Online Calibration of Deep Learning Sub-Models for Hybrid Numerical Modeling Systems

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.

Inversion of sea surface currents from satellite-derived SST-SSH synergies with 4DVarNets

Satellite altimetry is a unique way for direct observations of sea surface dynamics. This is however limited to the surface-constrained geostrophic component of sea surface velocities. Ageostrophic dynamics are however expected to be significant for horizontal scales below 100~km and time scale below 10~days. The assimilation of ocean general circulation models likely reveals only a fraction of this ageostrophic component. Here, we explore a learning-based scheme to better exploit the synergies between the observed sea surface tracers, especially sea surface height (SSH) and sea surface temperature (SST), to better inform sea surface currents. More specifically, we develop a 4DVarNet scheme which exploits a variational data assimilation formulation with trainable observations and {\em a priori} terms. An Observing System Simulation Experiment (OSSE) in a region of the Gulf Stream suggests that SST-SSH synergies could reveal sea surface velocities for time scales of 2.5-3.0 days and horizontal scales of 0.5$^\circ$-0.7$^\circ$, including a significant fraction of the ageostrophic dynamics ($\approx$ 47\%). The analysis of the contribution of different observation data, namely nadir along-track altimetry, wide-swath SWOT altimetry and SST data, emphasizes the role of SST features for the reconstruction at horizontal spatial scales ranging from \nicefrac{1}{20}$^\circ$ to \nicefrac{1}{4}$^\circ$.

Guided Unsupervised Learning by Subaperture Decomposition for Ocean SAR Image Retrieval

Spaceborne synthetic aperture radar (SAR) can provide accurate images of the ocean surface roughness day-or-night in nearly all weather conditions, being an unique asset for many geophysical applications. Considering the huge amount of data daily acquired by satellites, automated techniques for physical features extraction are needed. Even if supervised deep learning methods attain state-of-the-art results, they require great amount of labeled data, which are difficult and excessively expensive to acquire for ocean SAR imagery. To this end, we use the subaperture decomposition (SD) algorithm to enhance the unsupervised learning retrieval on the ocean surface, empowering ocean researchers to search into large ocean databases. We empirically prove that SD improve the retrieval precision with over 20% for an unsupervised transformer auto-encoder network. Moreover, we show that SD brings important performance boost when Doppler centroid images are used as input data, leading the way to new unsupervised physics guided retrieval algorithms.

Multimodal 4DVarNets for the reconstruction of sea surface dynamics from SST-SSH synergies

Due to the irregular space-time sampling of sea surface observations, the reconstruction of sea surface dynamics is a challenging inverse problem. While satellite altimetry provides a direct observation of the sea surface height (SSH), which relates to the divergence-free component of sea surface currents, the associated sampling pattern prevents from retrieving fine-scale sea surface dynamics, typically below a 10-day time scale. By contrast, other satellite sensors provide higher-resolution observations of sea surface tracers such as sea surface temperature (SST). Multimodal inversion schemes then arise as an appealing strategy. Though theoretical evidence supports the existence of an explicit relationship between sea surface temperature and sea surface dynamics under specific dynamical regimes, the generalization to the variety of upper ocean dynamical regimes is complex. Here, we investigate this issue from a physics-informed learning perspective. We introduce a trainable multimodal inversion scheme for the reconstruction of sea surface dynamics from multi-source satellite-derived observations. The proposed 4DVarNet schemes combine a variational formulation involving trainable observation and a priori terms with a trainable gradient-based solver. We report an application to the reconstruction of the divergence-free component of sea surface dynamics from satellite-derived SSH and SST data. An observing system simulation experiment for a Gulf Stream region supports the relevance of our approach compared with state-of-the-art schemes. We report relative improvement greater than 50% compared with the operational altimetry product in terms of root mean square error and resolved space-time scales. We discuss further the application and extension of the proposed approach for the reconstruction and forecasting of geophysical dynamics from irregularly-sampled satellite observations.

Guided deep learning by subaperture decomposition: ocean patterns from SAR imagery

Spaceborne synthetic aperture radar can provide meters scale images of the ocean surface roughness day or night in nearly all weather conditions. This makes it a unique asset for many geophysical applications. Sentinel 1 SAR wave mode vignettes have made possible to capture many important oceanic and atmospheric phenomena since 2014. However, considering the amount of data provided, expanding applications requires a strategy to automatically process and extract geophysical parameters. In this study, we propose to apply subaperture decomposition as a preprocessing stage for SAR deep learning models. Our data centring approach surpassed the baseline by 0.7, obtaining state of the art on the TenGeoPSARwv data set. In addition, we empirically showed that subaperture decomposition could bring additional information over the original vignette, by rising the number of clusters for an unsupervised segmentation method. Overall, we encourage the development of data centring approaches, showing that, data preprocessing could bring significant performance improvements over existing deep learning models.

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.

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.

Learning Variational Data Assimilation Models and Solvers

This paper addresses variational data assimilation from a learning point of view. Data assimilation aims to reconstruct the time evolution of some state given a series of observations, possibly noisy and irregularly-sampled. Using automatic differentiation tools embedded in deep learning frameworks, we introduce end-to-end neural network architectures for data assimilation. It comprises two key components: a variational model and a gradient-based solver both implemented as neural networks. A key feature of the proposed end-to-end learning architecture is that we may train the NN models using both supervised and unsupervised strategies. Our numerical experiments on Lorenz-63 and Lorenz-96 systems report significant gain w.r.t. a classic gradient-based minimization of the variational cost both in terms of reconstruction performance and optimization complexity. Intriguingly, we also show that the variational models issued from the true Lorenz-63 and Lorenz-96 ODE representations may not lead to the best reconstruction performance. We believe these results may open new research avenues for the specification of assimilation models in geoscience.