LatentFlow: A General Framework for Conditioning Stochastic Processes

2026-07-14

AI Summary

Existing stochastic process models often struggle to perform efficient conditional sampling under nonlinear observations, non-Gaussian likelihoods, or global constraints, typically requiring bespoke and complex algorithms. This work proposes the first universal conditioning framework that requires no training and avoids neural network approximations. The approach represents stochastic processes via deterministic mappings of tractable latent innovation variables, recasting conditional sampling as an inference problem in latent space. Exact solutions are achieved through backward-time stochastic differential equations (SDEs). The method accommodates a broad class of heterogeneous processes—including spatial priors, nonlinear dynamics, stochastic partial differential equations, extreme-value processes, and discrete-state processes—and enables high-fidelity conditional sampling on a single CPU within seconds, achieving both theoretical exactness and computational efficiency.

Abstract

Stochastic-process models are, as a rule, far easier to simulate than to condition. Non-linear observations, non-Gaussian likelihoods, black-box information, and global constraints all induce intractable conditional laws, requiring bespoke, model-specific constructions. We introduce LatentFlow, a single framework for conditioning stochastic processes, with no learned neural approximations and no training. Our starting point is to write the stochastic process as the deterministic image of a tractable latent innovation, $f_0 = T_{\vartheta}(ξ_0)$, with $ξ_0$ sampled from a simple reference distribution. This reduces process-level conditioning to latent-space inference: pull the likelihood back through $T_{\vartheta}$, sample the resulting latent law with a tractable guided probability flow, and push the samples forward. This construction is provably exact at the level of the target law; in practice, approximation enters only through finite terminal noising, Monte Carlo guidance, and time discretisation of the continuous-time dynamics, each of which is explicit and systematically reducible. As LatentFlow is training-free, conditioning reduces to solving a single reverse-time SDE. This enables conditional sampling in seconds on a single desktop CPU across model classes that have never shared a scalable method: classical spatial priors, nonlinear stochastic dynamics, mechanistic models from the physical and life sciences, stochastic PDEs, heavy-tails and extremes, point and discrete-state processes, and neural or simulator-defined processes.

Problem

• stochastic processes
• conditioning
• non-Gaussian likelihoods
• global constraints
• black-box information

Innovation

• LatentFlow
• stochastic process conditioning
• training-free inference
• reverse-time SDE
• latent-space inference