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

Paula Chen, Tingwei Meng, Zongren Zou, Jérôme Darbon, George Em Karniadakis

TL;DR

The paper addresses interpretability and efficiency in SciML by linking regularized learning with integral-type losses to a time-dependent Hamilton-Jacobi PDE via a generalized Hopf formula, enabling interpretation of incremental updates as PDE evolution. It develops a Riccati-based methodology for quadratic-regularized linear regression, showing the learning problem is equivalent to a time-dependent LQR and deriving ODEs for $P(t)$ and $q(t)$ whose solution yields the optimal parameters via $\boldsymbol{\theta}^* = q(t)$. Numerical experiments on streaming data for a boundary-value ODE and a 2D Poisson equation demonstrate memory and computational advantages and avoidance of catastrophic forgetting, with errors decreasing as more data are incorporated. The work opens avenues to extend the framework to nonconvex Hamiltonians, higher dimensions, and integration with existing high-dimensional HJ PDE solvers for broader SciML applications.

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.

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

TL;DR

The paper addresses interpretability and efficiency in SciML by linking regularized learning with integral-type losses to a time-dependent Hamilton-Jacobi PDE via a generalized Hopf formula, enabling interpretation of incremental updates as PDE evolution. It develops a Riccati-based methodology for quadratic-regularized linear regression, showing the learning problem is equivalent to a time-dependent LQR and deriving ODEs for and whose solution yields the optimal parameters via . Numerical experiments on streaming data for a boundary-value ODE and a 2D Poisson equation demonstrate memory and computational advantages and avoidance of catastrophic forgetting, with errors decreasing as more data are incorporated. The work opens avenues to extend the framework to nonconvex Hamiltonians, higher dimensions, and integration with existing high-dimensional HJ PDE solvers for broader SciML applications.

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.
Paper Structure (11 sections, 1 theorem, 19 equations, 4 figures)

This paper contains 11 sections, 1 theorem, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Connection between the generalized Hopf formula and learning problems
  3. Generalized Hopf formula for HJ PDEs with time-dependent Hamiltonians
  4. Connection to learning problems
  5. Regularized linear regression problems with quadratic integral-type losses
  6. Problem formulation and connection to LQR problems
  7. Riccati-based methodology
  8. Numerical examples
  9. A boundary-value ODE problem
  10. 2D Poisson equation
  11. Summary

Key Result

Proposition 1

If $H:(0,\infty)\times\mathbb{R}^n\to\mathbb{R}$ and $J:\mathbb{R}^n\to\mathbb{R}$ are continuous, $J$ is convex, and $H(t,\mathbf{p})$ is convex in $\mathbf{p}$ then the generalized Hopf formula eq:generalizedHopf is the viscosity solution to the time-dependent HJ PDE eq:HJPDE_timedependent.

Figures (4)

  • Figure 1: (See Section \ref{['sec:theory']}) Illustration of a connection between a regularized learning problem with integral-type loss (top), the generalized Hopf formula for HJ PDEs with time-dependent Hamiltonians (middle), and the corresponding optimal control problem (bottom). The colors indicate the associated quantities between each problem. For example, the optimal weights in the learning problem are equivalent to the momentum in the HJ PDE, which is related to the control in the optimal control problem (cyan). This color scheme is reused in the subsequent illustrations of our connection. The solid-line arrows denote direct equivalences. The dotted arrows represent additional mathematical relations.
  • Figure 2: (See Section \ref{['sec:theory']}) Mathematical formulation describing the connection between a regularized learning problem with integral-type loss (top), the generalized Hopf formula for HJ PDEs with time-dependent Hamiltonians (middle), and the corresponding optimal control problem (bottom). The content of this illustration matches that of Figure \ref{['fig:intro_connection_in_words']} by replacing each term in Figure \ref{['fig:intro_connection_in_words']} with its corresponding mathematical expression. The colors indicate the associated quantities between each problem. The solid-line arrows denote direct equivalences. The dotted arrows represent additional mathematical relations.
  • Figure 3: Continual learning of the solution $u$ and source term $f$ of the boundary-value ODE \ref{['eq:ode']} using our Riccati-based approach, where information of $f$ is treated as a flow with respect to $t$ and cannot be stored once visited. $--$: inferences of $u$, $f$ at different $t$; ---: exact values of $u$, $f$; $--$: where integration has advanced in $t$ so far. Our Riccati-based approach naturally coincides with the continual learning framework by allowing new information to be continuously incorporated into the learned model without requiring access to any of the previous information. Instead, all of the previous information is encoded in the solution to the corresponding HJ PDE, thus avoiding catastrophic forgetting.
  • Figure 4: Continual learning of the solution $u$ and source term $f$ of the 2D Poisson equation \ref{['eq:poisson']} using our Riccati-based approach. White dashed lines: where integration has advanced in $y$ so far. Information of $f$ is discretized in $x$ and then propagated along $y$. Hence, our approach only requires access to 1D slices of the domain instead of the entire domain, which highlights the potential memory benefits of our Riccati-based approach.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2