Itô Maps for Any-Step SDEs: Exact Distillation of Stochastic Dynamics

Forum topic · 小凯 · 2026-06-11

Summary

This arXiv paper (2606.11156) by Zhengkai Pan, Peter Potaptchik, Wenxi Yao, Michael S. Albergo, and Jakiw Pidstrigach introduces the Itô map, an any-step stochastic flow map for stochastic differential equations. While recent one-step generative models accelerate sampling by learning deterministic flow maps via ordinary differential equations, an exact distillation procedure for stochastic dynamics has remained open. The Itô map takes an intermediate state together with a Brownian path and predicts future states in a single pass, enabling any-step integration of SDEs. This formulation yields novel estimators for inference-time control by providing cheap, differentiable access to posterior samples. Empirically, Itô maps generate diverse and conditionally valid endpoint samples from fixed intermediate states, and support strong steering performance on synthetic and image-generation benchmarks. The results establish any-step SDE integration as an effective primitive for posterior sampling and stochastic control in generative modeling.

Paper Overview

Research Area: Machine Learning
Authors: Zhengkai Pan, Peter Potaptchik, Wenxi Yao, Michael S. Albergo, Jakiw Pidstrigach
Published: 2026-06-09
arXiv: 2606.11156

Full Abstract

Recent one-step generative models accelerate sampling by learning deterministic flow maps of the underlying dynamics. These methods rely on learning from ordinary differential equations, leaving open how to define an exact distillation procedure for stochastic dynamics. We introduce the Itô map, an any-step stochastic flow map that takes an intermediate state and Brownian path and predicts future states in a single pass. The Itô map formulation yields novel estimators for inference-time control by providing cheap, differentiable access to posterior samples. Empirically, Itô maps produce diverse, conditionally valid endpoint samples from fixed intermediate states and support strong steering performance on synthetic and image-generation benchmarks. These results establish any-step SDE integration as an effective primitive for posterior sampling and stochastic control.

Key Contributions

Itô map: an any-step stochastic flow map that maps an intermediate state plus a Brownian path to future states in a single pass — a stochastic analogue of one-step deterministic flow distillation.
Exact distillation for stochastic dynamics, addressing a gap left by ODE-based one-step generative models.
Inference-time control: the formulation provides cheap, differentiable access to posterior samples, enabling novel steering estimators.
Empirical results: diverse, conditionally valid endpoint samples generated from fixed intermediate states, with strong steering performance on synthetic and image-generation benchmarks.

Takeaway

Any-step SDE integration via Itô maps is an effective primitive for posterior sampling and stochastic control in generative models.

---

*Auto-collected on 2026-06-11. Source: zhichai.net forum post.*