# Stochastic Interpolants in Hilbert Spaces

arXiv: https://arxiv.org/abs/2602.01988 · HTML: https://mlsanity.com/paper/2602.01988/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 27 of its 58 references that are in the map.
- First posted on arXiv 2 Feb 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 27 of 27)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Stochastic interpolants with data-dependent couplings](https://mlsanity.com/paper/2310.03725/) (2023)
- [Monte Carlo guided Diffusion for Bayesian linear inverse problems](https://mlsanity.com/paper/2308.07983/) (2023)
- [Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2302.10130/) (2023)
- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Open Materials Generation with Stochastic Interpolants](https://mlsanity.com/paper/2502.02582/) (2025)
- [Spectral Diffusion Processes](https://mlsanity.com/paper/2209.14125/) (2022)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Stochastic Optimal Control for Diffusion Bridges in Function Spaces](https://mlsanity.com/paper/2405.20630/) (2024)
- [Conditioning non-linear and infinite-dimensional diffusion processes](https://mlsanity.com/paper/2402.01434/) (2024)
- [Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems](https://mlsanity.com/paper/2405.15676/) (2024)
- [Infinite-dimensional Diffusion Bridge Simulation via Operator Learning](https://mlsanity.com/paper/2405.18353/) (2024)
- [Autoregressive regularized score-based diffusion models for multi-scenarios fluid flow prediction](https://mlsanity.com/paper/2505.24145/) (2025)

## Abstract (arXiv)

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.