Generative Modeling of Random Fields from Limited Data via Constrained Latent Flow Matching

TL;DR

This work introduces constrained latent flow matching (c-LFM), a framework for learning distributions over random fields $p(\mathbf{u})$ from limited data by operating in the latent space of a pre-trained VAE that employs a DeepONet-based function decoder. By augmenting the ELBO with statistical and physical residuals $R(\cdot)$ and $F(\cdot)$ and applying latent-flow matching in the latent space, the method yields continuous function samples that respect domain knowledge and remain feasible under data scarcity. The approach is demonstrated on random-field reconstruction from sparse sensors and random-field inference from indirect data, and extended to real-world-inspired wind velocity field estimation and material property characterization, showing improved reconstruction accuracy, coherence, and feasibility compared to unconstrained baselines, especially in small-data regimes. These results highlight the practical impact of integrating physics/statistical constraints with latent-space generative modeling to produce reliable, continuous random-field samples when training data are scarce or indirect.

Abstract

Deep generative models are promising tools for science and engineering, but their reliance on abundant, high-quality data limits applicability. We present a novel framework for generative modeling of random fields (probability distributions over continuous functions) that incorporates domain knowledge to supplement limited, sparse, and indirect data. The foundation of the approach is latent flow matching, where generative modeling occurs on compressed function representations in the latent space of a pre-trained variational autoencoder (VAE). Innovations include the adoption of a function decoder within the VAE and integration of physical/statistical constraints into the VAE training process. In this way, a latent function representation is learned that yields continuous random field samples satisfying domain-specific constraints when decoded, even in data-limited regimes. Efficacy is demonstrated on two challenging applications: wind velocity field reconstruction from sparse sensors and material property inference from a limited number of indirect measurements. Results show that the proposed framework achieves significant improvements in reconstruction accuracy compared to unconstrained methods and enables effective inference with relatively small training datasets that is intractable without constraints.

Figures (17)

• Figure 1: c-LFM for modeling random fields from limited data. A VAE with function decoder learns a latent representation of the continuous random field, $\mathbf{U}(\mathbf{x}, \omega)$, from sparse observations. VAE loss terms are shown in purple boxes, where a residual error supplements limited data with statistical/physical constraints. Flow matching enables latent variable sampling.
• Figure 2: Demonstration of random field reconstruction from sparse data with c-LFM. (Top) True samples and statistics from a Gaussian Process along with (Bottom) generated samples and statistics using c-LFM with covariance constraint (left) versus standard LFM with no constraint (right). The statistical constraint allows for accurate covariance recovery from sparse sensors.
• Figure 3: Demonstration of random field inference from indirect data with c-LFM. (Top) true samples of $u(x)$ (observed) and $v(x)$ (unknown) and (Bottom) generated samples along with true (blue) versus generated (red) pointwise distributions. A physical constraint incorporating the Poisson equation allows for inference of $v(x)$ from limitied observations of $u(x)$ only.
• Figure 4: Wind velocity estimation (a) diagram along with comparisons of (b) mean and variance wind fields and (c) coherence for the true wind test data versus generated samples (with and without coherence constraint).
• Figure 5: Material property characterization (a) diagram, the (b) PDF at three distinct spatial coordinates, $\hat{\mathbf{x}}$, and (c) correlation with respect to $V(0.5, \hat{x_2})$ along horizontal slices versus the true material property, and (d) the relative L2 error in mean and variance field vs. number of train data.