Stochastic Interpolants in Hilbert Spaces

TL;DR

This paper develops a rigorous framework for stochastic interpolants in infinite-dimensional Hilbert spaces, enabling flexible generative modeling of function-valued data and complex PDE-based benchmarks.

Contribution

It extends stochastic interpolants from finite to infinite-dimensional spaces with theoretical foundations and practical applications.

Findings

Achieves state-of-the-art results on PDE benchmarks

Provides explicit error bounds and well-posedness proofs

Enables generative bridges between arbitrary functional distributions

Abstract

Although diffusion models have successfully extended to function-valued data, stochastic interpolants -- which offer a flexible way to bridge arbitrary distributions -- remain limited to finite-dimensional settings. This work bridges this gap by establishing a rigorous framework for stochastic interpolants in infinite-dimensional Hilbert spaces. We provide comprehensive theoretical foundations, including proofs of well-posedness and explicit error bounds. We demonstrate the effectiveness of the proposed framework for conditional generation, focusing particularly on complex PDE-based benchmarks. By enabling generative bridges between arbitrary functional distributions, our approach achieves state-of-the-art results, offering a powerful, general-purpose tool for scientific discovery.