Itô maps for any-step SDEs

2026-06-09

AI Summary

This work addresses the limitations of existing single-step generative models, which predominantly rely on ordinary differential equations and struggle to accurately model stochastic dynamical systems. The authors propose the Itô map—a theoretically grounded, exact flow mapping derived from Itô integral theory for stochastic differential equations—that enables one-step prediction of future states from intermediate states and Brownian paths with arbitrary step sizes. This approach supports end-to-end training and facilitates efficient, differentiable posterior sampling and stochastic control during inference. Experimental results demonstrate that the method generates diverse yet conditionally consistent terminal samples on both synthetic data and image generation tasks, significantly enhancing guided control capabilities.

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 a useful primitive for posterior sampling and stochastic control.

Problem

• stochastic differential equations
• any-step integration
• posterior sampling
• stochastic control
• flow maps

Innovation

• Itô map
• any-step SDE
• stochastic flow
• posterior sampling
• differentiable inference