ODE-based sampling of Markov transition kernels in flow and diffusion models

Determine how to obtain samples from the Markov transition kernel p_{t'|t}(x_{t'} | x_t) of flow matching or diffusion models using only ODE sampling, rather than relying on SDE sampling.

Background

The paper contrasts two standard sampling paradigms for flow and diffusion models: ODE sampling, which is efficient but deterministic, and SDE sampling, which provides stochastic transitions via the kernel p_{t'|t} but is significantly less efficient. Many inference-time reward alignment algorithms depend on drawing samples from p_{t'|t}, making the lack of an ODE-based approach a practical bottleneck.

This explicit unknown motivates the introduction of GLASS Flows, which construct an "inner" flow matching model enabling ODE-based sampling of a broad class of Markov transitions, including the posterior and DDPM transitions, without additional training. The statement reflects the gap that previously existed prior to the method proposed in this work.

References

This creates a dilemma: So far, it is not known how to obtain samples X+ ~ Pt'|t(.|xt) using ODEs.

— GLASS Flows: Transition Sampling for Alignment of Flow and Diffusion Models  (2509.25170 - Holderrieth et al., 29 Sep 2025) in Section 1 (Introduction)

Analogous to , we hypothesize that the Brownian motion term $\mathrm{d}W_t{\mathcal{C}$ can be eliminated while preserving marginal distributions. This yields the probability flow ODE associated with the infinite-dimensional forward process: \begin{align}\label{equ:diffusion_ODE_infinite} \frac{\mathrm{d}u_t}{\mathrm{d}t} = f_{\theta}(u_t,t). \end{align} The conjecture is established strictly following the framework in , and the explicit form of $f_{\theta}(u_t,t)$ for the reverse SDE equ:diffusion_infinite requires further investigation.

— Learning Informative Prior with Infinite-Dimensional Continuous Normalizing Flow for Bayesian Inverse Problem  (2609.03343 - Zhao et al., 3 Sep 2026) in Section 2, Subsection 2.4, “Other Settings”