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

2025-05-19

AI Summary

To address data sparsity, indirectness, and limited sample size in random field modeling for scientific and engineering applications, this paper proposes a domain-knowledge-integrated constrained generative framework. Methodologically, we design a physics- and statistics-constrained variational autoencoder (cVAE), pioneering the incorporation of hard constraints into latent-space learning; develop a functional decoder that inherently enforces prior physical or statistical laws in reconstructions; and integrate latent-space flow matching for efficient, controllable probabilistic modeling. Our key contributions are: (i) the first synergistic optimization of physical constraints and flow matching within the VAE latent space; and (ii) state-of-the-art performance on wind velocity field reconstruction and material property inversion—achieving high-fidelity random field generation and rigorous uncertainty quantification from only a few indirect observations, significantly outperforming unconstrained baselines.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Web Mining and Content Analysis: Web data generation and simulation
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

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.

Problem

• Generative modeling of random fields with limited data
• Incorporating domain knowledge to enhance sparse data
• Improving reconstruction accuracy with constrained latent flow matching

Innovation

• Latent flow matching for generative modeling
• Function decoder in VAE for continuous samples
• Physical constraints integrated into VAE training