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arXiv · 2601.06071

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

Majid Darehmiraki
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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.

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Majid Darehmiraki. 2025-12-29. Port--Hamiltonian Diffusion Models: A Control-Theoretic Perspective on Generative Modeling. https://arxiv.org/abs/2601.06071

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