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Artificial intelligence and deep learning are currently reshaping numerical simulation frameworks by introducing new modeling capabilities. These frameworks are extensively investigated in the context of model correction and parameterization where they demonstrate great potential and often outperform traditional physical models. Most of these efforts in defining hybrid dynamical systems follow {offline} learning strategies in which the neural parameterization (called here sub-model) is trained to output an ideal correction. Yet, these hybrid models can face hard limitations when defining what should be a relevant sub-model response that would translate into a good forecasting performance. End-to-end learning schemes, also referred to as online learning, could address such a shortcoming by allowing the deep learning sub-models to train on historical data. However, defining end-to-end training schemes for the calibration of neural sub-models in hybrid systems requires working with an optimization problem that involves the solver of the physical equations. Online learning methodologies thus require the numerical model to be differentiable, which is not the case for most modeling systems. To overcome this difficulty and bypass the differentiability challenge of physical models, we present an efficient and practical online learning approach for hybrid systems. The method, called EGA for Euler Gradient Approximation, assumes an additive neural correction to the physical model, and an explicit Euler approximation of the gradients. We demonstrate that the EGA converges to the exact gradients in the limit of infinitely small time steps. Numerical experiments are performed on various case studies, including prototypical ocean-atmosphere dynamics. Results show significant improvements over offline learning, highlighting the potential of end-to-end online learning for hybrid modeling.

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.

An assessment of variance in ocean current signal and noise shared by in situ observations (drifters) and a large gridded analysis (GlobCurrent) is sought as a function of day of the year for 1993-2015 and across a broad spectrum of current speed. Regardless of the division of collocations, it is difficult to claim that any synoptic assessment can be based on independent observations. Instead, a measurement model that departs from ordinary linear regression by accommodating error correlation is proposed. The interpretation of independence is explored by applying Fuller's (1987) concept of equation and measurement error to a division of error into shared (correlated) and unshared (uncorrelated) components, respectively. The resulting division of variance in the new model favours noise. Ocean current shared (equation) error is of comparable magnitude to unshared (measurement) error and the latter is, for GlobCurrent and drifters respectively, comparable to ordinary and reverse linear regression. Although signal variance appears to be small, its utility as a measure of agreement between two variates is highlighted. Sparse collocations that sample a dense grid permit a first order autoregressive form of measurement model to be considered, including parameterizations of analysis-in situ error cross-correlation and analysis temporal error autocorrelation. The former (cross-correlation) is an equation error term that accommodates error shared by both GlobCurrent and drifters. The latter (autocorrelation) facilitates an identification and retrieval of all model parameters. Solutions are sought using a prescribed calibration between GlobCurrent and drifters (by variance matching). Because the true current variance of GlobCurrent and drifters is small, signal to noise ratio is near zero at best. This is particularly evident for moderate current speed and meridional current component., Comment: 24 pages, 14 figures

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.

This paper addresses the data-driven identification of latent dynamical representations of partially-observed systems, i.e., dynamical systems for which some components are never observed, with an emphasis on forecasting applications, including long-term asymptotic patterns. Whereas state-of-the-art data-driven approaches rely on delay embeddings and linear decompositions of the underlying operators, we introduce a framework based on the data-driven identification of an augmented state-space model using a neural-network-based representation. For a given training dataset, it amounts to jointly learn an ODE (Ordinary Differential Equation) representation in the latent space and reconstructing latent states. Through numerical experiments, we demonstrate the relevance of the proposed framework w.r.t. state-of-the-art approaches in terms of short-term forecasting performance and long-term behaviour. We further discuss how the proposed framework relates to Koopman operator theory and Takens' embedding theorem.

The highly heterogeneous and biologically active continental shelf-seas are important components of the oceanic carbon sink. Carbon rich water from shelf-seas is exported at depth to the open ocean, a process known as the continental shelf pump, with open-ocean surface water moving (transported) onto the shelf driving the export at depth. Existing methods to study shelf-wide exchange focus on the wind or geostrophic currents, often ignoring their combined effect, spatial heterogeniety or any other ageostrophic components. Here we investigate the influence that wind, wave and current interactions can have on surface transport and carbon export across continental shelves. Using a 21 year global re-analysis dataset we confirm that geostrophic and wind driven Ekman processes are important for the transport of water onto shelf seas; but the dominance of each is location and season dependent. A global wave model re-analysis shows that one type of ageostrophic flow, Stokes drift due to waves, can also be significant. A regional case study using two submesocale model simulations identifies that up to 100% of the cross-shelf surface flow in European seas can be due to ageostrophic components. Using these results and grouping shelf-seas based on their observed carbon accumulation rates shows that differences in rates are consistent with imbalances between the processes driving atmosphere-ocean exchange at the surface and those driving carbon export at depth. Therefore expected future changes in wind and wave climate support the need to monitor cross-shelf transport and the size of the continental shelf-sea carbon pump. The results presented show that the Sea Surface Kinematics Multiscale monitoring satellite mission (SKIM), will be capable of providing measurements of the total cross-shelf current, which are now needed to enable routine monitoring of the global continental shelf-sea carbon pump., Comment: 22 pages, 3 tables, 5 figures

