# A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables

arXiv: https://arxiv.org/abs/2006.01789 · HTML: https://mlsanity.com/paper/2006.01789/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2020.

## Built on by (most cited first, 29 of 29)

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- [Weak neural variational inference for solving Bayesian inverse problems without forward models: applications in elastography](https://mlsanity.com/paper/2407.20697/) (2024)
- [DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling](https://mlsanity.com/paper/2502.06250/) (2025)
- [Efficient Deconvolution in Populational Inverse Problems](https://mlsanity.com/paper/2505.19841/) (2025)
- [PAD-Hand: Physics-Aware Diffusion for Hand Motion Recovery](https://mlsanity.com/paper/2603.26068/) (2026)
- [A Primer on Variational Inference for Physics-Informed Deep Generative Modelling](https://mlsanity.com/paper/2409.06560/) (2024)
- [Autoregressive regularized score-based diffusion models for multi-scenarios fluid flow prediction](https://mlsanity.com/paper/2505.24145/) (2025)
- [Adversarial Disentanglement by Backpropagation with Physics-Informed Variational Autoencoder](https://mlsanity.com/paper/2506.13658/) (2025)
- [PIRF: Physics-Informed Reward Fine-Tuning for Diffusion Models](https://mlsanity.com/paper/2509.20570/) (2025)
- [PILD: Physics-Informed Learning via Diffusion](https://mlsanity.com/paper/2601.21284/) (2026)
- [Physics-aware, deep probabilistic modeling of multiscale dynamics in the Small Data regime](https://mlsanity.com/paper/2102.04269/) (2021)
- [Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis](https://mlsanity.com/paper/2305.13879/) (2023)
- [Physics-integrated generative modeling using attentive planar normalizing flow based variational autoencoder](https://mlsanity.com/paper/2404.12267/) (2024)
- [Variational Bayesian surrogate modelling with application to robust design optimisation](https://mlsanity.com/paper/2404.14857/) (2024)
- [Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later](https://mlsanity.com/paper/2509.19929/) (2025)
- [Channel Knowledge Map Construction via Physics-Inspired Diffusion Model Without Prior Observations](https://mlsanity.com/paper/2512.02757/) (2025)
- [Multi-view Bayesian optimisation in an input-output reduced space for engineering design](https://mlsanity.com/paper/2501.01552/) (2025)
- [Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data](https://mlsanity.com/paper/2505.11308/) (2025)
- [A Multimodal Conditional Mixture Model with Distribution-Level Physics Priors](https://mlsanity.com/paper/2602.10451/) (2026)
- [GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media](https://mlsanity.com/paper/2602.14642/) (2026)
- [Multiphase-Diff: Diffusion-Based Generative Modeling for High-Contrast Multiphase Physical Systems with Sharp Interfaces](https://mlsanity.com/paper/2608.13669/) (2026)
- [PMosFM: Preconditioned Manifold Matching for One-Step Physics-Constrained Generation](https://mlsanity.com/paper/2609.40287/) (2026)
- [Characterizing and Mitigating Distributional Bias in Physics-Informed Generative Models](https://mlsanity.com/paper/2610.16582/) (2026)

## Abstract (arXiv)

The data-centric construction of inexpensive surrogates for fine-grained, physical models has been at the forefront of computational physics due to its significant utility in many-query tasks such as uncertainty quantification. Recent efforts have taken advantage of the enabling technologies from the field of machine learning (e.g. deep neural networks) in combination with simulation data. While such strategies have shown promise even in higher-dimensional problems, they generally require large amounts of training data even though the construction of surrogates is by definition a Small Data problem. Rather than employing data-based loss functions, it has been proposed to make use of the governing equations (in the simplest case at collocation points) in order to imbue domain knowledge in the training of the otherwise black-box-like interpolators. The present paper provides a flexible, probabilistic framework that accounts for physical structure and information both in the training objectives as well as in the surrogate model itself. We advocate a probabilistic (Bayesian) model in which equalities that are available from the physics (e.g. residuals, conservation laws) can be introduced as virtual observables and can provide additional information through the likelihood. We further advocate a generative model i.e. one that attempts to learn the joint density of inputs and outputs that is capable of making use of unlabeled data (i.e. only inputs) in a semi-supervised fashion in order to promote the discovery of lower-dimensional embeddings which are nevertheless predictive of the fine-grained model's output.