# A posteriori learning for quasi-geostrophic turbulence parametrization

arXiv: https://arxiv.org/abs/2204.03911 · HTML: https://mlsanity.com/paper/2204.03911/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 118 references that are in the map.
- First posted on arXiv 8 Apr 2022.

## Builds on (most cited first, 26 of 26)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
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- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Sub-seasonal forecasting with a large ensemble of deep-learning weather prediction models](https://mlsanity.com/paper/2102.05107/) (2021)
- [Physics-based Deep Learning](https://mlsanity.com/paper/2109.05237/) (2021)
- [Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High-Resolution Simulations](https://mlsanity.com/paper/1709.00037/) (2017)
- [DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)](https://mlsanity.com/paper/1911.09145/) (2019)
- [Using machine learning to correct model error in data assimilation and forecast applications](https://mlsanity.com/paper/2010.12605/) (2020)
- [Building high accuracy emulators for scientific simulations with deep neural architecture search](https://mlsanity.com/paper/2001.08055/) (2020)
- [Learning Variational Data Assimilation Models and Solvers](https://mlsanity.com/paper/2007.12941/) (2020)
- [Limits of End-to-End Learning](https://mlsanity.com/paper/1704.08305/) (2017)
- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)
- [Data-Driven Discovery of Coarse-Grained Equations](https://mlsanity.com/paper/2002.00790/) (2020)
- [Physical invariance in neural networks for subgrid-scale scalar flux modeling](https://mlsanity.com/paper/2010.04663/) (2020)
- [Embedded training of neural-network sub-grid-scale turbulence models](https://mlsanity.com/paper/2105.01030/) (2021)
- [Machine Learning Emulation of 3D Cloud Radiative Effects](https://mlsanity.com/paper/2103.11919/) (2021)
- [Learning Runge-Kutta Integration Schemes for ODE Simulation and Identification](https://mlsanity.com/paper/2105.04999/) (2021)

## Built on by (most cited first, 30 of 36)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Learning to correct spectral methods for simulating turbulent flows](https://mlsanity.com/paper/2207.00556/) (2022)
- [DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems](https://mlsanity.com/paper/2402.04467/) (2024)
- [Differentiable Programming for Earth System Modeling](https://mlsanity.com/paper/2208.13825/) (2022)
- [A Neural PDE Solver with Temporal Stencil Modeling](https://mlsanity.com/paper/2302.08105/) (2023)
- [Explaining the physics of transfer learning a data-driven subgrid-scale closure to a different turbulent flow](https://mlsanity.com/paper/2206.03198/) (2022)
- [Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systems](https://mlsanity.com/paper/2301.10391/) (2023)
- [Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges](https://mlsanity.com/paper/2306.05014/) (2023)
- [Gradient-free online learning of subgrid-scale dynamics with neural emulators](https://mlsanity.com/paper/2310.19385/) (2023)
- [Joint Parameter and Parameterization Inference with Uncertainty Quantification through Differentiable Programming](https://mlsanity.com/paper/2403.02215/) (2024)
- [What You See is Not What You Get: Neural Partial Differential Equations and The Illusion of Learning](https://mlsanity.com/paper/2411.15101/) (2024)
- [AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarios](https://mlsanity.com/paper/2302.10184/) (2023)
- [Multi-fidelity climate model parameterization for better generalization and extrapolation](https://mlsanity.com/paper/2309.10231/) (2023)
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- [Online Calibration of Deep Learning Sub-Models for Hybrid Numerical Modeling Systems](https://mlsanity.com/paper/2311.10665/) (2023)
- [Decoding complexity: how machine learning is redefining scientific discovery](https://mlsanity.com/paper/2405.04161/) (2024)
- [INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solvers](https://mlsanity.com/paper/2511.12764/) (2025)
- [Data-driven multiscale modeling for correcting dynamical systems](https://mlsanity.com/paper/2303.17496/) (2023)
- [On the choice of training data for machine learning of geostrophic mesoscale turbulence](https://mlsanity.com/paper/2307.00734/) (2023)
- [Extreme Event Prediction with Multi-agent Reinforcement Learning-based Parametrization of Atmospheric and Oceanic Turbulence](https://mlsanity.com/paper/2312.00907/) (2023)
- [Development of an offline and online hybrid model for the Integrated Forecasting System](https://mlsanity.com/paper/2403.03702/) (2024)
- [Guided Unconditional and Conditional Generative Models for Super-Resolution and Inference of Quasi-Geostrophic Turbulence](https://mlsanity.com/paper/2507.00719/) (2025)
- [Electron neural closure for turbulent magnetosheath simulations: energy channels](https://mlsanity.com/paper/2510.00282/) (2025)
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- [Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations](https://mlsanity.com/paper/2302.03286/) (2023)
- [Differentiable Turbulence: Closure as a partial differential equation constrained optimization](https://mlsanity.com/paper/2307.03683/) (2023)
- [Adjoint-based online learning of two-layer quasi-geostrophic baroclinic turbulence](https://mlsanity.com/paper/2411.14106/) (2024)
- [NORi: An ML-Augmented Ocean Boundary Layer Parameterization](https://mlsanity.com/paper/2512.04452/) (2025)
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## Abstract (arXiv)

The use of machine learning to build subgrid parametrizations for climate models is receiving growing attention. State-of-the-art strategies address the problem as a supervised learning task and optimize algorithms that predict subgrid fluxes based on information from coarse resolution models. In practice, training data are generated from higher resolution numerical simulations transformed in order to mimic coarse resolution simulations. By essence, these strategies optimize subgrid parametrizations to meet so-called textita priori criteria. But the actual purpose of a subgrid parametrization is to obtain good performance in terms of textita posteriori metrics which imply computing entire model trajectories. In this paper, we focus on the representation of energy backscatter in two dimensional quasi-geostrophic turbulence and compare parametrizations obtained with different learning strategies at fixed computational complexity. We show that strategies based on textita priori criteria yield parametrizations that tend to be unstable in direct simulations and describe how subgrid parametrizations can alternatively be trained end-to-end in order to meet textita posteriori criteria. We illustrate that end-to-end learning strategies yield parametrizations that outperform known empirical and data-driven schemes in terms of performance, stability and ability to apply to different flow configurations. These results support the relevance of differentiable programming paradigms for climate models in the future.