Fokker-Planck Score Learning: Efficient Free-Energy Estimation under Periodic Boundary Conditions

TL;DR

This paper introduces a physics-informed, score-based diffusion method that leverages periodic boundary conditions to efficiently estimate free energy profiles in molecular simulations, outperforming traditional sampling techniques.

Contribution

The authors develop a novel Fokker-Planck score learning framework that explicitly exploits periodic boundary conditions for more efficient free-energy estimation.

Findings

Up to tenfold increase in efficiency over umbrella sampling.

Effective reconstruction of free-energy profiles in benchmark and membrane permeation scenarios.

Neural network trained on non-equilibrium trajectories accurately encodes free-energy gradients.

Abstract

Accurate free-energy estimation is essential in molecular simulation, yet the periodic boundary conditions (PBC) commonly used in computer simulations have rarely been explicitly exploited. Equilibrium methods such as umbrella sampling, metadynamics, and adaptive biasing force require extensive sampling, while non-equilibrium pulling with Jarzynski's equality suffers from poor convergence due to exponential averaging. Here, we introduce a physics-informed, score-based diffusion framework: by mapping PBC simulations onto a Brownian particle in a periodic potential, we derive the Fokker-Planck steady-state score that directly encodes free-energy gradients. A neural network is trained on non-equilibrium trajectories to learn this score, providing a principled scheme to efficiently reconstruct the potential of mean force (PMF). On benchmark periodic potentials and small-molecule membrane permeation, our method is up to one order of magnitude more efficient than umbrella sampling.