Free energy Estimation on Any State Space

2026-05-29

AI Summary

Free energy estimation is crucial in physics and statistical inference, yet remains challenging in non-continuous state spaces such as discrete, multimodal, or autoregressive settings. This work proposes a generalized neural transport learning framework that, for the first time, extends efficient free energy estimation algorithms to arbitrary state spaces. The key innovation lies in uncovering a generalized dihedral group structure formed by infinitesimal time reversal and the generalized Doob h-transform, which is integrated with thermodynamic path integrals and group-theoretic analysis. Experimental results demonstrate that the proposed method significantly improves both efficiency and accuracy of free energy estimation across continuous, discrete, multimodal, and autoregressive configurations.

Abstract

Free energy estimation is a fundamental yet challenging problem, from physics to statistics. Classical approaches rely on thermodynamic transformations, ranging from direct estimation, quasistatic integration, to finite-time averaging. Recent work [He and Du et al., 2025] learns neural transports to significantly accelerate the efficiency in the finite-time regime. In this paper, we generalize this framework to arbitrary state spaces. Building on this view, we develop a generalized neural transport learning approach for efficient estimation. Experiments validate the effectiveness and efficiency of the proposed method beyond continuous settings, extending to discrete and multimodal spaces as well as autoregressive settings. Beyond free energy estimation, we establish algebraic identities and reveal a group-theoretic structure linking infinitesimal time reversal and generalized Doob's $h$-transforms, showing that their compositions form a generalized dihedral group.

Problem

• free energy estimation
• state space
• discrete spaces
• multimodal spaces
• autoregressive settings

Innovation

• neural transport
• free energy estimation
• arbitrary state spaces
• Doob's h-transform
• generalized dihedral group