Energy Generative Modeling: A Lyapunov-based Energy Matching Perspective

TL;DR

This paper introduces a unified Lyapunov-based control framework for static scalar energy generative models, enabling simultaneous training and sampling with theoretical guarantees.

Contribution

It unifies training and sampling in a single control-theoretic framework and derives finite-step stopping criteria for Langevin sampling.

Findings

Finite step stopping criterion for Langevin sampling.

No Lyapunov certificate exists for deterministic gradient flow.

Additive composition of energies preserves Gibbs measure and Lyapunov certificate.

Abstract

Generative models based on static scalar energy functions represent an emerging paradigm in which a single time independent potential drives sample generation through its gradient field, eliminating the need for time conditioning entirely. We unify the training and sampling phases of this paradigm, conventionally treated as separate procedures, within a single framework: density transport on the Wasserstein space, cast as a nonlinear control problem in which the Kullback Leibler (KL) divergence serves as a Lyapunov function. Training and sampling are then two instances of this same master dynamics, differing only in initial condition. Within this autonomous framework we develop two analytic results. First, since the Lyapunov certificate is asymptotic, we derive a finite step stopping criterion for Langevin sampling and prove that no Lyapunov certificate exists for the deterministic gradient flow on the same energy landscape. Second, the reformulation brings the toolkit of nonlinear control theory to bear on static scalar energy generative modeling, that is, we show that additive composition of trained scalar energies retains an explicit Gibbs invariant measure and inherits the closed-loop Lyapunov certificate. Beyond these immediate results, this reformulation bridges static scalar energy generative models with the full toolkit of nonlinear control theory, opening the door to barrier functions for constrained generation and contraction metrics for accelerated sampling. Experiments on synthetic distributions validate the theoretical predictions.