Random Process Flow Matching: Generative Implicit Representations of Multivariate Random Fields

2026-05-27

AI Summary

This work addresses the challenge of generating high-fidelity realizations and providing reliable uncertainty quantification for high-dimensional, high-frequency, or multi-dimensional random fields from only a single sparse observational sample. The authors propose RP Flow, a novel framework that uniquely integrates flow matching with Gaussian process posteriors. By leveraging neural implicit representations and random Fourier features, RP Flow learns a continuous field from sparse data that can be queried at arbitrary locations. Calibrated uncertainty estimates are achieved through ensemble sampling. This approach overcomes the conventional reliance of generative models on large-scale datasets, enabling accurate synthesis and traceable uncertainty quantification even under extreme sparsity, high frequencies, or high-dimensional settings.

Abstract

Generative modeling provides a powerful framework for learning data distributions. These models initially relied on probabilistic methods such as Gaussian Processes (GP) for uncertainty-aware predictions and shifted towards larger trainable models to learn more complex distributions. In this work, we introduce Random Process (RP) Flow, a Flow Matching-based framework that represents the vector field as a neural implicit function. Unlike modern generative methods, our setting involves a single observed field, from which only sparse measurements are available. RP Flow uses Random Fourier Features to learn an implicit signal representation that can be queried at any arbitrary location from a limited set of observations, while encoding uncertainty through ensemble sampling. We propose constructing a Bayesian posterior by GP regression in the source space to generate high-quality samples. Our empirical results demonstrate that this framework generates realistic samples along with calibrated uncertainty estimates, even under challenging conditions such as high frequency, high sparsity, or high dimensionality. These findings position RP Flow as a milestone towards generative models for reconstruction tasks where data is scarce and uncertainty must remain traceable.

Problem

• Generative modeling
• Multivariate Random Fields
• Sparse observations
• Uncertainty quantification
• Implicit representations

Innovation

• Flow Matching
• Neural Implicit Representation
• Random Fourier Features
• Gaussian Process Regression
• Uncertainty Quantification