Recently, I have several research directions whose details are listed as follows.
In this project, we proposed several neural network architectures to compute the viscosity solutions to high-dimensional Hamilton-Jacobi (HJ) partial differential equations (PDEs).
We encode the physics of HJ PDEs in these architectures. The HJ theory provides mathematical guarantees for the outputs of these neural networks.
If these architectures are implemented on the dedicated hardware designed for neural networks, they also enjoy the efficient computation provided by the hardware. Therefore, they provide possibilities to efficiently solve high-dimensional practical problems in real-time.
Since our representations do not involve any grid or discretization, these representations overcome the curse of dimensionality for certain HJ PDEs.
Cite:
(Preprint) 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. arXiv:2105.03336 (2021). link
J. Darbon and 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, 2021, 425:109907. link
J. Darbon, GP. Langlois and T. Meng, Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures. Research in the Mathematical Sciences, 2020, 7(3):20. link
In this project, we proposed analytical solutions and efficient algorithms for certain high-dimensional non-smooth optimal control problems. These problems may be regarded as building blocks for more complicated non-smooth optimal control problems.
Cite:
(Preprint) P. Chen, J. Darbon and T. Meng, Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems. arXiv:2110.02541 (2021). link
(Preprint) P. Chen, J. Darbon and T. Meng, Lax-Oleinik-type formulas and efficient algorithms for certain high-dimensional optimal control problems. arXiv:2109.14849 (2021). link
This project investigates the connections of Hamilton-Jacobi (HJ) partial differential equations (PDEs) and variational models in imaging sciences. To be specific, we studied image decomposition models and image denoising models with non-additive noise. These new connections open new doors to develop efficient solvers for certain high-dimensional HJ PDEs using algorithms developed in imaging sciences. Moreover, they provide possibilities to understand some algorithms in imaging sciences from the perspective of HJ theory.
Cite:
(Preprint) J. Darbon, T. Meng and E. Resmerita, On Hamilton-Jacobi PDEs and image denoising models with certain non-additive noise. arXiv:2105.13997 (2021). link
J. Darbon and T. Meng, On decomposition models in imaging sciences and multi-time Hamilton-Jacobi partial differential equations. SIAM Journal on Imaging Sciences, 2020, 13(2): 971-1014. link