Provably accurate data assimilation for fully Bayesian predictions of geophysical dynamics through reduced-order models
Abstract
This proposal concerns the development of a systematic and rigorous framework for data assimilation (DA) for inherently infinite-dimensional geophysical flows. Accurate and robust dynamical predictions of the nonlinear, multi-scale, intermittent processes in geophysical fluid dynamics (GFD), including the atmosphere-ocean dynamics, are invariably crucial in operational settings. Despite the widespread use and many successes of physics-informed deep neural networks (DNN), the fundamental problem of sensitivity to the initial condition in dynamical prediction of complex flows cannot be overcome by any forward model alone. Thus, the need for data assimilation in dynamic predictions has been recognized long ago. However, it has proven difficult to derive DA techniques for nonlinear dynamics that would be capable of providing meaningful and reliable estimates of the state and the error statistics beyond idealized bounds on the asymptotic accuracy of point estimates (usually the filtering mean or the maximum a posteriori estimates). All currently available methods (e.g. nudging, EnKF or 3D/4Dvar) are based on ad-hoc approximations and provide only point estimates for the state. Moreover, DA in geophysical applications is compounded by the infinite dimensionality of the underlying PDE dynamics that is only partially observed, while the predictions are necessarily based on finite- dimensional models. Despite the operational need for a reliable near-real-time predictions in geophysical flows, provably robust and accurate DA algorithms capable of a meaningful uncertainty quantification of the error statistics do not exist. We will develop a DA framework which, for the first time, will be capable of providing reliable, fully Bayesian es- timates of the non-Gaussian error statistics (i.e. the full probability distribution) in the estimates of various classes of geophysical dynamics, as well as rigorous bounds on the filter accuracy. Our method is based on finite-dimensional, realistic observations and reduced-order dynamics with a precise quantification of model error. This will be achieved by exploiting a hidden, hitherto unrecognized, conditionally Gaussian structure which is inherent in many dissipative GFD flowsdue to the quadratic nature of the nonlinearities. This allows one to #enslave# small scale, unobserved degrees of freedom to the observed components through so-called inertial manifolds or determining maps, leading to conditional Gaussianity. Importantly, this novel and long overdue framework will allow us to extend the rigorous analysis to account for different realistic types of observations ranging from Eulerian volume averaged or unstructured nodal, to trajectory-based (Lagrangian) observations. In particular, such aunified and general framework will allow us to systematically consider the effects of spatio-temporally inhomogeneous, and possiblylocalized, observations on the DA estimates, as well as design and analyze hybrid DA techniquesthat combine partial Eulerian and Lagrangian observations, in order to recover the three-dimensional, time-dependent structure of oceanic or coupled atmoshpere-ocean flows.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Nov 09, 2024
- Source ID
- N000142412699
Entities
People
- Michal Branicki
Organizations
- Office of Naval Research
- United States Navy
- University of Edinburgh