Onboard probabilistic forecasting and data assimilation is challenging for unmanned autonomous platforms. Due to the operational constraints, efficient adaptive reduced order models (ROMs) are needed. To extend the duration for which Dynamic Mode Decomposition (DMD) predictions are accurate, we utilize and augment incremental methods that update the reduced order state but also adapt the DMD. Our adaptive ROM methods are dynamic and stochastic. They update the state, parameters, and basis functions, in response to the changing forecasts, possibly computed in remote centers, and to observations made by the autonomous platforms and by other assets. For the latter, to allow learning even when observations are sparse and multivariate, we employ Bayesian data assimilation. Specifically, we extend the Gaussian Mixture Model – Dynamically Orthogonal (GMM-DO) filter to stochastic DMD forecasts and Bayesian GMM updates of the DMD coefficients, state, and parameters, learning from the limited gappy observation data sets.

Accurate numerical simulation and modeling of ocean dynamics is playing an increasingly large role in scientific ocean applications. However, resolving these dynamics with traditional computational techniques can often be prohibitively expensive, necessitating the creation of next-generation high-order ocean models. In this work, we apply the local discontinuous Galerkin (LDG) and hybridizable discontinuous Galerkin (HDG) finite element methodology to discretize the ocean equations with a free-surface. We provide comparison of the strengths and weaknesses of the two formulations in terms of accuracy, efficiency, and scalability, and provide detailed discussion of numerical choices and their consequences as they relate to ocean modeling. We verify our methodology with numerical experiments and results from nonhydrostatic gravity wave theory.

Accurate and computationally efficient acoustic models are needed for varied marine applications. In this paper, we focus our attention on forward models, which are essential to inverse problems such as imaging and mapping. First, we introduce new dynamically orthogonal (DO) equations for the acoustic wave equation in full generality, allowing for stochastic and spatially heterogeneous parameters. These equations may be spatially discretized and integrated in time numerically. Alternatively, the DO equations may be discretized themselves, admitting a non-intrusive reduced-order approach to solve the stochastic wave equation. We demonstrate the latter with a test case of an acoustic pulse traveling through the ocean with an uncertain sound speed. Second, we adapt the spatially discrete DO approach, typically used to reduce the stochastic dimension, to efficient reduced-order modeling of deterministic 3D acoustic propagation. We solve the 3D parabolic wave equation and show that low-rank solutions rapidly converge to the full-rank solution. Together, these approaches offer novel ways to solve stochastic and deterministic problems with strong or weak scattering at a reduced computational cost.

There are many significant challenges for unmanned autonomous platforms at sea including predicting the likely scenarios for the ocean environment, quantifying regional uncertainties, and updating forecasts of the evolving dynamics using their observations. Due to the operational constraints such as onboard power, memory, bandwidth, and space limitations, efficient adaptive reduced order models (ROMs) are needed for onboard predictions. In the first part, several reduced order modeling schemes for regional ocean forecasting onboard autonomous platforms at sea are described, investigated, and evaluated. We find that Dynamic Mode Decomposition (DMD), a data-driven dimensionality reduction algorithm, can be used for accurate predictions for short periods in ocean environments. We evaluate DMD methods for ocean PE simulations by comparing and testing several schemes including domain splitting, adjusting training size, and utilizing 3D inputs. Three new approaches that combine uncertainty with DMD are also investigated and found to produce practical and accurate results, especially if we employ either an ensemble of DMD forecasts or the DMD of an ensemble of forecasts. We also demonstrate some results from projecting/compressing high-fidelity forecasts using schemes such as POD projection and K-SVD for sparse representation due to showing promise for distributing forecasts efficiently to remote vehicles. In the second part, we combine DMD methods with the GMM-DO filter to produce DMD forecasts with Bayesian data assimilation that can quickly and efficiently be computed onboard an autonomous platform. We compare the accuracy of our results to traditional DMD forecasts and DMD with Ensemble Kalman Filter (EnKF) forecast results and show that in Root Mean Square Error (RMSE) sense as well as error field sense, that the DMD with GMM-DO errors are smaller and the errors grow slower in time than the other mentioned schemes. We also showcase the DMD of the ensemble method with GMM-DO. We conclude that due to its accurate and computationally efficient results, it could be readily applied onboard autonomous platforms. Overall, our contributions developed and integrated stochastic DMD forecasts and efficient Bayesian GMM-DO updates of the DMD state and parameters, learning from the limited gappy observation data sets.

The past decade has seen an increasing use of autonomous vehicles (propelled AUVs, ocean gliders, solar-vehicles, etc.) in marine applications. For the operation of these vehicles, efficient methods for path planning are critical. Path planning, in the most general sense, corresponds to a set of rules to be provided to an autonomous robot for navigating from one configuration to another in some optimal fashion. Increasingly, having autonomous vehicles that optimally collect/harvest external fields from highly dynamic environments has grown in relevance. Autonomously maximizing the harvest in minimum time is our present path planning objective. Such optimization has numerous impactful applications. For instance, in the case of energy optimal path planning where long endurance and low power are crucial, it is important to be able to optimally harvest energy (solar, wind, wave, thermal, etc.) along the way and/or leverage the environment (winds, currents, etc.) to reduce energy expenditure. Similarly, autonomous marine cleanup or collection vehicles, tasked with harvesting plastic waste, oil spills, or seaweed fields, need to be able to plan paths that maximize the amount of material harvested in order to optimize the cleanup or collection process. In this work, we develop an exact partial differential equation-based methodology that predicts harvest-time optimal paths for autonomous vehicles navigating in dynamic environments. The governing differential equations solve the multi-objective optimization problem of navigating a vehicle autonomously in a highly dynamic flow field to any destination with the goal of minimizing travel time while also maximizing the amount harvested by the vehicle. Using Hamilton-Jacobi theory for reachability, our methodology computes the exact set of Pareto optimal solutions to the multi-objective path planning problem. This is completed by numerically solving a reachability problem for the autonomous vehicle in an augmented state space consisting of the vehicle’s position in physical space as well as its harvest state. Our approach is applicable to path planning in various environments, however we primarily present examples of navigating in dynamic ocean flows. The following cases, in particular, are studied. First, we validate our methodology using a benchmark case of planning paths through a region with a harvesting field present in a halfspace, as this case admits a semi-analytical solution that we compare to the results of our method. We next consider a more complex unsteady environment as we solve for harvest-time optimal missions in a quasi-geostrophic double-gyre ocean flow field. Following this, we provide harvest-time optimal paths to the highly relevant issue of collecting harmful algae blooms. Our final case considers an application to next generation offshore aquaculture technologies. In particular, we consider in this case path planning of an offshore moving fish farm that accounts for optimizing fish growth. Overall, we find that our exact planning equations and efficient schemes are promising to address several pressing challenges for our planet.