Initializing Doob's Lagrangian with the Committor: A Freidlin–Wentzell Perspective on Rare Event Sampling

Sampling rare transitions between metastable states is a fundamental challenge in computational chemistry because physically relevant transition paths occur on timescales far exceeding those accessible to direct simulation. Variational path optimization techniques based on minimizing a stochastic path action have been developed for decades, but most of these approaches require propagating the dynamics of multiple replicas of the system of interest. Doob's Lagrangian (DL) casts the problem as the minimization of an action over controlled path measures, with a neural control field and a training objective that does not require simulating the dynamics, but, in practice, it is sensitive to initialization and unstable under a misspecified transit time $T$. In this work, we establish a mathematical equivalence between the DL framework and Freidlin–Wentzell (FW) large-deviation theory, showing that minimizing the Doob action over both the control field and $T$ is asymptotically equivalent to computing the quasipotential governing Arrhenius–Kramers transition rates. This equivalence motivates a principled initialization strategy based on the stationary committor function, for which we prove, when a single transition channel dominates, that the initial control lies exponentially close in $T$ to the optimal Doob control field and that the joint $(v,T)$ optimization is stable. On alanine dipeptide and chignolin at finite temperature, the committor initialization combined with the optimization of $T$ removes the collapse of the transit time to unphysically small values observed under linear initialization, and recovers physically meaningful transition paths where the original DL method fails.