Hugo Frezat

Hugo Frezat, Julien Le Sommer, Ronan Fablet et al.

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 $\textit{a priori}$ criteria. But the actual purpose of a subgrid parametrization is to obtain good performance in terms of $\textit{a posteriori}$ 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 $\textit{a priori}$ 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 $\textit{a posteriori}$ 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.

Hugo Frezat, Ronan Fablet, Guillaume Balarac et al.

In this paper, we propose a generic algorithm to train machine learning-based subgrid parametrizations online, i.e., with \textit{a 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.

Etienne Meunier, Said Ouala, Hugo Frezat et al.

We present a differentiable extension of the VEROS ocean model, enabling automatic differentiation through its dynamical core. We describe the key modifications required to make the model fully compatible with JAX autodifferentiation framework and evaluate the numerical consistency of the resulting implementation. Two illustrative applications are then demonstrated: (i) the correction of an initial ocean state through gradient-based optimization, and (ii) the calibration of unknown physical parameters directly from model observations. These examples highlight how differentiable programming can facilitate end-to-end learning and parameter tuning in ocean modeling. Our implementation is available online.

Fei Er Yan, Hugo Frezat, Julien Le Sommer et al.

For reasons of computational constraint, most global ocean circulation models used for Earth System Modeling still rely on parameterizations of sub-grid processes, and limitations in these parameterizations affect the modeled ocean circulation and impact on predictive skill. An increasingly popular approach is to leverage machine learning approaches for parameterizations, regressing for a map between the resolved state and missing feedbacks in a fluid system as a supervised learning task. However, the learning is often performed in an `offline' fashion, without involving the underlying fluid dynamical model during the training stage. Here, we explore the `online' approach that involves the fluid dynamical model during the training stage for the learning of baroclinic turbulence and its parameterization, with reference to ocean eddy parameterization. Two online approaches are considered: a full adjoint-based online approach, related to traditional adjoint optimization approaches that require a `differentiable' dynamical model, and an approximately online approach that approximates the adjoint calculation and does not require a differentiable dynamical model. The online approaches are found to be generally more skillful and numerically stable than offline approaches. Others details relating to online training, such as window size, machine learning model set up and designs of the loss functions are detailed to aid in further explorations of the online training methodology for Earth System Modeling.

Hugo Frezat, Thomas Gastine, Alexandre Fournier

The use of machine learning to represent subgrid-scale (SGS) dynamics is now well established in weather forecasting and climate modelling. Recent advances have demonstrated that SGS models trained via ``online'' end-to-end learning -- where the dynamical solver operating on the filtered equations participates in the training -- can outperform traditional physics-based approaches. Most studies, however, have focused on idealised periodic domains, neglecting the mechanical boundaries present e.g. in planetary interiors. To address this issue, we consider two-dimensional quasi-geostrophic turbulent flow in an axisymmetric bounded domain that we model using a pseudo-spectral differentiable solver, thereby enabling online learning. We examine three configurations, varying the geometry (between an exponential container and a spherical shell) and the rotation rate. Flow is driven by a prescribed analytical forcing, allowing for precise control over the energy injection scale and an exact estimate of the power input. We evaluate the accuracy of the online-trained SGS model against the reference direct numerical simulation using integral quantities and spectral diagnostics. In all configurations, we show that an SGS model trained on data spanning only one turnover time remains stable and accurate over integrations at least a hundred times longer than the training period. Moreover, we demonstrate the model's remarkable ability to reproduce slow processes occurring on time scales far exceeding the training duration, such as the inward drift of jets in the spherical shell. These results suggest a promising path towards developing SGS models for planetary and stellar interior dynamics, including dynamo processes.

Hugo Frezat, Julien Le Sommer, Ronan Fablet et al.

Modeling the subgrid-scale dynamics of reduced models is a long standing open problem that finds application in ocean, atmosphere and climate predictions where direct numerical simulation (DNS) is impossible. While neural networks (NNs) have already been applied to a range of three-dimensional flows with success, two dimensional flows are more challenging because of the backscatter of energy from small to large scales. We show that learning a model jointly with the dynamical solver and a meaningful \textit{a posteriori}-based loss function lead to stable and realistic simulations when applied to quasi-geostrophic turbulence.

Hugo Frezat, Guillaume Balarac, Julien Le Sommer et al.

In this paper we present a new strategy to model the subgrid-scale scalar flux in a three-dimensional turbulent incompressible flow using physics-informed neural networks (NNs). When trained from direct numerical simulation (DNS) data, state-of-the-art neural networks, such as convolutional neural networks, may not preserve well known physical priors, which may in turn question their application to real case-studies. To address this issue, we investigate hard and soft constraints into the model based on classical transformation invariances and symmetries derived from physical laws. From simulation-based experiments, we show that the proposed transformation-invariant NN model outperforms both purely data-driven ones as well as parametric state-of-the-art subgrid-scale models. The considered invariances are regarded as regularizers on physical metrics during the a priori evaluation and constrain the distribution tails of the predicted subgrid-scale term to be closer to the DNS. They also increase the stability and performance of the model when used as a surrogate during a large-eddy simulation. Moreover, the transformation-invariant NN is shown to generalize to regimes that have not been seen during the training phase.