# Enforcing Statistical Constraints in Generative Adversarial Networks for Modeling Chaotic Dynamical Systems

arXiv: https://arxiv.org/abs/1905.06841 · HTML: https://mlsanity.com/paper/1905.06841/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 May 2019.

## Built on by (most cited first, 30 of 33)

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- [PARC: Physics-Aware Recurrent Convolutional Neural Networks to Assimilate Meso-scale Reactive Mechanics of Energetic Materials](https://mlsanity.com/paper/2204.07234/) (2022)
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- [Binned Spectral Power Loss for Improved Prediction of Chaotic Systems](https://mlsanity.com/paper/2502.00472/) (2025)
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- [Generating Physical Dynamics under Priors](https://mlsanity.com/paper/2409.00730/) (2024)
- [Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network](https://mlsanity.com/paper/2507.08749/) (2025)
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- [ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Prediction](https://mlsanity.com/paper/2302.05783/) (2023)
- [Physics-Constrained Generative Adversarial Networks for 3D Turbulence](https://mlsanity.com/paper/2212.00217/) (2022)
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- [Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models](https://mlsanity.com/paper/2506.20771/) (2025)
- [Robust Estimation for Nonparametric Families via Generative Adversarial Networks](https://mlsanity.com/paper/2202.01269/) (2022)
- [Generative Adversarial Network for Probabilistic Forecast of Random Dynamical System](https://mlsanity.com/paper/2111.03126/) (2021)
- [A multiscale and multicriteria Generative Adversarial Network to synthesize 1-dimensional turbulent fields](https://mlsanity.com/paper/2307.16580/) (2023)
- [Neural Network Methods for Radiation Detectors and Imaging](https://mlsanity.com/paper/2311.05726/) (2023)
- [Neural Partial Differential Equations with Functional Convolution](https://mlsanity.com/paper/2303.07194/) (2023)
- [Jacobian-Scaled K-means Clustering for Physics-Informed Segmentation of Reacting Flows](https://mlsanity.com/paper/2305.01539/) (2023)
- [Physics Encoded Blocks in Residual Neural Network Architectures for Digital Twin Models](https://mlsanity.com/paper/2411.11497/) (2024)
- [Latent Generative Modeling of Random Fields from Limited Training Data](https://mlsanity.com/paper/2505.13007/) (2025)
- [Synergizing Transport-Based Generative Models and Latent Geometry for Stochastic Closure Modeling](https://mlsanity.com/paper/2602.17089/) (2026)
- [Hybrid Generative Models for Two-Dimensional Datasets](https://mlsanity.com/paper/2106.00203/) (2021)

## Abstract (arXiv)

Simulating complex physical systems often involves solving partial differential equations (PDEs) with some closures due to the presence of multi-scale physics that cannot be fully resolved. Therefore, reliable and accurate closure models for unresolved physics remains an important requirement for many computational physics problems, e.g., turbulence simulation. Recently, several researchers have adopted generative adversarial networks (GANs), a novel paradigm of training machine learning models, to generate solutions of PDEs-governed complex systems without having to numerically solve these PDEs. However, GANs are known to be difficult in training and likely to converge to local minima, where the generated samples do not capture the true statistics of the training data. In this work, we present a statistical constrained generative adversarial network by enforcing constraints of covariance from the training data, which results in an improved machine-learning-based emulator to capture the statistics of the training data generated by solving fully resolved PDEs. We show that such a statistical regularization leads to better performance compared to standard GANs, measured by (1) the constrained model's ability to more faithfully emulate certain physical properties of the system and (2) the significantly reduced (by up to 80%) training time to reach the solution. We exemplify this approach on the Rayleigh-Benard convection, a turbulent flow system that is an idealized model of the Earth's atmosphere. With the growth of high-fidelity simulation databases of physical systems, this work suggests great potential for being an alternative to the explicit modeling of closures or parameterizations for unresolved physics, which are known to be a major source of uncertainty in simulating multi-scale physical systems, e.g., turbulence or Earth's climate.