Goal-oriented learning of stochastic dynamical systems using error bounds on path-space observables

TL;DR

This paper develops a novel error bound and goal-oriented learning approach for stochastic dynamical systems, improving the accuracy of path-dependent observable predictions like reaction rates, with theoretical guarantees and practical benefits.

Contribution

Introduces an error bound for path-space observables and employs it as a variational loss for goal-oriented surrogate modeling of stochastic systems.

Findings

Enhanced accuracy in predicting first hitting time statistics.

Robustness to distributional shifts in training data.

Error bounds applicable to broad class of observables.

Abstract

The governing equations of stochastic dynamical systems often become cost-prohibitive for numerical simulation at large scales. Surrogate models of the governing equations, learned from data of the high-fidelity system, are routinely used to predict key observables with greater efficiency. However, standard choices of loss function for learning the surrogate model fail to provide error guarantees in path-dependent observables, such as reaction rates of molecular dynamical systems. This paper introduces an error bound for path-space observables and employs it as a novel variational loss for the goal-oriented learning of a stochastic dynamical system. We show the error bound holds for a broad class of observables, including mean first hitting times on unbounded time domains. We derive an analytical gradient of the goal-oriented loss function by leveraging the formula for Frechet derivatives of expected path functionals, which remains tractable for implementation in stochastic gradient descent schemes. We demonstrate that surrogate models of overdamped Langevin systems developed via goal-oriented learning achieve improved accuracy in predicting the statistics of a first hitting time observable and robustness to distributional shift in the data.