# Machine Learning for Stochastic Parameterization: Generative Adversarial Networks in the Lorenz '96 Model

arXiv: https://arxiv.org/abs/1909.04711 · HTML: https://mlsanity.com/paper/1909.04711/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2019.

## Built on by (most cited first, 30 of 45)

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- [DiffESM: Conditional Emulation of Temperature and Precipitation in Earth System Models with 3D Diffusion Models](https://mlsanity.com/paper/2409.11601/) (2024)
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- [The Need for Ethical, Responsible, and Trustworthy Artificial Intelligence for Environmental Sciences](https://mlsanity.com/paper/2112.08453/) (2021)
- [Broadband Ground Motion Synthesis via Generative Adversarial Neural Operators: Development and Validation](https://mlsanity.com/paper/2309.03447/) (2023)
- [Controlled abstention neural networks for identifying skillful predictions for regression problems](https://mlsanity.com/paper/2104.08236/) (2021)
- [On the modern deep learning approaches for precipitation downscaling](https://mlsanity.com/paper/2207.00808/) (2022)
- [A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization](https://mlsanity.com/paper/2208.09104/) (2022)
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- [CEBoosting: Online Sparse Identification of Dynamical Systems with Regime Switching by Causation Entropy Boosting](https://mlsanity.com/paper/2304.07863/) (2023)
- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [A Machine Learning Tutorial for Operational Meteorology, Part I: Traditional Machine Learning](https://mlsanity.com/paper/2204.07492/) (2022)
- [CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilation](https://mlsanity.com/paper/2404.06749/) (2024)
- [Deep Learning Based Cloud Cover Parameterization for ICON](https://mlsanity.com/paper/2112.11317/) (2021)
- [Using Probabilistic Machine Learning to Better Model Temporal Patterns in Parameterizations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2203.14814/) (2022)
- [Embed and Emulate: Learning to estimate parameters of dynamical systems with uncertainty quantification](https://mlsanity.com/paper/2211.01554/) (2022)
- [Defining error accumulation in ML atmospheric simulators](https://mlsanity.com/paper/2405.14714/) (2024)
- [Combining data assimilation and machine learning to estimate parameters of a convective-scale model](https://mlsanity.com/paper/2109.02953/) (2021)
- [Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models](https://mlsanity.com/paper/2506.20771/) (2025)
- [Controlled abstention neural networks for identifying skillful predictions for classification problems](https://mlsanity.com/paper/2104.08281/) (2021)
- [History-Based, Bayesian, Closure for Stochastic Parameterization: Application to Lorenz '96](https://mlsanity.com/paper/2210.14488/) (2022)
- [Statistically Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution](https://mlsanity.com/paper/2307.16422/) (2023)
- [Navigating the Noise: Bringing Clarity to ML Parameterization Design with O(100) Ensembles](https://mlsanity.com/paper/2309.16177/) (2023)
- [Controllable Probabilistic Forecasting with Stochastic Decomposition Layers](https://mlsanity.com/paper/2512.18815/) (2025)
- [Machine learning-based conditional mean filter: a generalization of the ensemble Kalman filter for nonlinear data assimilation](https://mlsanity.com/paper/2106.07908/) (2021)

## Abstract (arXiv)

Stochastic parameterizations account for uncertainty in the representation of unresolved sub-grid processes by sampling from the distribution of possible sub-grid forcings. Some existing stochastic parameterizations utilize data-driven approaches to characterize uncertainty, but these approaches require significant structural assumptions that can limit their scalability. Machine learning models, including neural networks, are able to represent a wide range of distributions and build optimized mappings between a large number of inputs and sub-grid forcings. Recent research on machine learning parameterizations has focused only on deterministic parameterizations. In this study, we develop a stochastic parameterization using the generative adversarial network (GAN) machine learning framework. The GAN stochastic parameterization is trained and evaluated on output from the Lorenz '96 model, which is a common baseline model for evaluating both parameterization and data assimilation techniques. We evaluate different ways of characterizing the input noise for the model and perform model runs with the GAN parameterization at weather and climate timescales. Some of the GAN configurations perform better than a baseline bespoke parameterization at both timescales, and the networks closely reproduce the spatio-temporal correlations and regimes of the Lorenz '96 system. We also find that in general those models which produce skillful forecasts are also associated with the best climate simulations.