Port--Hamiltonian Diffusion Models: A Control-Theoretic Perspective on Generative Modeling
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911365566824448 |
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| author | Darehmiraki, Majid |
| author_facet | Darehmiraki, Majid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06071 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Port--Hamiltonian Diffusion Models: A Control-Theoretic Perspective on Generative Modeling Darehmiraki, Majid Mathematical Physics 60H10, 34H05 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. |
| title | Port--Hamiltonian Diffusion Models: A Control-Theoretic Perspective on Generative Modeling |
| topic | Mathematical Physics 60H10, 34H05 |
| url | https://arxiv.org/abs/2601.06071 |