Stochastic Interpolants in Hilbert Spaces
This provides a powerful, general-purpose tool for scientific discovery by enabling generative bridges between arbitrary functional distributions.
The paper tackles the limitation of stochastic interpolants to finite-dimensional settings by establishing a rigorous framework for them in infinite-dimensional Hilbert spaces, achieving state-of-the-art results on complex PDE-based benchmarks.
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.