Stochastic Lifting for Generating Trajectories of Stochastic Physical Systems

Flow and diffusion models generate high-quality samples in many modalities; however, many network evaluations are required during inference due to numerical integration of an underlying differential equation. Flow maps alleviate this problem by learning the solution map of the differential equation directly, enabling few-step sampling. Yet, current methods are restricted to approximating the solution map of ODEs. These methods can be used to learn the transition kernel of an SDE, thereby obtaining a solution map that recovers the marginal distributions of the process (weak convergence) rather than the solution path (strong convergence). We propose Strong Stochastic Flow Maps (SSFMs) as a novel framework for learning the strong solution map of additive-noise SDEs, directly generalizing deterministic flow maps to the stochastic setting. Further, a polynomial approximation to Brownian motion is introduced and shown to converge pathwise. These results enable a simulation-free training objective for the solution map of diffusion models. We demonstrate that SSFMs outperform previous stochastic flow map methods on image generation and enable few-step sampling of molecular systems.

Diffusion models achieve strong generation quality, diversity, and distribution coverage, but their performance often comes with expensive inference. In this work, we propose Stochastic Transition-Map Distillation (STMD), a teacher-free framework for accelerating diffusion model inference while preserving probabilistic sample generation. In contrast to score-based diffusion models, whose denoising parametrization models the mean of the posterior distribution, STMD distills the full transition map associated with the sampling stochastic differential equation (SDE). We parameterize these SDE transitions with a conditional Mean Flow model, yielding a one- or few-step stochastic sampler that retains the transition structure of the underlying diffusion process. This perspective is especially useful for downstream tasks that require stochastic inference, such as diffusion posterior sampling, inverse problems, and energy-based fine-tuning. Compared to recent distillation methods, STMD requires no pretrained teacher, bi-level optimization, or trajectory simulation and caching, enabling efficient and scalable training. We derive convergence bounds for our method in the Wasserstein distance, providing a strong theoretical foundation for our approach, and validate STMD on various image generation examples on the MNIST, CIFAR-10, and CelebA datasets.

We develop a kernel method for generative modeling within the stochastic interpolant framework, replacing neural network training with linear systems. The drift of the generative SDE is $\hat b_t(x) = \nablaφ(x)^\topη_t$, where $η_t \in \mathbb{R}^P$ solves a $P\times P$ system computable from data, with $P$ independent of the data dimension $d$. Since estimates are inexact, the diffusion coefficient $D_t$ affects sample quality; the optimal $D_t^*$ from Girsanov diverges at $t=0$, but this poses no difficulty and we develop an integrator that handles it seamlessly. The framework accommodates diverse feature maps: scattering transforms, pretrained generative models, etc, enabling generation and model combination without neural network training. We demonstrate the approach on financial time series, turbulence, and image generation.