Goal-oriented learning of stochastic dynamical systems using error bounds on path-space observables
This work addresses the need for reliable surrogate models in fields like molecular dynamics, though it is incremental as it builds on existing variational loss methods.
The paper tackles the problem of learning surrogate models for stochastic dynamical systems without error guarantees for path-dependent observables, and introduces a goal-oriented loss function that improves accuracy in predicting first hitting time statistics and robustness to distributional shift.
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.