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.

翻译：尽管扩散模型已成功扩展至函数值数据领域，但随机插值——这一为任意分布间构建桥梁的灵活方法——目前仍局限于有限维场景。本研究通过建立无限维希尔伯特空间中随机插值的严格框架，填补了这一空白。我们提供了完整的理论基础，包括解的存在唯一性证明及显式误差界。我们通过聚焦于复杂的偏微分方程基准测试，展示了所提框架在条件生成任务中的有效性。该方法能够实现任意函数分布间的生成式桥梁构建，取得了最先进的性能，为科学发现提供了一个强大且通用的工具。