Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for
  continual scientific machine learning
Paula Chen Tingwei Meng Zongren Zou Jérôme Darbon George Em Karniadakis
We address two major challenges in scientific machine learning (SciML):
interpretability and computational efficiency. We increase the interpretability
of certain learning processes by establishing a new theoretical connection
between optimization problems arising from SciML and a generalized Hopf
formula, which represents the viscosity solution to a Hamilton-Jacobi partial
differential equation (HJ PDE) with time-dependent Hamiltonian. Namely, we show
that when we solve certain regularized learning problems with integral-type
losses, we actually solve an optimal control problem and its associated HJ PDE
with time-dependent Hamiltonian. This connection allows us to reinterpret
incremental updates to learned models as the evolution of an associated HJ PDE
and optimal control problem in time, where all of the previous information is
intrinsically encoded in the solution to the HJ PDE. As a result, existing HJ
PDE solvers and optimal control algorithms can be reused to design new
efficient training approaches for SciML that naturally coincide with the
continual learning framework, while avoiding catastrophic forgetting. As a
first exploration of this connection, we consider the special case of linear
regression and leverage our connection to develop a new Riccati-based
methodology for solving these learning problems that is amenable to continual
learning applications. We also provide some corresponding numerical examples
that demonstrate the potential computational and memory advantages our
Riccati-based approach can provide.