Most learning results are limited in interpretability and generalization over different computational grid resolutions, initial and boundary conditions, domain geometries, and physical or problem-specific parameters.  We review how we simultaneously address these challenges using our novel and versatile methodology of unified neural partial delay differential equations with applications to idealized ocean and chaotic systems. We augment existing/low-fidelity dynamical models directly in their partial differential equation (PDE) forms with both Markovian and non-Markovian neural network (NN) closure parameterizations. The melding of the existing models with NNs in the continuous spatiotemporal space followed by numerical discretization automatically allows for the desired generalizability. The Markovian term is designed to enable extraction of its analytical form and thus provides interpretability. The non-Markovian terms allow accounting for inherently missing time delays needed to represent the real world. Our flexible modeling framework provides full autonomy for the design of the unknown closure terms such as using any linear-, shallow-, or deep-NN architectures, selecting the span of the input function libraries, and using either or both Markovian and non-Markovian closure terms, all in accord with prior knowledge. We obtain adjoint PDEs in the continuous form, thus enabling direct implementation across differentiable and non-differentiable computational physics codes, different ML frameworks, and treatment of nonuniformly-spaced spatiotemporal training data. We first demonstrate the generalized neural closure models (gnCMs) framework using sets of experiments based on advecting nonlinear waves, shocks, and ocean acidification models. We then discuss and highlight applications to progressively more chaotic systems, emphasizing the need for adapted learning schemes. Our learned gnCMs discover missing chaotic physics, find leading numerical error terms, discriminate among candidate functional forms in an interpretable fashion, achieve generalization, and compensate for the lack of complexity in simpler models.

Reliable acoustic predictions remain challenging due to the sparse and heterogeneous data, as well as to the complex ocean physics, sea surface and seabed processes, multiscale interactions, and large dimensions.These complexities lead to several sources of uncertainty. Predicting the full probability distributions of the ocean-acoustic-seabed fields then allows robust informative modeling, inference, and decision-making. In this work, we integrate our acoustic stochastic Dynamically-Orthogonal Parabolic Equations (DO-PEs) and Gaussian Mixture Model-DO (GMM-DO) frameworks with the MSEAS primitive equation ocean modeling system to enable unprecedented probabilistic forecasting and learning of ocean physics and acoustic pressure and transmission loss (TL) fields, accounting for uncertainties in the ocean, acoustics, bathymetry, and seabed fields. We demonstrate the use of this system for low to mid-frequency propagation with real ocean data assimilation in three regions. The first sea experiment takes place in the western Mediterranean Sea where we showcase the system’s performance in predicting ocean and acoustic probability densities, and assimilating sparse TL and sound speed data for joint ocean physics-acoustics-source depth inversion in deep ocean conditions with steep ridges. In the second application, we simulate stochastic acoustic propagation in Massachusetts Bay around Stellwagen Bank and use our GMM-DO Bayesian inference system to assimilate TL data for acoustic and source depth inversion in shallow dynamics with strong internal waves. Finally, in the third experiment in the New York Bight, we employ our system as a novel probabilistic approach for broadband acoustic modeling and inversion. Overall, our results mark significant progress toward end-to-end ocean acoustic systems for new ocean exploration and management, risk analysis, and advanced operations.

