Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learning

Paula Chen, Tingwei Meng, Zongren Zou, Jérôme Darbon, George Em Karniadakis
Proceedings of the 6th Annual Learning for Dynamics & Control Conference, PMLR 242:1-12, 2024.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v242-chen24a, title = {Leveraging {H}amilton-{J}acobi {PDE}s with time-dependent {H}amiltonians for continual scientific machine learning}, author = {Chen, Paula and Meng, Tingwei and Zou, Zongren and Darbon, J\'{e}r\^{o}me and Karniadakis, George Em}, booktitle = {Proceedings of the 6th Annual Learning for Dynamics & Control Conference}, pages = {1--12}, year = {2024}, editor = {Abate, Alessandro and Cannon, Mark and Margellos, Kostas and Papachristodoulou, Antonis}, volume = {242}, series = {Proceedings of Machine Learning Research}, month = {15--17 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v242/chen24a/chen24a.pdf}, url = {https://proceedings.mlr.press/v242/chen24a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learning %A Paula Chen %A Tingwei Meng %A Zongren Zou %A Jérôme Darbon %A George Em Karniadakis %B Proceedings of the 6th Annual Learning for Dynamics & Control Conference %C Proceedings of Machine Learning Research %D 2024 %E Alessandro Abate %E Mark Cannon %E Kostas Margellos %E Antonis Papachristodoulou %F pmlr-v242-chen24a %I PMLR %P 1--12 %U https://proceedings.mlr.press/v242/chen24a.html %V 242 %X 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.
APA
Chen, P., Meng, T., Zou, Z., Darbon, J. & Karniadakis, G.E.. (2024). Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learning. Proceedings of the 6th Annual Learning for Dynamics & Control Conference, in Proceedings of Machine Learning Research 242:1-12 Available from https://proceedings.mlr.press/v242/chen24a.html.

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