# Latent Generative Modeling of Random Fields from Limited Training Data

arXiv: https://arxiv.org/abs/2505.13007 · HTML: https://mlsanity.com/paper/2505.13007/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 54 references that are in the map.
- First posted on arXiv 19 May 2025.

## Builds on (most cited first, 18 of 18)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Solving Inverse Problems with Latent Diffusion Models via Hard Data Consistency](https://mlsanity.com/paper/2307.08123/) (2023)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [Enforcing Statistical Constraints in Generative Adversarial Networks for Modeling Chaotic Dynamical Systems](https://mlsanity.com/paper/1905.06841/) (2019)
- [End-To-End Latent Variational Diffusion Models for Inverse Problems in High Energy Physics](https://mlsanity.com/paper/2305.10399/) (2023)
- [PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physics](https://mlsanity.com/paper/2106.02993/) (2021)
- [Training-Free Constrained Generation With Stable Diffusion Models](https://mlsanity.com/paper/2502.05625/) (2025)
- [Highly-scalable, physics-informed GANs for learning solutions of stochastic PDEs](https://mlsanity.com/paper/1910.13444/) (2019)
- [Inverse Estimation of Elastic Modulus Using Physics-Informed Generative Adversarial Networks](https://mlsanity.com/paper/2006.05791/) (2020)
- [Generative Modeling of Microweather Wind Velocities for Urban Air Mobility](https://mlsanity.com/paper/2503.02690/) (2025)

## Built on by (most cited first, 1 of 1)

- [Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Sampling](https://mlsanity.com/paper/2605.12754/) (2026)

## Abstract (arXiv)

The ability to accurately model random fields plays a critical role in science and engineering for problems involving uncertain, spatially-varying quantities such as heterogeneous material properties and turbulent flows. Deep generative models offer a powerful tool for sampling high- or infinite-dimensional uncertainties like random fields, but their reliance on large, dense training datasets limits their applicability in contexts where sufficient data is difficult or expensive to obtain. In this work, we propose a latent-space approach to generative modeling of random fields that incorporates domain knowledge to supplement limited training data. A constraint-aware variational autoencoder (VAE) with a function decoder is first used to learn compact latent representations of continuous functions that adhere to known physical or statistical constraints, even when training data is sparse or indirect. Generative modeling is then performed in the learned latent space, decoupling constraint enforcement from the sampling process. This decoupling enables expressive multi-step generative methods to be deployed in data-limited settings where existing constrained multi-step approaches are not directly applicable. The richer latent distributions captured by the generative model also overcome limitations of standard VAEs, which rely on simple parametric priors and struggle to represent complex, multimodal, or heavy-tailed distributions over functions. Efficacy is demonstrated on two challenging applications: wind velocity field reconstruction from sparse sensors and material property inference from indirect measurements. Results show the effectiveness of incorporating domain knowledge constraints for data-limited problems and the improved sample quality and robustness of the latent generative modeling approach versus directly sampling a constrained VAE.