser Boston, Boston, MA, 1997, https://doi.org/10.1007/978-0-8176-4755-1. · Zbl 0890.49011 |
| [8] | R. E. Bellman, Adaptive Control Processes: A Guided Tour, Princeton University Press, Princeton, NJ, 1961. · Zbl 0103.12901 |
| [9] | D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, London, 1982, https://doi.org/10.1016/C2013-0-10366-2. · Zbl 0572.90067 |
| [10] | D. P. Bertsekas, Reinforcement Learning and Optimal Control, Athena Scientific, Belmont, MA, 2019. |
| [11] | O. Bokanowski, N. Gammoudi, and H. Zidani, Optimistic planning algorithms for state-constrained optimal control problems, Comput. Math. Appl., 109 (2022), pp. 158-179, https://doi.org/10.1016/j.camwa.2022.01.016. · Zbl 1524.49062 |
| [12] | O. Bokanowski, J. Garcke, M. Griebel, and I. Klompmaker, An adaptive sparse grid semi-Lagrangian scheme for first order Hamilton-Jacobi-Bellman equations, J. Sci. Comput., 55 (2013), pp. 575-605. · Zbl 1269.65076 |
| [13] | M. Chen, J. F. Fisac, S. Sastry, and C. J. Tomlin, Safe sequential path planning of multi-vehicle systems via double-obstacle Hamilton-Jacobi-Isaacs variational inequality, in Proceedings of the 2015 European Control Conference (ECC), IEEE, Piscataway, NJ, 2015, pp. 3304-3309. |
| [14] | M. Chen, Q. Hu, J. F. Fisac, K. Akametalu, C. Mackin, and C. J. Tomlin, Reachability-based safety and goal satisfaction of unmanned aerial platoons on air highways, J. Guidance Control Dynam., 40 (2017), pp. 1360-1373. |
| [15] | M. Chen and C. J. Tomlin, Exact and efficient Hamilton-Jacobi reachability for decoupled systems, in Proceedings of the 2015 54th IEEE Conference on Decision and Control (CDC), 2015, pp. 1297-1303. |
| [16] | P. Chen, J. Darbon, and T. Meng, Hopf-type Representation Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems, preprint, https://arxiv.org/abs/2110.02541, 2021. |
| [17] | P. Chen, J. Darbon, and T. Meng, Lax-Oleinik-type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems, preprint, https://arxiv.org/abs/2109.14849, 2021. |
| [18] | A. R. Conn, N. I. M. Gould, and P. L. Toint, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds, SIAM J. Numer. Anal., 28 (1991), pp. 545-572, https://doi.org/10.1137/0728030. · Zbl 0724.65067 |
| [19] | M. Coupechoux, J. Darbon, J.-M. Kélif, and M. Sigelle, Optimal trajectories of a UAV base station using Lagrangian mechanics, in Proceedings of the IEEE INFOCOM 2019-IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS), 2019, pp. 626-631. |
| [20] | J. Darbon, On convex finite-dimensional variational methods in imaging sciences and Hamilton-Jacobi equations, SIAM J. Imaging Sci., 8 (2015), pp. 2268-2293, https://doi.org/10.1137/130944163. · Zbl 1330.35076 |
| [21] | J. Darbon, P. M. Dower, and T. Meng, Neural Network Architectures Using Min Plus Algebra for Solving Certain High Dimensional Optimal Control Problems and Hamilton-Jacobi PDEs, preprint, https://arxiv.org/abs/2105.03336, 2021. · Zbl 1507.49019 |
| [22] | J. Darbon, G. P. Langlois, and T. Meng, Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures, Res. Math. Sci., 7 (2020), 20, https://doi.org/10.1007/s40687-020-00215-6. · Zbl 1445.35119 |
| [23] | J. Darbon and T. Meng, On decomposition models in imaging sciences and multi-time Hamilton-Jacobi partial differential equations, SIAM J. Imaging Sci., 13 (2020), pp. 971-1014, https://doi.org/10.1137/19M1266332. · Zbl 1455.35051 |
| [24] | J. Darbon and T. Meng, On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton-Jacobi partial differential equations, J. Comput. Phys., 425 (2021), 109907, https://doi.org/10.1016/j.jcp.2020.109907. · Zbl 07508503 |
