Leveraging Multi-time Hamilton-Jacobi PDEs for Certain Scientific Machine Learning Problems

Understand

Hamilton-Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences.

• By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multi-time case.
• In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multi-time Hopf formula, which corresponds to a representation of the solution to certain multi-time HJ PDEs.
• Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem.