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Online learning of subgrid-scale models for quasi-geostrophic turbulence in planetary interiors

This paper introduces an online end-to-end machine learning framework for subgrid-scale modeling in bounded, rotating quasi-geostrophic turbulence that successfully reproduces long-term dynamical behaviors and outperforms classical closure schemes, thereby enabling efficient long-term simulations of planetary interior processes beyond the reach of direct numerical methods.

Original authors: Hugo Frezat, Thomas Gastine, Alexandre Fournier

Published 2026-07-29

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CC BY 4.0

Original authors: Hugo Frezat, Thomas Gastine, Alexandre Fournier

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the inside of a planet like Earth or Jupiter as a giant, churning pot of soup, but instead of vegetables, it's made of super-hot, swirling liquid metal or gas. This soup is spinning incredibly fast, and that spin creates invisible forces that organize the chaos into giant, river-like streams of wind or current called "jets." Scientists want to understand how these planetary engines work because they create the magnetic fields that protect us from space radiation and drive the weather. But there's a catch: these flows happen on scales as tiny as a meter and as huge as the planet itself, all at once. Trying to simulate every single tiny swirl on a computer is like trying to count every grain of sand on a beach while the tide is coming in; it takes so much computing power that even the fastest supercomputers can't do it for the millions of years these planets have been spinning.

To get around this, scientists use "subgrid-scale models." Think of these as a clever shortcut. Instead of tracking every single tiny grain of sand, the computer simulates the big waves and uses a smart guess to figure out how the tiny grains are pushing and pulling on the big waves. For decades, these guesses have been based on simple math formulas. But recently, researchers have started trying to teach computers to learn these guesses themselves using machine learning, much like how a video game AI learns to play better by watching thousands of hours of gameplay. The big question is: can a computer learn to guess the tiny details accurately enough to run a simulation of a planet's interior for a long time without the whole thing falling apart?

This paper, written by Hugo Frezat, Thomas Gastine, and Alexandre Fournier, tackles that exact question for a specific type of planetary flow called "quasi-geostrophic turbulence." The team built a special computer program that can "learn" the missing tiny details while it is running the simulation, a method they call "online learning." They tested this on a simplified model of a planet's interior—a spinning, ring-shaped container filled with fluid. They created three different scenarios: one with a simple exponential shape and two with spherical shapes (like a hollow ball), each spinning at different speeds. In these simulations, the fluid naturally formed giant east-west jets, ripples known as Rossby waves, and, in the spherical cases, these jets slowly drifted inward over time.

The researchers trained their machine learning model using data from a super-detailed, high-resolution simulation that ran for just one "turnover time" (the time it takes for a big swirl to go around once). Then, they let the model run a "coarse" simulation—one that ignores the tiny details to save time—using their learned guess to fill in the gaps. The results were impressive. Even though the model was only trained for a short period, it successfully predicted the behavior of the system for 100 times longer than the training window. It accurately reproduced the energy of the flows, the patterns of the jets, and even the slow, mysterious inward drift of the jets in the spherical containers.

Crucially, the paper argues against the idea that you can just train a model once and then plug it into a simulation later (an "offline" approach). The authors found that for this specific type of spinning fluid with solid walls, offline training led to unstable, crashing simulations. The "online" method, where the model learns while the simulation is running, was the only way to keep things stable. They also compared their AI model to older, standard methods like "hyperdiffusivity" (which acts like extra friction) and the "Leith closure" (a classic formula for turbulence). While the older methods worked okay if you kept the grid resolution high, they failed miserably when the researchers tried to make the simulation even coarser to save more time. The AI model, however, thrived on the coarse grid, running about 11 times faster than the high-resolution simulation and 5 times faster than the older methods, all while keeping the physics correct.

The study suggests that this online learning approach is a powerful tool for exploring long-term planetary processes that were previously impossible to simulate. It shows that a machine learning model, trained on a tiny slice of time, can capture the complex physics needed to predict how planetary jets move and merge over millions of years. While the current tests were limited to specific, simplified setups, the success opens the door to using these techniques to study more complex, real-world scenarios, potentially helping us understand the deep, slow changes happening inside our own planet and others.

Technical Summary: Online Learning of Subgrid-Scale Models for Quasi-Geostrophic Turbulence in Planetary Interiors

Problem Statement
Numerical simulations of geophysical fluid processes in planetary interiors, such as the geodynamo, are constrained by the vast separation of time and length scales. Direct Numerical Simulations (DNS) resolving all relevant scales (e.g., viscous boundary layers of ~1 meter over geological timescales) are computationally infeasible. Traditional approaches rely on subgrid-scale (SGS) parameterizations, such as eddy-diffusivity, hyperdiffusivity, or similarity models. However, these methods often struggle with anisotropy, non-local spectral interactions, and the specific challenges of bounded domains. While machine learning (ML) has shown promise in atmospheric and oceanic applications, existing "offline" learning approaches—where SGS terms are learned independently of the solver dynamics—have largely been restricted to idealized periodic domains. These offline methods fail in bounded geometries due to commutation errors arising from non-homogeneous grids near boundaries, often leading to unstable simulations when coupled back into the solver.