| [25] | J. Darbon, T. Meng, and E. Resmerita, On Hamilton-Jacobi PDEs and image denoising models with certain nonadditive noise, J. Math. Imaging Vis., 64 (2022), pp. 408-441, https://doi.org/10.1007/s10851-022-01073-3. · Zbl 07557140 |
| [26] | J. Darbon and S. Osher, Algorithms for overcoming the curse of dimensionality for certain Hamilton-Jacobi equations arising in control theory and elsewhere, Res. Math. Sci., 3 (2016), 19, https://doi.org/10.1186/s40687-016-0068-7. · Zbl 1348.49026 |
| [27] | D. Delahaye, S. Puechmorel, P. Tsiotras, and E. Féron, Mathematical models for aircraft trajectory design: A survey, in Air Traffic Management and Systems, Springer, Japan, 2014, pp. 205-247. |
| [28] | J. Denk and G. Schmidt, Synthesis of a walking primitive database for a humanoid robot using optimal control techniques, in Proceedings of the IEEE-RAS International Conference on Humanoid Robots, 2001, pp. 319-326. |
| [29] | B. Djeridane and J. Lygeros, Neural approximation of PDE solutions: An application to reachability computations, in Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 3034-3039, https://doi.org/10.1109/CDC.2006.377184. |
| [30] | S. Dolgov, D. Kalise, and K. K. Kunisch, Tensor decomposition methods for high-dimensional Hamilton-Jacobi-Bellman equations, SIAM J. Sci. Comput., 43 (2021), pp. A1625-A1650, https://doi.org/10.1137/19M1305136. · Zbl 1471.65184 |
| [31] | P. M. Dower, W. M. McEneaney, and H. Zhang, Max-plus fundamental solution semigroups for optimal control problems, in 2015 Proceedings of the Conference on Control and Its Applications, SIAM, Philadelphia, 2015, pp. 368-375 https://doi.org/10.1137/1.9781611974072.51. · Zbl 07875214 |
| [32] | A. El Khoury, F. Lamiraux, and M. Taix, Optimal motion planning for humanoid robots, in Proceedings of the 2013 IEEE International Conference on Robotics and Automation, 2013, pp. 3136-3141. |
| [33] | L. C. Evans, An Introduction to Mathematical Optimal Control Theory Version 0.2, lecture notes, Department of Mathematics University of California, Berkeley, 1983, https://math.berkeley.edu/ evans/control.course.pdf. |
| [34] | F. Fahroo and I. M. Ross, Costate estimation by a Legendre pseudospectral method, J. Guidance Control Dynam., 24 (2001), pp. 270-277. |
| [35] | F. Fahroo and I. M. Ross, Direct trajectory optimization by a Chebyshev pseudospectral method, J. Guidance Control Dynam., 25 (2002), pp. 160-166. |
| [36] | M. Fallon, S. Kuindersma, S. Karumanchi, M. Antone, T. Schneider, H. Dai, C. P. D’Arpino, R. Deits, M. DiCicco, D. Fourie, T. Koolen, P. Marion, M. Posa, A. Valenzuela, K.-T. Yu, J. Shah, K. Iagnemma, R. Tedrake, and S. Teller, An architecture for online affordance-based perception and whole-body planning, J. Field Robotics, 32 (2015), pp. 229-254. |
| [37] | S. Feng, E. Whitman, X. Xinjilefu, and C. G. Atkeson, Optimization based full body control for the Atlas robot, in Proceedings of the 2014 IEEE-RAS International Conference on Humanoid Robots, 2014, pp. 120-127. |
| [38] | W. H. Fleming and W. M. McEneaney, A max-plus-based algorithm for a Hamilton-Jacobi-Bellman equation of nonlinear filtering, SIAM J. Control Optim., 38 (2000), pp. 683-710, https://doi.org/10.1137/S0363012998332433. · Zbl 0949.35039 |
| [39] | K. Fujiwara, S. Kajita, K. Harada, K. Kaneko, M. Morisawa, F. Kanehiro, S. Nakaoka, and H. Hirukawa, An optimal planning of falling motions of a humanoid robot, in Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2007, pp. 456-462. |
| [40] | J. Garcke and A. Kröner, Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids, J. Sci. Comput., 70 (2017), pp. 1-28. · Zbl 1434.49020 |
| [41] | D. Garg, Advances in Global Pseudospectral Methods for Optimal Control, Ph.D. thesis, University of Florida, Gainesville, FL, 2011. |
