Dynamical Partition Functions of Stochastic Dynamics via Variational Flows

Nonequilibrium thermodynamics is governed by the dynamical partition function, and its computation in high-dimensional continuous-state dynamics is a longstanding challenge. The Feynman-Kac formalism provides a rigorous representation for generating functions of arbitrary path observables; however, practical evaluation beyond low dimensions or the weak-noise limit is hindered by the curse of dimensionality and the exponentially growing replica demands of trajectory-based methods. Here we develop a mesh-free neural variational framework that realizes the Feynman-Kac theorem with generative flow models, recasting tilted stochastic evolution as a time-dependent optimization problem. This approach enables the direct computation of both finite-time and asymptotic trajectory thermodynamics in a unified manner. The method applies to general observables, enabling the evaluation of work, entropy production, and current fluctuations. We demonstrate the accuracy and scalability of this method in various nonequilibrium systems including high-dimensional cases.