The authors have withdrawn this article [arXiv admin]., Comment: The authors have withdrawn this article [arXiv admin]

New parameterizations for the spectra dissipation of wind-generated waves are proposed. The rates of dissipation have no predetermined spectral shapes and are functions of the wave spectrum and wind speed and direction, in a way consistent with observation of wave breaking and swell dissipation properties. Namely, the swell dissipation is nonlinear and proportional to the swell steepness, and dissipation due to wave breaking is non-zero only when a non-dimensional spectrum exceeds the threshold at which waves are observed to start breaking. An additional source of short wave dissipation due to long wave breaking is introduced to represent the dissipation of short waves due to longer breaking waves. Several degrees of freedom are introduced in the wave breaking and the wind-wave generation term of Janssen (J. Phys. Oceanogr. 1991). These parameterizations are combined and calibrated with the Discrete Interaction Approximation of Hasselmann et al. (J. Phys. Oceangr. 1985) for the nonlinear interactions. Parameters are adjusted to reproduce observed shapes of directional wave spectra, and the variability of spectral moments with wind speed and wave height. The wave energy balance is verified in a wide range of conditions and scales, from gentle swells to major hurricanes, from the global ocean to coastal settings. Wave height, peak and mean periods, and spectral data are validated using in situ and remote sensing data. Some systematic defects are still present, but the parameterizations yield the best overall results to date. Perspectives for further improvement are also given., Comment: revised version for Journal of Physical Oceanography

Satellite Synthetic Aperture Radar (SAR) observations can provide a global view of ocean swell fields when using a specific "wave mode" sampling. A methodology is presented to routinely derive integral properties of the longer wavelength (swell) portion of the wave spectrum from SAR Level 2 products, and both monitor and predict their evolution across ocean basins. SAR-derived estimates of swell height, and energy-weighted peak period and direction, are validated against buoy observations, and the peak directions are used to project the peak periods in one dimension along the corresponding great circle route, both forward and back in time, using the peak period group velocity. The resulting real time dataset of great circle-projected peak periods produces two-dimensional maps that can be used to monitor and predict the spatial extent, and temporal evolution, of individual ocean swell fields as they propagate from their source region to distant coastlines. The methodology is found to be consistent with the dispersive arrival of peak swell periods at a mid-ocean buoy. The simple great circle propagation method cannot project the swell heights in space like the peak periods, because energy evolution along a great circle is a function of the source storm characteristics and the unknown swell dissipation rate. A more general geometric optics model is thus proposed for the far field of the storms. This model is applied here to determine the attenuation over long distances. For one of the largest recorded storms, observations of 15 s period swells are consistent with a constant dissipation rate that corresponds to a 3300 km e-folding scale for the energy. In this case, swell dissipation is a significant term in the wave energy balance at global scales., Comment: 14 pages. Submitted to Journal of Geophysical Research (revised)

Global observations of ocean swell, from satellite Synthetic Aperture Radar data, are used to estimate the dissipation of swell energy for a number of storms. Swells can be very persistent with energy e-folding scales exceeding 20,000 km. For increasing swell steepness this scale shrinks systematically, down to 2800 km for the steepest observed swells, revealing a significant loss of swell energy. This value corresponds to a normalized energy decay in time {\ss} = 4.2 x 10-6 s -1 . Many processes may be responsible for this dissipation. Because no particular trend is found with wind magnitude and direction, the increase of dissipation rate in dissipation with swell steepness is interpreted as a laminar to turbulent transition of the boundary layer, with a threshold Reynolds number of the order of 100,000. These observations of swell evolution open the way for more accurate wave forecasting models, and provides a constraint on swell-induced air-sea fluxes of momentum and energy., Comment: Article soumis \`a GRL le 12/09/2008 apr\`es rejet par Nature Geoscience (soumis en avril 2008). Modifi\'e en Janvier 2009 et accept\'e en f\'evrier 2009