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Stochastic Interpolants in Hilbert Spaces

James Boran Yu, RuiKang OuYang, Julien Horwood, José Miguel Hernández-Lobato

TL;DR

This work addresses the gap in bridging arbitrary functional distributions by formulating Stochastic Interpolants directly in infinite-dimensional Hilbert spaces. It introduces a time-changed Conditional Bridge SDE that enables conditional generation between coupled function-valued distributions, and proves existence and strong uniqueness of solutions under $H_C$-Lipschitz conditions with two data-assumption regimes. A Wasserstein-$\mathcal{W}_2$ error bound ties learning accuracy of the velocity and denoiser to distributional closeness, while a regularising time change stabilises training and sampling. Empirically, the method achieves competitive or state-of-the-art results on PDE-based forward and inverse problems, highlighting the practical impact for scientific discovery in function spaces. The framework rigorously addresses measure-theoretic and well-posedness challenges unique to infinite dimensions and offers a principled pathway for functional generative modeling.

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.

Stochastic Interpolants in Hilbert Spaces

TL;DR

This work addresses the gap in bridging arbitrary functional distributions by formulating Stochastic Interpolants directly in infinite-dimensional Hilbert spaces. It introduces a time-changed Conditional Bridge SDE that enables conditional generation between coupled function-valued distributions, and proves existence and strong uniqueness of solutions under -Lipschitz conditions with two data-assumption regimes. A Wasserstein- error bound ties learning accuracy of the velocity and denoiser to distributional closeness, while a regularising time change stabilises training and sampling. Empirically, the method achieves competitive or state-of-the-art results on PDE-based forward and inverse problems, highlighting the practical impact for scientific discovery in function spaces. The framework rigorously addresses measure-theoretic and well-posedness challenges unique to infinite dimensions and offers a principled pathway for functional generative modeling.

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.
Paper Structure (78 sections, 14 theorems, 129 equations, 2 figures, 4 tables, 2 algorithms)

This paper contains 78 sections, 14 theorems, 129 equations, 2 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Stochastic Differential Equations and Stochastic Interpolants in finite-dimensional spaces
  5. SDEs in $\mathbb{R}^d$.
  6. Stochastic Interpolants.
  7. Gaussian Measures and Stochastic Processes in Hilbert Spaces
  8. Hilbert Spaces.
  9. Gaussian Measures in Hilbert Spaces.
  10. Cameron-Martin Spaces.
  11. Stochastic Processes in $H$.
  12. Challenges in Extending SIs to Infinite Dimensions
  13. Choice of Gaussian Noise.
  14. No Lebesgue Measure.
  15. Well-Posedness of SDEs.
  16. ...and 63 more sections

Key Result

Theorem 2

Suppose that for any $\overline{t} \in (0, 1)$, the CB-SDE (eqn:cbsde) admits a unique (in law) solution on $[0, \overline{t}]$ for $\mu_{0}$-every initial condition $x_{0}$ and the associated infinite-dimensional Fokker-Planck equation is uniquely solvable. Then, for any $t \in (0, 1)$, the law of

Figures (2)

  • Figure 1: Evolution of predictions for forward (a) and inverse (b) tasks on a randomly chosen example in the test set. For SDE inference, we generate 150 predictions to visualise the distribution of samples.
  • Figure 2: Evolution of predictions from the ODE (top) and a single sample of the SDE (bottom), for forward and inverse tasks. We randomly select an example from the test set for Darcy flow and Navier-Stokes. We set the RBF length scale $\ell = 0.02$.

Theorems & Definitions (36)

  • Definition 1
  • Theorem 2
  • proof
  • Proposition 3
  • proof
  • Remark 3.1
  • Remark 3.2
  • Theorem 4: Existence
  • proof : Proof
  • Theorem 5: Uniqueness
  • ...and 26 more