Reliable acoustic predictions remain challenging due to the sparse and heterogeneous data, as well as to the complex ocean physics, sea surface and seabed processes, multiscale interactions, and large dimensions. These complexities lead to several sources of uncertainty. Predicting the full probability distributions of the ocean-acoustic-seabed fields then allows robust informative modeling, inference, and decision-making. In this work, we integrate our acoustic stochastic Dynamically-Orthogonal Parabolic Equations (DO-PEs) and Gaussian Mixture Model-DO (GMM-DO) frameworks with the MSEAS primitive equation ocean modeling system to enable unprecedented probabilistic forecasting and learning of ocean physics and acoustic pressure and transmission loss (TL) fields, accounting for uncertainties in the ocean, acoustics, bathymetry, and seabed fields. We demonstrate the use of this system for low to mid-frequency propagation with real ocean data assimilation in three regions. The first sea experiment takes place in the western Mediterranean Sea where we showcase the system’s performance in predicting ocean and acoustic probability densities, and assimilating sparse TL and sound speed data for joint ocean physics-acoustics-source depth inversion in deep ocean conditions with steep ridges. In the second application, we simulate stochastic acoustic propagation in Massachusetts Bay around Stellwagen Bank and use our GMM-DO Bayesian inference system to assimilate TL data for acoustic and source depth inversion in shallow dynamics with strong internal waves. Finally, in the third experiment in the New York Bight, we employ our system as a novel probabilistic approach for broadband acoustic modeling and inversion. Overall, our results mark significant progress toward end-to-end ocean acoustic systems for new ocean exploration and management, risk analysis, and advanced operations.

Accurate sea ice models are essential to predict the complex evolution of rapidly changing sea ice conditions and study impacts on climate, wildlife, and navigation. However, numerical models for sea ice contain various uncertainties associated with initial conditions and forcing (wind, ocean), as well as with parameter values and parameterizations, functional forms of the constitutive relations, and state variables such as sea ice thickness and concentration, all of which limit predictive capabilities. In this work, we first develop new stochastic partial differential equation (PDE)-based Sea Ice Dynamically Orthogonal equations and schemes for efficient uncertainty propagation and probabilistic predictions. These equations and schemes preserve nonlinearities in the underlying spatiotemporal dynamics and evolve the non-Gaussianity of the statistics with a lower computational cost than Monte Carlo methods commonly used in sea ice data assimilation and sensitivity analysis. We then use the Gaussian Mixture Model (GMM)-DO filter for sea ice Bayesian nonlinear data assimilation and learning. Assimilating noisy and sparse measurements, we provide posterior probability distributions for not only the sea ice velocities, thickness, and concentration, but also for the external forcing, parameters, and even functional forms of the sea ice model. The equations and schemes are evaluated using stochastic test cases, in which we showcase the ability to evolve non-Gaussian statistics and capture complex nonlinear dynamics efficiently. We demonstrate the stochastic convergence of the probabilistic predictions to the stochastic subspace size and coefficient samples. Finally, we highlight the principled joint nonlinear inference and learning of the sea ice state and dynamics.

In the Gulf of Mexico, the Loop Current (LC) is a salient oceanographic feature, being a warm-water current originating from the Caribbean Sea, traveling northward, protruding into the Gulf, and eventually departing through the Florida Strait. This dynamic stream influences the intensity of hurricanes, the vitality of coastal and estuarine ecosystems, the efficiency of petroleum exploration and extraction, and the prosperity of the fishing sector. These intricacies, in turn, profoundly affect the region’s overall economic framework. Given its overarching impact, the compelling question arises: How can we accurately forecast the likelihood of future Loop Current phenomena and anticipate the evolving oceanic conditions within the Gulf? We develop and utilize differential Dynamical-Orthogonal primitive-equations (DO-PEs) for efficient and high-resolution stochastic ocean forecasting in regions with complex ocean dynamics. We can then perform the equivalent of massive ensemble simulations of 10⁶ members in a stochastic subspace while ensuring that the initial statistics respect the physical processes, modeled complex dynamics, and uncertainty in initial conditions of the Gulf of Mexico. We demonstrate the convergence of DO probabilistic forecasts as the number of modes increases, effectively replicating full-order Monte Carlo ensemble simulations. Our analysis quantifies predictability boundaries, predictive capacities, and variability from initial and boundary conditions. Additionally, we formulate ensemble hindcasts for varying periods and LC states. We complete some targeted multi-resolution modeling experiments. We also compute Mutual Information (MI) and correlation fields to determine the most informative observations and their impacts on ocean predictions. Ultimately, our work can be used to provide rich probabilistic forecasts and inform the design of shorter- and longer-term observing campaigns.