Port--Hamiltonian Diffusion Models: A Control-Theoretic Perspective on Generative Modeling

TL;DR

This paper introduces a control-theoretic framework for diffusion models by embedding them into port-Hamiltonian systems, providing structural insights and stability guarantees for generative modeling.

Contribution

It establishes a novel port-Hamiltonian systems perspective on diffusion models, linking score functions to Hamiltonian energy and ensuring stability independently of score accuracy.

Findings

Diffusion processes can be modeled as port-Hamiltonian systems.

The reverse diffusion process is interpreted as feedback-controlled PH dynamics.

Stability guarantees are achieved regardless of score estimation errors.

Abstract

Diffusion models have recently achieved remarkable success in generative modeling, yet they are commonly formulated as black-box stochastic systems with limited interpretability and few structural guarantees. In this paper, we establish a control-theoretic foundation for diffusion models by embedding them within the port--Hamiltonian (PH) systems framework. We show that the score function can be interpreted as the gradient of a learnable Hamiltonian energy, allowing both the forward and reverse diffusion processes to be formulated as structured PH dynamics. The reverse-time generative process is further interpreted as a feedback-controlled PH system, where dissipation plays a fundamental role in stabilizing sampling dynamics. This formulation yields intrinsic stability guarantees that are independent of score estimation accuracy. A simple analytical example illustrates the proposed framework.