Methodology
The authors investigate the potential of "online" end-to-end learning for SGS modeling in a two-dimensional quasi-geostrophic (QG) turbulence system within a rapidly rotating annular domain. The study utilizes three distinct configurations varying the container geometry (exponential vs. spherical shell) and the Ekman number ( $E$ ) to explore different dynamical regimes, including zonal jet formation, Rossby wave propagation, and jet migration.

Differentiable Solver: The authors developed a differentiable spectral solver based on a pseudo-spectral Fourier–Chebyshev method. This allows for the integration of the solver within the training loop, enabling the computation of exact gradients via automatic differentiation (using the JAX library).
Coarse-Graining Strategy: To generate training data, the authors employ a composite transform that maps high-resolution DNS fields to reduced-resolution fields. Crucially, in the radial direction, they utilize a Galerkin projection onto a basis that explicitly preserves the no-slip boundary conditions, rather than simple Chebyshev truncation, to avoid violating physical constraints.
Online Learning Framework: Unlike offline approaches that minimize the error between predicted and true SGS terms ( $\tau$ ), this study formulates the problem as minimizing the discrepancy between the time evolution of the coarse variables ( $\bar{z}$ ) and the reference DNS variables ( $z$ ). The neural network model $M$ acts as an additive source term in the coarse-grained equations. The loss function is defined as a kinetic energy mismatch metric over a training window of one eddy turnover time ( $t_L$ ).
Architecture: The model $M$ operates in spectral space to integrate seamlessly with the pseudo-spectral solver. It utilizes a low-complexity variant of ConvNext architectures, split into two paths: one for the axisymmetric zonal velocity ( $m=0$ ) and another for non-axisymmetric vorticity ( $m>0$ ). The network employs complex-valued operations with real-valued trainable layers.

Key Results
The study evaluates the trained models against three configurations and compares them against classical closure schemes (hyperdiffusivity and the Leith eddy-viscosity model) and under-resolved simulations.

Stability and Long-Term Dynamics: Models trained on a single turnover time ( $t_L$ ) successfully reproduce global integrated diagnostics (Reynolds number, energy budgets) and energy spectra over integration times exceeding the training window by two orders of magnitude (100 $t_L$ ).
Jet Migration: In the spherical shell configurations, the system exhibits a slow, quasi-periodic inward drift of zonal jets, a phenomenon occurring on timescales $\sim 10 t_L$ . The online-trained model accurately reproduces this migration, whereas classical hyperdiffusivity and under-resolved simulations fail to capture this long-term behavior.
Spectral Accuracy: The learned model correctly captures the anisotropic spectral properties of zonostrophic turbulence, including the $k^{-5/3}$ residual spectrum and the $k^{-5}$ zonal spectrum, as well as the Rhines scale.
Radial Coarsening: A critical finding is that while classical models (hyperdiffusivity, Leith) perform adequately when only the azimuthal direction is coarsened, their accuracy degrades significantly when the radial resolution is also reduced. The online-trained neural network, however, maintains accuracy even with a five-fold reduction in radial resolution ( $N_s$ from 321 to 65).
Computational Efficiency: By allowing for coarser radial grids and larger time steps (relaxing the CFL constraint), the learned model achieves a speed-up of approximately 5–7 times compared to classical closures and an 11-fold reduction in time-to-solution compared to DNS on a single NVIDIA A-100 GPU.

Significance and Claims
The paper claims that online learning offers a robust solution for SGS modeling in bounded, rotating geophysical flows where offline methods fail due to commutation errors. The primary contribution is demonstrating that a neural network trained on a short temporal window can implicitly learn to correct not only the missing SGS fluxes but also numerical artifacts (such as commutation errors) inherent to the coarse-graining process in non-homogeneous domains.

The authors emphasize that this approach preserves the physical structure and conservation properties of the underlying numerical solver, distinguishing it from purely data-driven black-box approaches. The results suggest that this framework enables the investigation of long-term geophysical processes, such as jet migration and merging, which are currently beyond the reach of direct numerical simulations. The authors note that while the current study is limited to $Re \sim 10^4$ due to dense matrix constraints, the methodology is scalable to higher Reynolds numbers ( $Re \sim 10^6$ ) via sparse implementations, and future work could extend the framework to convection-driven flows involving temperature perturbations.

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