Infinite-dimensional Diffusion Bridge Simulation via Operator Learning

TL;DR

This paper introduces a novel operator learning approach combined with score matching to efficiently simulate infinite-dimensional diffusion bridges, overcoming intractability issues and enabling resolution-independent modeling.

Contribution

It presents the first method for infinite-dimensional diffusion bridge simulation that is discretization equivariant and adaptable to any resolution without retraining.

Findings

Effective on synthetic and real biological data

Achieves high accuracy and resolution independence

Outperforms existing finite-dimensional methods

Abstract

The diffusion bridge, which is a diffusion process conditioned on hitting a specific state within a finite period, has found broad applications in various scientific and engineering fields. However, simulating diffusion bridges for modeling natural data can be challenging due to both the intractability of the drift term and continuous representations of the data. Although several methods are available to simulate finite-dimensional diffusion bridges, infinite-dimensional cases remain under explored. This paper presents a method that merges score matching techniques with operator learning, enabling a direct approach to learn the infinite-dimensional bridge and achieving a discretization equivariant bridge simulation. We conduct a series of experiments, ranging from synthetic examples with closed-form solutions to the stochastic nonlinear evolution of real-world biological shape data. Our method demonstrates high efficacy, particularly due to its ability to adapt to any resolution without extra training.