Functional Adjoint Sampler: Scalable Sampling on Infinite Dimensional Spaces

Learning-based methods for sampling from the Gibbs distribution in finite-dimensional spaces have progressed quickly, yet theory and algorithmic design for infinite-dimensional function spaces remain limited. This gap persists despite their strong potential for sampling the paths of conditional diffusion processes, enabling efficient simulation of trajectories of diffusion processes that respect rare events or boundary constraints. In this work, we present the adjoint sampler for infinite-dimensional function spaces, a stochastic optimal control-based diffusion sampler that operates in function space and targets Gibbs-type distributions on infinite-dimensional Hilbert spaces. Our Functional Adjoint Sampler (FAS) generalizes Adjoint Sampling (Havens et al., 2025) to Hilbert spaces based on a SOC theory called stochastic maximum principle, yielding a simple and scalable matching-type objective for a functional representation. We show that FAS achieves superior transition path sampling performance across synthetic potential and real molecular systems, including Alanine Dipeptide and Chignolin.

Scientists often need to understand an entire process, not just a single outcome. In chemistry, for example, a molecule may remain in one stable shape for a long time before moving to another stable shape. These rare transitions can reveal how reactions occur or how molecules fold, but direct simulation is expensive because most simulated trajectories stay near already stable states. This paper introduces Functional Adjoint Sampler, a method for generating samples that are continuous objects, such as paths over time. A path is one example of a function: it describes how a system changes. Most existing sampling methods first break such objects into a fixed grid of points and then work only with that finite representation. Our method formulates the sampling problem directly at the level of functions, and uses a finite approximation only for computation. We apply this idea to transition path sampling, where the goal is to generate realistic paths between two prescribed states. Our method enforces these endpoint conditions by construction and produces reliable transition paths on synthetic potentials and molecular systems, including alanine dipeptide and chignolin. These results suggest that function-space sampling can help study rare events more practically.