Stochastic Interpolants in Hilbert Spaces

2026-02-02

AI Summary

This work addresses the limitations of existing stochastic interpolation methods, which are confined to finite-dimensional spaces and thus struggle to model generative tasks between arbitrary distributions in function spaces. For the first time, the authors extend stochastic interpolation theory to infinite-dimensional Hilbert spaces, establishing a rigorous mathematical framework grounded in functional analysis and stochastic differential equations. They provide well-posedness guarantees and explicit error bounds, thereby overcoming dimensional constraints and enabling controllable generation under complex conditions. The proposed method achieves state-of-the-art performance on PDE-driven function space benchmark tasks, offering a general and efficient tool for generating high-dimensional continuous distributions in scientific computing.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation

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.

Problem

• stochastic interpolants
• Hilbert spaces
• infinite-dimensional
• generative modeling
• function-valued data

Innovation

• stochastic interpolants
• Hilbert spaces
• infinite-dimensional generative modeling
• conditional generation
• function-valued data