Random Process Flow Matching: Generative Implicit Representations of Multivariate Random Fields
For practitioners in scientific and engineering domains dealing with scarce field data, this framework provides a generative model that combines reconstruction with uncertainty quantification.
RP Flow introduces a Flow Matching-based generative model for multivariate random fields that learns an implicit neural representation from sparse observations, generating realistic samples with calibrated uncertainty estimates even under high frequency, sparsity, or dimensionality.
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.