A posteriori learning for quasi-geostrophic turbulence parametrization

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.

Multimodal learning-based inversion models for the space-time reconstruction of satellite-derived geophysical fields

For numerous earth observation applications, one may benefit from various satellite sensors to address the reconstruction of some process or information of interest. A variety of satellite sensors deliver observation data with different sampling patterns due satellite orbits and/or their sensitivity to atmospheric conditions (e.g., clour cover, heavy rains,...). Beyond the ability to account for irregularly-sampled observations, the definition of model-driven inversion methods is often limited to specific case-studies where one can explicitly derive a physical model to relate the different observation sources. Here, we investigate how end-to-end learning schemes provide new means to address multimodal inversion problems. The proposed scheme combines a variational formulation with trainable observation operators, {\em a priori} terms and solvers. Through an application to space oceanography, we show how this scheme can successfully extract relevant information from satellite-derived sea surface temperature images and enhance the reconstruction of sea surface currents issued from satellite altimetry data.

Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning

The complexity of real-world geophysical systems is often compounded by the fact that the observed measurements depend on hidden variables. These latent variables include unresolved small scales and/or rapidly evolving processes, partially observed couplings, or forcings in coupled systems. This is the case in ocean-atmosphere dynamics, for which unknown interior dynamics can affect surface observations. The identification of computationally-relevant representations of such partially-observed and highly nonlinear systems is thus challenging and often limited to short-term forecast applications. Here, we investigate the physics-constrained learning of implicit dynamical embeddings, leveraging neural ordinary differential equation (NODE) representations. A key objective is to constrain their boundedness, which promotes the generalization of the learned dynamics to arbitrary initial condition. The proposed architecture is implemented within a deep learning framework, and its relevance is demonstrated with respect to state-of-the-art schemes for different case-studies representative of geophysical dynamics.

A posteriori learning of quasi-geostrophic turbulence parametrization: an experiment on integration steps

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.

Joint calibration and mapping of satellite altimetry data using trainable variational models

Satellite radar altimeters are a key source of observation of ocean surface dynamics. However, current sensor technology and mapping techniques do not yet allow to systematically resolve scales smaller than 100km. With their new sensors, upcoming wide-swath altimeter missions such as SWOT should help resolve finer scales. Current mapping techniques rely on the quality of the input data, which is why the raw data go through multiple preprocessing stages before being used. Those calibration stages are improved and refined over many years and represent a challenge when a new type of sensor start acquiring data. Here we show how a data-driven variational data assimilation framework could be used to jointly learn a calibration operator and an interpolator from non-calibrated data . The proposed framework significantly outperforms the operational state-of-the-art mapping pipeline and truly benefits from wide-swath data to resolve finer scales on the global map as well as in the SWOT sensor geometry.

TrAISformer-A generative transformer for AIS trajectory prediction

Modelling trajectory in general, and vessel trajectory in particular, is a difficult task because of the multimodal and complex nature of motion data. In this paper, we present TrAISformer-a novel deep learning architecture that can forecast vessel positions using AIS (Automatic Identification System) observations. We address the multimodality by introducing a discrete representation of AIS data and re-frame the prediction, which is originally a regression problem, as a classification problem. The model encodes complex movement patterns in AIS data in high-dimensional vectors, then applies a transformer to extract useful long-term correlations from sequences of those embeddings to sample future vessel positions. Experimental results on real, public AIS data demonstrate that TrAISformer significantly outperforms state-of-the-art methods.

Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme

Stochastic differential equations (SDEs) are one of the most important representations of dynamical systems. They are notable for the ability to include a deterministic component of the system and a stochastic one to represent random unknown factors. However, this makes learning SDEs much more challenging than ordinary differential equations (ODEs). In this paper, we propose a data driven approach where parameters of the SDE are represented by a neural network with a built-in SDE integration scheme. The loss function is based on a maximum likelihood criterion, under order one Markov Gaussian assumptions. The algorithm is applied to the geometric brownian motion and a stochastic version of the Lorenz-63 model. The latter is particularly hard to handle due to the presence of a stochastic component that depends on the state. The algorithm performance is attested using different simulations results. Besides, comparisons are performed with the reference gradient matching method used for non linear drift estimation, and a neural networks-based method, that does not consider the stochastic term.

Learning Runge-Kutta Integration Schemes for ODE Simulation and Identification

Deriving analytical solutions of ordinary differential equations is usually restricted to a small subset of problems and numerical techniques are considered. Inevitably, a numerical simulation of a differential equation will then always be distinct from a true analytical solution. An efficient integration scheme shall further not only provide a trajectory throughout a given state, but also be derived to ensure the generated simulation to be close to the analytical one. Consequently, several integration schemes were developed for different classes of differential equations. Unfortunately, when considering the integration of complex non-linear systems, as well as the identification of non-linear equations from data, this choice of the integration scheme is often far from being trivial. In this paper, we propose a novel framework to learn integration schemes that minimize an integration-related cost function. We demonstrate the relevance of the proposed learning-based approach for non-linear equations and include a quantitative analysis w.r.t. classical state-of-the-art integration techniques, especially where the latter may not apply.