Infinite-dimensional Diffusion Bridge Simulation via Operator Learning

2024-05-28

AI Summary

Simulating diffusion bridges in infinite-dimensional spaces—arising from continuous representations of natural data—is hindered by intractable drift terms and the absence of closed-form solutions for conditional processes. Method: We propose the first end-to-end framework integrating score matching with neural operator learning (e.g., Fourier Neural Operator), directly learning discretization-equivariant infinite-dimensional bridge processes without explicitly solving stochastic differential equations. Contribution/Results: The method enables zero-shot generalization across arbitrary spatial resolutions. Evaluated on closed-form synthetic benchmarks and real-world biological morphological evolution tasks, it achieves significantly higher path fidelity compared to baseline methods, improves sampling efficiency by multiple-fold, and seamlessly adapts to multi-scale discretizations without retraining.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Diffusion Models for Vision
• Machine Learning: Evolutionary Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Simulating infinite-dimensional diffusion bridges efficiently
• Overcoming intractable drift and continuous data challenges
• Achieving resolution-adaptive discretization equivariant simulations

Innovation

• Combines score matching with operator learning
• Learns infinite-dimensional bridge directly
• Achieves discretization equivariant simulation