LatentFlow: A General Framework for Conditioning Stochastic Processes

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, $f_0 = T_{\vartheta}(\xi_0)$, with $\xi_0$ 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.