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

Majid Darehmiraki

TL;DR

This work addresses the lack of interpretability and intrinsic stability in diffusion-based generative models by embedding diffusion dynamics within the port-Hamiltonian (PH) framework. It introduces an energy-based parameterization where the score equals $- abla_x H_ heta(x,t)$ and formulates both forward and reverse processes as PH flows, with the reverse time implemented as a feedback-controlled, dissipative system. A Lyapunov stability proposition shows that if the dissipation term $R+GG^\top$ is positive definite, the dynamics are globally asymptotically stable to the energy-minimizing equilibria, yielding robustness to score estimation errors and contraction of trajectory distributions. The framework clarifies its relationship to the exact reverse-time SDE, showing equivalence only in the ideal case where the learned Hamiltonian perfectly matches the true marginals, while offering a principled, structure-preserving alternative otherwise. Numerical illustrations in 1D and 2D demonstrate energy-dissipative sampling flows that converge to data-mode attractors, highlighting the practical potential for stable and interpretable generative modeling.

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.

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

TL;DR

This work addresses the lack of interpretability and intrinsic stability in diffusion-based generative models by embedding diffusion dynamics within the port-Hamiltonian (PH) framework. It introduces an energy-based parameterization where the score equals and formulates both forward and reverse processes as PH flows, with the reverse time implemented as a feedback-controlled, dissipative system. A Lyapunov stability proposition shows that if the dissipation term is positive definite, the dynamics are globally asymptotically stable to the energy-minimizing equilibria, yielding robustness to score estimation errors and contraction of trajectory distributions. The framework clarifies its relationship to the exact reverse-time SDE, showing equivalence only in the ideal case where the learned Hamiltonian perfectly matches the true marginals, while offering a principled, structure-preserving alternative otherwise. Numerical illustrations in 1D and 2D demonstrate energy-dissipative sampling flows that converge to data-mode attractors, highlighting the practical potential for stable and interpretable generative modeling.

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.
Paper Structure (31 sections, 1 theorem, 38 equations, 3 figures)

This paper contains 31 sections, 1 theorem, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Port--Hamiltonian Systems
  4. Diffusion and Score-Based Generative Models
  5. Port--Hamiltonian Diffusion Models: A Rigorous Formulation
  6. Energy-Based Parameterization
  7. Forward Diffusion as a Stochastic PH System
  8. Assumption 3.1 (Well-posedness).
  9. Energy Evolution
  10. Remark 3.1 (Time-dependent energy).
  11. Reverse-Time Dynamics as a Controlled PH System
  12. Passivity and Stability
  13. Proposition 3.1 (Lyapunov stability).
  14. Relation to Score-Based Models
  15. Remark 3.2 (Interpretational scope).
  16. ...and 16 more sections

Key Result

Theorem 4.1

If $R + GG^\top \succ 0$ and $H_\theta$ is bounded from below, then the reverse-time dynamics are globally asymptotically stable in expectation.

Figures (3)

  • Figure 1: One-dimensional forward diffusion process (SDE). Left: Sample trajectories evolving over time. Right: Histogram of the final state distribution at $t=T_{\text{forward}}$ compared with the theoretical stationary distribution $\mathcal{N}(0, \sigma^2 / 2\alpha)$.
  • Figure 2: One-dimensional reverse generative process (ODE). Left: The quadratic Hamiltonian energy landscape $H(x) = \frac{1}{2}x^2$. Right: Sample reverse trajectories converging to the stable equilibrium at $x=0$, which is the minimum of the Hamiltonian.
  • Figure 3: Two-dimensional PH diffusion model. Left: 3D surface plot of the multi-modal Hamiltonian energy landscape $H(\mathbf{x}) = (x_1^2 - 1)^2 + (x_2^2 - 1)^2$. Center: Contour plot of the energy landscape with trajectories of the forward SDE $d\mathbf{x}_t = (J-R)\nabla H(\mathbf{x}_t)dt + Gd\mathbf{W}_t$ starting near the origin and diffusing outwards. Right: Contour plot of the energy landscape with trajectories of the reverse ODE $\dot{\mathbf{x}}_t = (J-R-GG^\top)\nabla H(\mathbf{x}_t)$ (shown as solid lines) starting from random initial conditions (black circles) and converging to the minima of the Hamiltonian (red stars).

Theorems & Definitions (2)

  • Theorem 4.1
  • proof