This paper addresses the data-driven identification of latent representations of partially observed dynamical systems, i.e., dynamical systems for which some components are never observed, with an emphasis on forecasting applications and long-term asymptotic patterns. Whereas state-of-the-art data-driven approaches rely in general 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 reconstructing the latent states and learning an ordinary differential equation representation in this space. Through numerical experiments, we demonstrate the relevance of the proposed framework with respect to state-of-the-art approaches in terms of short-term forecasting errors and long-term behavior. We further discuss how the proposed framework relates to the Koopman operator theory and Takens’ embedding theorem.

The forecasting and reconstruction of oceanic dynamics is a crucial challenge. While model driven strategies are still the state-of-the-art approaches in the reconstruction of spatio-temporal dynamics. The ever increasing availability of data collections in oceanography raised the relevance of data-driven approaches as computationally efficient representations of spatio-temporal fields reconstruction. This tools proved to outperform classical state-of-the-art interpolation techniques such as optimal interpolation and DINEOF in the retrievement of fine scale structures while still been computationally efficient comparing to model based data assimilation schemes. However, coupling this data-driven priors to classical filtering schemes limits their potential representativity. From this point of view, the recent advances in machine learning and especially neural networks and deep learning can provide a new infrastructure for dynamical modeling and interpolation within a data-driven framework. In this work we adress this challenge and develop a novel Neural-Network-based (NN-based) Kalman filter for spatio-temporal interpolation of sea surface dynamics. Based on a data-driven probabilistic representation of spatio-temporal fields, our approach can be regarded as an alternative to classical filtering schemes such as the ensemble Kalman filters (EnKF) in data assimilation. Overall, the key features of the proposed approach are two-fold: (i) we propose a novel architecture for the stochastic representation of two dimensional (2D) geophysical dynamics based on a neural networks, (ii) we derive the associated parametric Kalman-like filtering scheme for a computationally-efficient spatio-temporal interpolation of Sea Surface Temperature (SST) fields. We illustrate the relevance of our contribution for an OSSE (Observing System Simulation Experiment) in a case-study region off South Africa. Our numerical experiments report significant improvements in terms of reconstruction performance compared with operational and state-of-the-art schemes (e.g., optimal interpolation, Empirical Orthogonal Function (EOF) based interpolation and analog data assimilation).

Defining end-to-end (or online) 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, 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 show significant improvements over offline learning, highlighting the potential of end-to-end learning for hybrid modeling.

Ocean deoxygenation, driven by climate change, poses significant challenges to marine ecosystems and can profoundly alter nutrient and carbon cycling. Quantifying the rate and regional patterns of deoxygenation relies on spatio-temporal interpolation tools to fill gaps in observational coverage of dissolved oxygen. However, this task is challenging due to the sparsity of observations, and classical interpolation methods often lead to high uncertainty and biases, typically underestimating long-term deoxygenation trends. In this work, we develop a novel gridded dissolved oxygen product by integrating direct oxygen observations with machine-learning-based emulated oxygen estimates derived from temperature and salinity profiles. The gridded product is then generated through optimal interpolation of both the observed and emulated data. The resulting product (Ouala et al., 2025; https://doi.org/10.5281/zenodo.15478088) shows strong agreement with baseline climatology and captures well-known patterns of seasonal variability and long-term deoxygenation trends. It also outperforms current state-of-the-art products by more accurately capturing dissolved oxygen variability at synoptic and decadal scales, and by reducing uncertainty around long-term changes. This study highlights the potential of combining machine learning with classical interpolation methods to generate improved gridded biogeochemical products, enhancing our ability to study and understand ocean biogeochemical processes and their variability under a changing climate.

Bernard O Koopman proposed an alternative view of dynamical systems based on linear operator theory, in which the time evolution of a dynamical system is analogous to the linear propagation of an infinite-dimensional vector of observables. In the last few years, several works have shown that finite-dimensional approximations of this operator can be extremely useful for several applications, such as prediction, control, and data assimilation. In particular, a Koopman representation of a dynamical system with a finite number of dimensions will avoid all the problems caused by nonlinearity in classical state-space models. In this work, the identification of finite-dimensional approximations of the Koopman operator and its associated observables is expressed through the inversion of an unknown augmented linear dynamical system. The proposed framework can be regarded as an extended dynamical mode decomposition that uses a collection of latent observables. The use of a latent dictionary applies to a large class of dynamical regimes, and it provides new means for deriving appropriate finite-dimensional linear approximations to high-dimensional nonlinear systems.

Within the geosciences community, data-driven techniques have encountered a great success in the last few years. This is principally due to the success of machine learning techniques in several image and signal processing domains. However, when considering the data-driven simulation of ocean and atmospheric fields, the application of these methods is still an extremely challenging task due to the fact that the underlying dynamics usually depend on several complex hidden variables, which makes the learning and simulation process much more challenging.In this work, we aim to extract Ordinary Differential Equations (ODE) from partial observations of a system. We propose a novel neural network architecture guided by physical and mathematical considerations of the underlying dynamics. Specifically, our architecture is able to simulate the dynamics of the system from a single initial condition even if the initial condition does not lie in the attractor spanned by the training data. We show on different case studies the effectiveness of the proposed framework both in capturing long term asymptotic patterns of the dynamics of the system and in addressing data assimilation issues which relates to the short term forecasting performance of our model.