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The framework of deep operator network (DeepONet) has been widely exploited thanks to its capability of solving high dimensional partial differential equations.
E. Hopf, Generalized solutions of non-linear equations of first order, Journal of Mathematics and Mechanics 14 (6) (1965) 951–973
1965
Earlier work this paper cites.
R. Bellman, Dynamic Programming, Science 153 (3731) (1966) 34–37
1966
Earlier work this paper cites.
M. Bardi, L. C. Evans, On Hopf’s formulas for solutions of Hamilton–Jacobi equations, Nonlinear Analysis: Theory, Methods & Applications 8 (11) (1984) 1373–1381
1984
Earlier work this paper cites.
M. Crandall, P.-L. Lions, Two approximations of solutions of Hamilton–Jacobi equations, Mathematics of Computations 43 (167) (1984) 1–19
1984
Earlier work this paper cites.
S. Osher, C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations, SIAM Journal on Numerical Analysis 28 (4) (1991) 907–922
1991
Earlier work this paper cites.
M. G. Crandall, H. Ishii, P.-L. Lions, User’s Guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society 27 (1) (1992) 1–67
1992
Earlier work this paper cites.
T. Chen, H. Chen, Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems, IEEE Transactions on Neural Networks 6 (4) (1995) 911–917
1995
Earlier work this paper cites.
G. Barles, Convergence of numerical schemes for degenerate parabolic equations arising in finance theory, Numerical Methods in Finance 13 (1) (1997)
1997
Earlier work this paper cites.
M. Bardi, I. C. Dolcetta, et al., Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Vol. 12, Springer, 1997
1997
Earlier work this paper cites.
J. Nocedal, S. J. Wright, Numerical optimization, Springer, 1999
1999
Earlier work this paper cites.
G.-S. Jiang, D. Peng, Weighted ENO schemes for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (6) (2000) 2126–2143
2000
Earlier work this paper cites.
G. Barles, E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton–Jacobi–Bellman equations, ESAIM: Mathematical Modelling and Numerical Analysis 36 (1) (2002) 33–54
2002
Earlier work this paper cites.
S. Lenhart, J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC, 2007
2007
Earlier work this paper cites.
W.-S. Lin, L.-H. Chang, P.-C. Yang, Adaptive critic anti-slip control of wheeled autonomous robot, IET Control theory & Applications 1 (1) (2007) 51–57
2007
Earlier work this paper cites.
P. A. Forsyth, G. Labahn, Numerical methods for controlled Hamilton–Jacobi–Bellman pdes in finance, Journal of Computational Finance 11 (2) (2007) 1
2007
Cited alongside, same era.
I. M. Mitchell, Scalable calculation of reach sets and tubes for nonlinear systems with terminal integrators: a mixed implicit explicit formulation, in: Proceedings of the 14th International Conference on Hybrid Systems: Computation and Control, 2011, pp. 103–112
2011
Cited alongside, same era.
Y. Achdou, G. Barles, H. Ishii, G. L. Litvinov, Hamilton–Jacobi Equations: Approximations, Numerical Analysis and Applications (2013)
2013
Cited alongside, same era.
Y. Achdou, G. Barles, H. Ishii, G. L. Litvinov, G. Barles, An introduction to the theory of viscosity solutions for first-order Hamilton–Jacobi equations and applications, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications: Cetraro, Italy 2011, Editors: Paola Loreti, Nicoletta Anna Tchou (2013) 49–109
2013
J. Darbon, G. P. Langlois, T. Meng, Overcoming the curse of dimensionality for some Hamilton–Jacobi partial differential equations via neural network architectures, Research in the Mathematical Sciences 7 (3) (2020) 1–50
2020
Later among the works it cites.
2020
Later among the works it cites.
W. H. Sandholm, H. V. Tran, S. Arigapudi, Hamilton-Jacobi equations with semilinear costs and state constraints, with applications to large deviations in games, Mathematics of Operations Research (2021)
2021
Later among the works it cites.
I. Yegorov, P. M. Dower, Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations, Applied Mathematics & Optimization 83 (1) (2021) 1–49
2021
Later among the works it cites.
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Cited alongside, same era.
J. Darbon, S. Osher, Algorithms for overcoming the curse of dimensionality for certain Hamilton–Jacobi equations arising in control theory and elsewhere, Research in the Mathematical Sciences 3 (1) (2016) 1–26
2016
Cited alongside, same era.
M. Aliyu, A modified-secant iterative method for solving the Hamilton–Jacobi–Bellman-Isaac equations in non-linear optimal control, IET Control Theory & Applications 10 (16) (2016) 2136–2141
2016
Cited alongside, same era.
A. l. m. G. Baydin, B. A. Pearlmutter, A. A. Radul, J. M. Siskind, Automatic differentiation in machine learning: a survey, J. Mach. Learn. Res. 18 (2017) Paper No. 153, 43
2017
Cited alongside, same era.
A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, A. Lerer, Automatic differentiation in Pytorch (2017)
2017
Cited alongside, same era.
J. Han, A. Jentzen, W. E, Solving high-dimensional partial differential equations using deep learning, Proceedings of the National Academy of Sciences 115 (34) (2018) 8505–8510
2018
Cited alongside, same era.
doi:10.1016/j.jcp.2018.08.029
J. Sirignano, K. Spiliopoulos, DGM: a deep learning algorithm for solving partial differential equations , J. Comput. Phys. 375 (2018) 1339–1364 · 2018
Cited alongside, same era.
Y. T. Chow, J. Darbon, S. Osher, W. Yin, Algorithm for overcoming the curse of dimensionality for state-dependent Hamilton–Jacobi equations, Journal of Computational Physics 387 (2019) 376–409
2019
Cited alongside, same era.
2019
Cited alongside, same era.
2021
Later among the works it cites.
2021
Later among the works it cites.
J. Darbon, T. Meng, On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton–Jacobi partial differential equations, Journal of Computational Physics 425 (2021) 109907
2021
Later among the works it cites.
L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via deepONet based on the universal approximation theorem of operators, Nature Machine Intelligence 3 (3) (2021) 218–229
2021
Later among the works it cites.
S. Wang, H. Wang, P. Perdikaris, Learning the solution operator of parametric partial differential equations with physics-informed DeepONets, Science Advances 7 (40) (2021) eabi8605
2021
Later among the works it cites.
H. V. Tran, Hamilton-Jacobi Equations: Theory and Applications, Vol. 213, American Mathematical Soc., 2021
2021
Later among the works it cites.
L. C. Evans, Partial Differential Equations, Vol. 19, American Mathematical Society, 2022
2022
Later among the works it cites.
Y. Kim, I. Yang, On representation formulas for optimal control: A Lagrangian perspective, IET Control Theory & Applications 16 (16) (2022) 1633–1644
2022
Later among the works it cites.
2022
Later among the works it cites.