# Gradient-free online learning of subgrid-scale dynamics with neural emulators

arXiv: https://arxiv.org/abs/2310.19385 · HTML: https://mlsanity.com/paper/2310.19385/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 106 references that are in the map.
- First posted on arXiv 30 Oct 2023.

## Builds on (most cited first, 30 of 31)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [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)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [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)
- [NeuralSim: Augmenting Differentiable Simulators with Neural Networks](https://mlsanity.com/paper/2011.04217/) (2020)
- [Scientific multi-agent reinforcement learning for wall-models of turbulent flows](https://mlsanity.com/paper/2106.11144/) (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)
- [Learning to correct spectral methods for simulating turbulent flows](https://mlsanity.com/paper/2207.00556/) (2022)
- [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)
- [Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons](https://mlsanity.com/paper/2202.06988/) (2022)
- [Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks](https://mlsanity.com/paper/1808.03620/) (2018)
- [A posteriori learning for quasi-geostrophic turbulence parametrization](https://mlsanity.com/paper/2204.03911/) (2022)
- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [OceanNet: A principled neural operator-based digital twin for regional oceans](https://mlsanity.com/paper/2310.00813/) (2023)
- [Combining Machine Learning with Knowledge-Based Modeling for Scalable Forecasting and Subgrid-Scale Closure of Large, Complex, Spatiotemporal Systems](https://mlsanity.com/paper/2002.05514/) (2020)
- [Differentiable Programming for Earth System Modeling](https://mlsanity.com/paper/2208.13825/) (2022)
- [Differentiable hybrid neural modeling for fluid-structure interaction](https://mlsanity.com/paper/2303.12971/) (2023)
- [Deep Reinforcement Learning for Turbulence Modeling in Large Eddy Simulations](https://mlsanity.com/paper/2206.11038/) (2022)
- [Bridging observation, theory and numerical simulation of the ocean using Machine Learning](https://mlsanity.com/paper/2104.12506/) (2021)
- [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)
- [Differentiable physics-enabled closure modeling for Burgers' turbulence](https://mlsanity.com/paper/2209.11614/) (2022)

## Built on by (most cited first, 5 of 5)

- [Online Decision-Focused Learning](https://mlsanity.com/paper/2505.13564/) (2025)
- [INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solvers](https://mlsanity.com/paper/2511.12764/) (2025)
- [ADDA: a Modular Framework for Representing, Simulating and Assimilating Dynamics with End-to-end Differentiability](https://mlsanity.com/paper/2608.23297/) (2026)
- [A general physics-constrained method for the modelling of equation's closure terms with sparse data](https://mlsanity.com/paper/2505.03783/) (2025)
- [Mamba-Assisted Non-Markovian Closure for Reduced-Order Modeling](https://mlsanity.com/paper/2606.05371/) (2026)

## Abstract (arXiv)

In this paper, we propose a generic algorithm to train machine learning-based subgrid parametrizations online, i.e., with textita posteriori loss functions, but for non-differentiable numerical solvers. The proposed approach leverages a neural emulator to approximate the reduced state-space solver, which is then used to allow gradient propagation through temporal integration steps. We apply this methodology on a chaotic two-timescales Lorenz-96 system and a single layer quasi-geostrophic system with zonal dynamics, known to be highly unstable with offline strategies. Using our algorithm, we are able to train a parametrization that recovers most of the benefits of online strategies without having to compute the gradient of the original solver. We found that training the neural emulator and parametrization components separately with different loss quantities is necessary in order to minimize the propagation of approximation biases. Experiments on emulator architectures with different complexities also indicates that emulator performance is key in order to learn an accurate parametrization. This work is a step towards learning parametrization with online strategies for climate models.