Predictive dynamical models for marine ecosystems are used for a variety of needs. Due to the sparse measurements and limited understanding of the myriad of ocean processes, there is however significant uncertainty. There is model uncertainty in the parameter values, functional forms with diverse parameterizations, and level of complexity needed, and thus in the state variable fields. We develop a Bayesian model learning methodology that allows interpolation in the space of candidate dynamical models and discovery of new models from noisy, sparse, and indirect observations, all while estimating state variable fields and parameter values, as well as the joint probability distributions of all learned quantities. We address the challenges of high-dimensional and multidisciplinary dynamics governed by partial differential equations (PDEs) by using state augmentation and the computationally efficient Gaussian Mixture Model – Dynamically Orthogonal filter. Our innovations include stochastic formulation parameters and stochastic complexity parameters to unify candidate models into a single general model as well as stochastic expansion parameters within piecewise function approximations to generate dense candidate model spaces. These innovations allow handling many compatible and embedded candidate models, possibly none of which are accurate, and learning elusive unknown functional forms that augment these models. Our new Bayesian methodology is generalizable and interpretable. It seamlessly and rigorously discriminates among existing models, but also extrapolates out of the space of models to discover new ones. We perform a series of twin experiments based on flows past a ridge coupled with three-to-five component ecosystem models, including flows with chaotic advection. We quantify the learning skill, and evaluate convergence and the sensitivity to hyper-parameters. Our PDE framework successfully discriminates among functional forms and model complexities, and learns in the absence of prior knowledge by searching in dense function spaces. The probabilities of known, uncertain, and unknown model formulations, and of biogeochemical-physical fields and parameters, are updated jointly using Bayes’ law. Non-Gaussian statistics, ambiguity, and biases are captured. The parameter values and the model formulations that best explain the noisy, sparse, and indirect data are identified. When observations are sufficiently informative, model complexity and model functions are discovered.

The state of the ocean evolves and its dynamics involves transitions occurring at multiple scales. For efficient and rapid interdisciplinary forecasting, ocean observing and prediction systems must have the same behavior and adapt to the ever-changing dynamics. This chapter sets the basis of a distributed system for real-time interdisciplinary ocean field and uncertainty forecasting with adaptive modeling and adaptive sampling. The scientific goal is to couple physical and biological oceanography with ocean acoustic measurements. The technical goal is to build a dynamic modeling and instrumentation system based on advanced infrastructures, distributed/grid computing, and efficient information retrieval and visualization interfaces, from which all these are incorporated into the Poseidon system. Importantly, the Poseidon system combines a suite of modern legacy physical models, acoustic models, and ocean current monitoring data assimilation schemes with innovative modeling and adaptive sampling methods. The legacy systems are encapsulated at the binary level using software component methodologies. Measurement models are utilized to link the observed data to the dynamical model variables and structures. With adaptive sampling, the data acquisition is dynamic and aims to minimize the predicted uncertainties, maximize the optimized sampling of key dynamics, and maintain overall coverage. With adaptive modeling, model improvements dynamically select the best model structures and parameters among different physical or biogeochemical parameterizations. The dynamic coupling of models and measurements discussed here, and embodied in the Poseidon system, represents a Dynamic Data-Driven Applications Systems (DDDAS). Technical and scientific progress is highlighted based on examples in Massachusetts Bay, Monterey Bay, and the California Current System.

In January 2023, Rishab Jain, Sruthi Sentil, and Patrick Wahlig, high school seniors who joined MSEAS during summer 2022 as RSI scholars, recently entered the Regeneron Science Talent Search (STS), the nation’s oldest and most prestigious science and mathematics competition for high school seniors, and were named top 300 scholars out of 1,949 students in the 2023 competition. Each scholar will receive $2,000, and their schools will also receive $2,000 to use toward STEM-related activities. Rishab’s research with MSEAS was on “GlioMod: Spatiotemporal-Aware Glioblastoma Multiforme Tumor Growth Modeling with Deep Encoder-Decoder Networks,” and was advised by Abhinav Gupta and Wael Hajj Ali; Sruthi’s research was on “Machine Learning-Informed Ocean Source Localization In An Ocean Waveguide Environment,” and was advised by Wael Hajj Ali; and Patrick’s research was on “Towards a System for Modeling the Impacts of Ocean Acidification on Sea Scallops (Placopecten magellanicus) in Massachusetts Bay,” and was advised by Pat Haley.

This is an extraordinary accomplishment deserving of much celebration, so congrats Rishab, Sruthi, and Patrick!

Finite element discretizations of problems in computational physics often rely on adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. However, these spatial refinement strategies are often heuristic and rely on domain-specific knowledge or trial-and-error. We treat the process of adaptive mesh refinement as a local, sequential decision-making problem under incomplete information, formulating AMR as a partially observable Markov decision process. Using a deep reinforcement learning approach, we train policy networks for AMR strategy directly from numerical simulation. The training process does not require an exact solution or a high-fidelity ground truth to the partial differential equation at hand, nor does it require a pre-computed training dataset. The local nature of our reinforcement learning formulation allows the policy network to be trained inexpensively on much smaller problems than those on which they are deployed. The methodology is not specific to any particular partial differential equation, problem dimension, or numerical discretization, and can flexibly incorporate diverse problem physics. To that end, we apply the approach to a diverse set of partial differential equations, using a variety of high-order discontinuous Galerkin and hybridizable discontinuous Galerkin finite element discretizations. We show that the resultant deep reinforcement learning policies are competitive with common AMR heuristics, generalize well across problem classes, and strike a favorable balance between accuracy and cost such that they often lead to a higher accuracy per problem degree of freedom.

Whether due to the sheer size of a computational domain, the fine resolution required, or the multiples scales and stochasticity of the dynamics, the dimensionality of a system must often be reduced so that problems of interest become computationally tractable. In this paper, we develop retractions for time-integration schemes that efficiently and accurately evolve the dynamics of a system’s low-rank approximation. Through differential geometry, we analyze the error incurred at each time-step due to the high-order curvature of the manifold of fixed-rank matrices. We first obtain a novel, explicit, computationally inexpensive set of algorithms that we refer to as perturbative retractions and show that the set converges to an ideal retraction that projects optimally and exactly to the manifold of fixed-rank matrices by reducing what we define as the projection-retraction error. Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low-rank approximation of the full-rank solution. Using perturbative retractions, we then develop a new class of integration techniques that we refer to as dynamically orthogonal Runge–Kutta (DORK) schemes. DORK schemes integrate along the nonlinear manifold, updating the subspace upon which we project the system’s dynamics as it is integrated. Through numerical test cases, we demonstrate our schemes for matrix addition, real-time data compression, and deterministic and stochastic partial differential equations. We find that DORK schemes are highly accurate by incorporating knowledge of the dynamic, nonlinear manifold’s high-order curvature, and they are computationally efficient by limiting the growing rank needed to represent the evolving dynamics.