  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. As a first
exploration of this connection, we develop the relation between the regularized
linear regression problem and the Linear Quadratic Regulator (LQR). We then
leverage our theoretical connection to adapt standard LQR solvers (namely,
those based on the Riccati ordinary differential equations) to design new
training approaches for machine learning. Finally, we provide some numerical
examples that demonstrate the versatility and possible computational advantages
of our Riccati-based approach in the context of continual learning,
post-training calibration, transfer learning, and sparse dynamics
identification.