| [42] | S. Gaubert, W. McEneaney, and Z. Qu, Curse of dimensionality reduction in max-plus based approximation methods: Theoretical estimates and improved pruning algorithms, in Proceedings of the 2011 50th IEEE Conference on Decision and Control and European Control Conference, 2011, pp. 1054-1061. |
| [43] | S. Greydanus, M. Dzamba, and J. Yosinski, Hamiltonian Neural Networks, in Advances in Neural Information Processing Systems 32, H. Wallach et al., eds., Curran Associates, Inc., 2019, https://proceedings.neurips.cc/paper/2019/file/26cd8ecadce0d4efd6cc8a8725cbd1f8-Paper.pdf. |
| [44] | E. Hairer, M. Hochbruck, A. Iserles, and C. Lubich, Geometric numerical integration, Oberwolfach Rep., 3 (2006), pp. 805-882. · Zbl 1109.65301 |
| [45] | J. Han, A. Jentzen, and W. E, Solving high-dimensional partial differential equations using deep learning, Proc. Natl. Acad. Sci. USA, 115 (2018), pp. 8505-8510, https://doi.org/10.1073/pnas.1718942115. · Zbl 1416.35137 |
| [46] | M. Hofer, M. Muehlebach, and R. D’Andrea, Application of an approximate model predictive control scheme on an unmanned aerial vehicle, in Proceedings of the 2016 IEEE International Conference on Robotics and Automation (ICRA), 2016, pp. 2952-2957. |
| [47] | M. B. Horowitz, A. Damle, and J. W. Burdick, Linear Hamilton Jacobi Bellman equations in high dimensions, in Proceedings of the 53rd IEEE Conference on Decision and Control, 2014, pp. 5880-5887. |
| [48] | C. Huré, H. Pham, and X. Warin, Some Machine Learning Schemes for High-Dimensional Nonlinear PDEs, preprint, https://arxiv.org/abs/1902.01599v1, 2019. · Zbl 1440.60063 |
| [49] | C. Huré, H. Pham, A. Bachouch, and N. Langrené, Deep neural networks algorithms for stochastic control problems on finite horizon: Convergence analysis, SIAM J. Numer. Anal., 59 (2021), pp. 525-557, https://doi.org/10.1137/20M1316640. · Zbl 1466.65007 |
| [50] | F. Jiang, G. Chou, M. Chen, and C. J. Tomlin, Using Neural Networks to Compute Approximate and Guaranteed Feasible Hamilton-Jacobi-Bellman PDE Solutions, preprint, https://arxiv.org/abs/1611.03158, 2016. |
| [51] | L. Jin, S. Li, J. Yu, and J. He, Robot manipulator control using neural networks: A survey, Neurocomputing, 285 (2018), pp. 23-34, https://doi.org/10.1016/j.neucom.2018.01.002. |
| [52] | P. Jin, Z. Zhang, I. G. Kevrekidis, and G. E. Karniadakis, Learning Poisson Systems and Trajectories of Autonomous Systems via Poisson Neural Networks, preprint, https://arxiv.org/abs/2012.03133, 2020. |
| [53] | P. Jin, Z. Zhang, A. Zhu, Y. Tang, and G. E. Karniadakis, SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems, Neural Networks, 132 (2020), pp. 166-179. · Zbl 1475.68316 |
| [54] | D. Kalise, S. Kundu, and K. Kunisch, Robust feedback control of nonlinear PDEs by numerical approximation of high-dimensional Hamilton-Jacobi-Isaacs equations, SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1496-1524, https://doi.org/10.1137/19M1262139. · Zbl 1443.49041 |
| [55] | D. Kalise and K. Kunisch, Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs, SIAM J. Sci. Comput., 40 (2018), pp. A629-A652, https://doi.org/10.1137/17M1116635. · Zbl 1385.49022 |
| [56] | W. Kang and L. C. Wilcox, Mitigating the curse of dimensionality: Sparse grid characteristics method for optimal feedback control and HJB equations, Comput. Optim. Appl., 68 (2017), pp. 289-315. · Zbl 1383.49045 |
| [57] | Y. H. Kim, F. L. Lewis, and D. M. Dawson, Intelligent optimal control of robotic manipulators using neural networks, Automatica, 36 (2000), pp. 1355-1364, https://doi.org/10.1016/S0005-1098(00)00045-5. · Zbl 1002.93039 |
| [58] | D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, in Proceedings of the 3rd International Conference on Learning Representations (ICLR 2015), San Diego, CA, 2015, Conference Track Proceedings, 2015. |
| [59] | M. R. Kirchner, M. J. Debord, and J. P. Hespanha, A Hamilton-Jacobi formulation for optimal coordination of heterogeneous multiple vehicle systems, in Proceedings of the 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2020, pp. 11623-11630. |
| [60] | S. Kuindersma, R. Deits, M. Fallon, A. Valenzuela, H. Dai, F. Permenter, T. Koolen, P. Marion, and R. Tedrake, Optimization-based locomotion planning, estimation, and control design for the Atlas humanoid robot, Autonomous Robots, 40 (2016), pp. 429-455. |
| [61] | K. Kunisch, S. Volkwein, and L. Xie, HJB-POD-based feedback design for the optimal control of evolution problems, SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 701-722, https://doi.org/10.1137/030600485. · Zbl 1058.35061 |
| [62] | P. Lambrianides, Q. Gong, and D. Venturi, A new scalable algorithm for computational optimal control under uncertainty, J. Comput. Phys., 420 (2020), 109710, https://doi.org/10.1016/j.jcp.2020.109710. · Zbl 07506628 |
| [63] | D. Lee and C. J. Tomlin, A Hopf-Lax formula in Hamilton-Jacobi analysis of reach-avoid problems, IEEE Control Syst. Lett., 5 (2020), pp. 1055-1060. |
| [64] | D. Lee and C. J. Tomlin, A Computationally Efficient Hamilton-Jacobi-based Formula for State-Constrained Optimal Control Problems, preprint, https://arxiv.org/abs/2106.13440, 2021. |
| [65] | F. Lewis, D. Dawson, and C. Abdallah, Robot Manipulator Control: Theory and Practice, 2nd ed., revised and expanded, Control Engineering, Marcel Dekker, 2004. |
| [66] | A. Li, S. Bansal, G. Giovanis, V. Tolani, C. Tomlin, and M. Chen, Generating robust supervision for learning-based visual navigation using Hamilton-Jacobi reachability, in Learning for Dynamics and Control, PMLR, 2020, pp. 500-510. |
| [67] | F. Lin and R. D. Brandt, An optimal control approach to robust control of robot manipulators, IEEE Trans. Robotics Automation, 14 (1998), pp. 69-77. |
| [68] | L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nature Mach. Intell., 3 (2021), pp. 218-229. |
| [69] | V. R. Makkapati, J. Ridderhof, P. Tsiotras, J. Hart, and B. van Bloemen Waanders, Desensitized trajectory optimization for hypersonic vehicles, in Proceedings of the 2021 IEEE Aerospace Conference (50100), 2021, pp. 1-10, https://doi.org/10.1109/AERO50100.2021.9438511. |
| [70] | W. M. McEneaney, A curse-of-dimensionality-free numerical method for solution of certain HJB PDEs, SIAM J. Control Optim., 46 (2007), pp. 1239-1276, https://doi.org/10.1137/040610830. · Zbl 1251.65168 |
| [71] | W. M. McEneaney, Max-plus Methods for Nonlinear Control and Estimation, Systems Control Found. Appl., Birkhäuser Boston, Boston, MA, 2006. · Zbl 1103.93005 |
| [72] | W. M. McEneaney, A. Deshpande, and S. Gaubert, Curse-of-complexity attenuation in the curse-of-dimensionality-free method for HJB PDEs, in Proceedings of the 2008 American Control Conference, IEEE, Piscataway, NJ, 2008, pp. 4684-4690. |
| [73] | W. M. McEneaney and L. J. Kluberg, Convergence rate for a curse-of-dimensionality-free method for a class of HJB PDEs, SIAM J. Control Optim., 48 (2009), pp. 3052-3079, https://doi.org/10.1137/070681934. · Zbl 1218.35074 |
| [74] | X. Meng and G. E. Karniadakis, A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems, J. Comput. Phys., 401 (2020), 109020. · Zbl 1454.76006 |
| [75] | K. R. Meyer and D. C. Offin, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Appl. Math. Sci. 90, Springer, Cham, 1992, https://doi.org/https://doi.org/10.1007/978-3-319-53691-0. · Zbl 1372.70002 |
