Nonequilibrium free energy calculations from time-reversed path ensembles through stochastic optimal control*

As free energy calculations come to occupy central roles in research fronts throughout chemistry, physics, biology, and data science, it is timely to address computational cost-efficiency challenges for applications in and out of thermodynamic equilibrium.  For equilibrium applications, free energy calculations relying on extensive sampling of Boltzmann--Gibbs conformational ensembles can prove prohibitively expensive when screening vast swaths of chemical space; for nonequilibrium applications, burgeoning interest in driven systems and active matter has opened a new market for methods to quantify free energy distortions due to persistent currents originating from thermal gradients and active forces.  Both sets of applications are addressed by the fluctuation relations of Jarzynski, Kawasaki, and Crooks, which relate free energy differences to path-wise work/heat averages and suggest ways to enable (in nonequilibrium applications) or accelerate (in equilibrium applications) free energy calculations by reweighing trajectory data from short, irreversible transformations.  Yet, many numerical implementations of fluctuation relations have been fraught with numerical issues due to statistically inefficient choices of the nonequilibrium switching protocol.  In our work, we build from Jarzynski's observed connection between time-reversed driven trajectory ensembles and statistically optimal nonequilibrium averages [C. Jarzynski, Phys. Rev. E  73, 046105 (2006)] to characterize switching protocols that render rare, time-reversed trajectories typical via optimally controlled overdamped Langevin dynamics.  For several test diffusions featuring both additive and multiplicative noise, we adaptively learn optimal switching protocols by fitting a Hilbert or neural-network ansatz to experimental or simulated trajectory data, from which we estimate equilibrium free energy landscapes and their nonequilibrium distortions.