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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