Score-Based Langevin Surrogates Miss Escape Rates

Giorgini has shown that a cheap score-based Langevin surrogate can match the invariant measure and the short-lag correlations, thus reducing the need for dynamics that are expensive to simulate. While this surrogate matches statistics of typical behavior, we ask whether matching the invariant measure and the short-lag correlations still holds for rare events. A rare-event quantity of interest to researchers in molecular dynamics and climate is the escape rate: how often the system jumps from one metastable state to another. We construct an irreversible overdamped Langevin system with a circulation on the barrier between two metastable states. This construction is designed so that we are likely to see a failure to match the escape rate, because the circulation preserves the invariant measure, while short-lag correlations mostly see the metastable states and need not show this circulation. Our contribution is a concrete example where this surrogate, despite matching the invariant measure exactly by construction, and matching on short-lag autocorrelations, does not match the escape rate. Even when the antisymmetric part of the short-lag correlations is nonzero in the surrogate, the escape rate still does not match. For applications that can benefit from cheap learned surrogates, but still need the escape rate, checking the invariant measure and the short-lag correlations need not be enough.