| [76] | T. Nakamura-Zimmerer, Q. Gong, and W. Kang, Adaptive deep learning for high-dimensional Hamilton-Jacobi-Bellman equations, SIAM J. Sci. Comput., 43 (2021), pp. A1221-A1247, https://doi.org/10.1137/19M1288802. · Zbl 1467.49028 |
| [77] | T. Nakamura-Zimmerer, Q. Gong, and W. Kang, QRnet: Optimal regulator design with LQR-augmented neural networks, IEEE Control Syst. Lett., 5 (2021), pp. 1303-1308, https://doi.org/10.1109/LCSYS.2020.3034415. |
| [78] | K. N. Niarchos and J. Lygeros, A neural approximation to continuous time reachability computations, in Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 6313-6318, https://doi.org/10.1109/CDC.2006.377358. |
| [79] | D. Onken, L. Nurbekyan, X. Li, S. W. Fung, S. Osher, and L. Ruthotto, A Neural Network Approach for High-dimensional Optimal Control, preprint, https://arxiv.org/abs/2104.03270, 2021. |
| [80] | C. Parzani and S. Puechmorel, On a Hamilton-Jacobi-Bellman approach for coordinated optimal aircraft trajectories planning, Optimal Control Appl. Methods, 39 (2018), pp. 933-948. · Zbl 1396.49019 |
| [81] | M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), pp. 686-707. · Zbl 1415.68175 |
| [82] | A. V. Rao, A survey of numerical methods for optimal control, Adv. Astronautical Sci., 135 (2009), pp. 497-528. |
| [83] | C. Reisinger and Y. Zhang, Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems, Anal. Appl., 18 (2020), pp. 951-999, https://doi.org/10.1142/S0219530520500116. · Zbl 1456.82804 |
| [84] | D. R. Robinson, R. T. Mar, K. Estabridis, and G. Hewer, An efficient algorithm for optimal trajectory generation for heterogeneous multi-agent systems in non-convex environments, IEEE Robotics Automation Lett., 3 (2018), pp. 1215-1222. |
| [85] | I. M. Ross and M. Karpenko, A review of pseudospectral optimal control: From theory to flight, Ann. Rev. Control, 36 (2012), pp. 182-197, https://doi.org/10.1016/j.arcontrol.2012.09.002. |
| [86] | V. R. Royo and C. Tomlin, Recursive Regression with Neural Networks: Approximating the HJI PDE Solution, preprint, https://arxiv.org/abs/1611.02739, 2016. |
| [87] | A. Rucco, G. Notarstefano, and J. Hauser, An efficient minimum-time trajectory generation strategy for two-track car vehicles, IEEE Trans. Control Syst. Tech., 23 (2015), pp. 1505-1519. |
| [88] | A. Rucco, P. Sujit, A. P. Aguiar, J. B. De Sousa, and F. L. Pereira, Optimal rendezvous trajectory for unmanned aerial-ground vehicles, IEEE Trans. Aerospace Electron. Syst., 54 (2017), pp. 834-847. |
| [89] | J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, J. Comput. Phys., 375 (2018), pp. 1339-1364, https://doi.org/10.1016/j.jcp.2018.08.029. · Zbl 1416.65394 |
| [90] | S. Spedicato and G. Notarstefano, Minimum-time trajectory generation for quadrotors in constrained environments, IEEE Trans. Control Syst. Tech., 26 (2017), pp. 1335-1344. |
| [91] | E. Todorov, Efficient computation of optimal actions, Proc. Natl. Acad. Sci. USA, 106 (2009), pp. 11478-11483. · Zbl 1203.68327 |
| [92] | I. Yegorov and P. M. Dower, Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations, Appl. Math. Optim., 83 (2017), pp. 1-49. · Zbl 1461.49028 |
| [93] | Z. Zhang, Y. Shin, and G. E. Karniadakis, GFINNs: GENERIC formalism informed neural networks for deterministic and stochastic dynamical systems, Phil. Trans. R. Soc. A, 380 (2022), 20210207, https://doi.org/10.1098/rsta.2021.0207. |
| [94] | M. Zhou, J. Han, and J. Lu, Actor-critic method for high dimensional static Hamilton-Jacobi-Bellman partial differential equations based on neural networks, SIAM J. Sci. Comput., 43 (2021), pp. A4043-A4066, https://doi.org/10.1137/21M1402303. · Zbl 1481.65203 |
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