Past Event: Oden Institute Seminar

Bayesian Learning of Stochastic Dynamical Models: State, Parameters, also Model Structures?

Pierre F.J. Lermusiaux, Department of Mechanical Engineering, Massachusetts Institute of Technology

Abstract

In contemporary ocean science and engineering, modeling systems that integrate understanding of complex multi-scale phenomena and utilize efficient numerics are paramount. Advances in this area require that we quantify and predict uncertainties and reduce these by nonlinear data assimilation; the results determine critical observation needs and inspire model improvements. The study of interdisciplinary coastal ocean dynamics is a curiosity-driven intellectual challenge in itself. It is also vital to society for multiple industries and human activities, and ultimately for the assessment of human impacts on the ocean’s health and climate. In this presentation, we first highlight recent results by our research group, including high-order Finite-Element schemes for biogeochemical ocean dynamics and exact time-optimal path planning for swarms of ocean vehicles using new level-set equations. We then address a holistic challenge in ocean Bayesian estimation: predict the pdfs of large nonlinear ocean systems using stochastic PDEs, assimilate data using Bayes’ law with these pdfs, predict the future data that optimally reduce uncertainties and learn the model structures themselves. Specifically, stochastic Dynamically Orthogonal (DO) equations and their adaptive stochastic subspace are employed to predict prior probabilities for the state and parameters. At assimilation times, the DO realizations are fit to semiparametric Gaussian Mixture Models (GMMs) using the Expectation-Maximization algorithm and the Bayesian Information Criterion. Bayes’ Law is then efficiently carried out analytically within the evolving stochastic subspace. The advantages of respecting nonlinear dynamics and preserving non-Gaussian statistics are brought to light. This Bayesian approach is then employed for adaptive sampling, i.e. predicting the optimal future data, and for adaptive modeling, i.e. predicting the optimal model structures. The latter is obtained by computing marginal likelihoods for candidate model structures within a model probability framework. The result is a holistic methodology for Bayesian model inference for high-dimensional nonlinear stochastic dynamical systems with limited data. Examples are provided using time-dependent ocean and fluid flows, including cavity, double-gyre and sudden-expansion flows with jets and eddies. The Bayesian model inference is illustrated by the estimation of obstacle shapes and of biogeochemical reaction equations. Biography/Research Interests Dr. Lermusiaux is an Associate Professor of Mechanical Engineering and Ocean Science and Engineering at MIT. He received a Fulbright Foundation Fellowship, was awarded the Wallace Prize at Harvard (1993), and the Ogilvie Young Investigator Lecture in Ocean Eng. at MIT (1998). He was awarded the MIT Doherty Chair in Ocean Utilization (2009-2011) and the 2010 Ruth and Joel Spira Award for Distinguished Teaching by the School of Eng. at MIT. He has made outstanding contributions in data assimilation, as well as in ocean modeling and uncertainty predictions. His research thrusts include understanding and modeling complex physical and interdisciplinary oceanic dynamics and processes. With his group, he creates, develops and utilizes new mathematical models and computational methods for ocean predictions and dynamical diagnostics, for optimization and control of autonomous ocean observation systems, for uncertainty quantification and prediction, and for data assimilation and data-model comparisons. He has participated in many national and international sea exercises. He has served on numerous committees, and has organized several major meetings. He is associate editor of three journals. He has more than seventy refereed publications.