Supervised Guidance Training for Infinite-Dimensional Diffusion Models

Score-based diffusion models have recently been extended to infinite-dimensional function spaces, with uses such as inverse problems arising from partial differential equations. In the Bayesian formulation of inverse problems, the aim is to sample from a posterior distribution over functions obtained by conditioning a prior on noisy observations. While diffusion models provide expressive priors in function space, the theory of conditioning them to sample from the posterior remains open. We address this, assuming that either the prior lies in the Cameron-Martin space, or is absolutely continuous with respect to a Gaussian measure. We prove that the models can be conditioned using an infinite-dimensional extension of Doob's $h$-transform, and that the conditional score decomposes into an unconditional score and a guidance term. As the guidance term is intractable, we propose a simulation-free score matching objective (called *Supervised Guidance Training*) enabling efficient and stable posterior sampling. We illustrate the theory with numerical examples on Bayesian inverse problems in function spaces. In summary, our work offers the first function-space method for fine-tuning trained diffusion models to accurately sample from a posterior.

Many scientific problems involve recovering an unknown value from data comprising of noisy observed measurements. For example, maybe we take a temperature reading once a day and would like to infer the temperature for the whole day. The problem is, there are many solutions that could explain the same data. A principled way to handle this uncertainty is Bayesian inference, which produces a range of solutions with probabilities for each one, called a posterior distribution, to explain how plausible each solution is. Diffusion models are powerful tools for learning such distributions, but until now there was no mathematically rigorous way to condition them on observations when the unknown quantity is a function rather than a finite-dimensional vector. We introduce a new algorithm that takes a pretrained model and guides it with the observed measurements, to get samples of solutions. Our method allows scientists to reuse a single pre-trained model across many different observation settings, simply by training a lightweight guidance network. Experiments on PDE-based inverse problems and shape reconstruction demonstrate that our method can accurately sample from posterior distributions, giving a principled way to adapt diffusion models for Bayesian inference in scientific applications.