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. Parametrization the data-driven models The proposed framework, is tested with a dimension of the augmented state space d E = 40 i.e. , 20 latent states are concatenated to the observed variable. In this experiment, the quadratic part of the NODE model f θ is convolutional and corresponds to the true non-linear interactions in (20) with trainable weights. Such representation significantly accelerates the computation of the constraint C 1 in (18) 1 . Similar to the Lorenz 63 experiment, both the constrained and unconstrained versions of the model are compared to the sparse regression technique, to a stacked bidirectional LSTM (RNN) and to the Latent-ODE model [13]). The SR model is based on a second-order polynomial representation. The state-space of the model is built on a stacked delay embedding of the 20 observed states. For each observed state, the dimension of the embedding is computed using the FNN method and the lag is chosen according to both the mutual information and the correlation techniques. The time integration of the SR model is carried using the LOSDA solver [109]. The RNN model includes 10 LSTM layers with 100 hidden units. The input sequence of the model is an observation sequence of size 20. Regarding the Latent-ODE model, the recognition network is an RNN with 100 hidden units. The dimension of the latent space is set to 40 and the dynamical model is a linear quadratic neural ODE. The decoder network is a fully-connected neural network with one hidden layer and 100 hidden units. for classical chaotic ODEs (Lorenz 63 and 96), the proposed methodology matches state-of-the-art short-term forecasting performances while ensuring realistic long-term patterns for partially-observed settings. We also report a more complex case study, in which we have considered low-resolution observations of a subdomain of SWE dynamics with unknown boundary conditions. The results from this study support the potential of the proposed technique to derive faithful data-based representations of ocean flows, especially ocean surface dynamics. Such dynamics involve, similar to the PSWE case study, but at a higher level of complexity, unseen components, missing forcing, unknown boundary conditions and unresolved spatio-temporal scales. In all numerical experiments, both the data-driven model formulation in an augmented space and the boundedness constraints are key features to capture the dynamics underlying the observations.