Stochastic Interpolants in Hilbert Spaces

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.

💡 Research Summary

The paper addresses a fundamental limitation of current generative models—most notably diffusion models and stochastic interpolants (SIs)—which are formulated in finite‑dimensional Euclidean spaces. While diffusion models have been successfully applied to function‑valued data by discretising the underlying function on a grid, this approach ignores the infinite‑dimensional nature of the data and can suffer from deteriorating performance as the discretisation becomes finer. Moreover, the theory of SIs, which provides a flexible bridge between arbitrary source and target distributions, has never been rigorously extended beyond finite dimensions.

The authors propose a comprehensive framework for stochastic interpolants directly in separable Hilbert spaces, thereby removing the reliance on Lebesgue measure and isotropic Gaussian noise. The key ingredients are:

1. Trace‑class Gaussian noise – a Gaussian measure (N(0,C)) with a trace‑class covariance operator (C) is used as the reference measure. This ensures that samples lie almost surely in the Hilbert space and avoids the Feldman‑Hajek singularity problem that would arise with time‑varying covariances.

2. Hilbert‑space stochastic interpolant – defined as
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