Data-Driven Discovery of Closure Models

Abstract

Derivation of reduced order representations of dynamical systems requires the modeling of the truncated dynamics on the retained dynamics. In its most general form, this so-called closure model has to account for memory effects. In this work, we present a framework of operator inference to extract the governing dynamics of closure from data in a compact, non-Markovian form. We employ sparse polynomial regression and artificial neural networks to extract the underlying operator. For a special class of non-linear systems, observability of the closure in terms of the resolved dynamics is analyzed and theoretical results are presented on the compactness of the memory. The proposed framework is evaluated on examples consisting of linear to nonlinear systems with and without chaotic dynamics, with an emphasis on predictive performance on unseen data.

Deep learning of dynamics and signal-noise decomposition with time-stepping constraints

“…Neural networks have been used to model dynamical systems for decades [8,14,34], although recent advances in computing power, data volumes, and deep learning architectures have dramatically improved their capabilities. Reccurrent neural networks naturally model sequential processes and have been used for forecasting [60,37,28,39,38] and closure models for reduced order models [61,36]. Deep learning approaches have also been used recently to find coordinate transformations that make strongly nonlinear systems approximately linear, related to Koopman operator theory [57,64,63,32,29].…”

Deep Learning of Markov Model Based Machines for Determination of Better Treatment Option Decisions for Infertile Women

“…There were also successful methods in deep learning algorithms which score patients in ICU (Intensive Care Unit) for their severity and to predict mortality without using any model based assumptions in scoring systems (42) and for other medical applications, for example detection of worms through endoscopy (43), ophthalmology studies (44), cardiovascular studies (45), Parkinson's disease data (46), medical scoring systems (47). Deep learning procedures involved in various levels of abstraction for ranking system models can be found in (48,49), applications for mathematical models, parameter computations and stability of algorithms are found in (50)(51)(52)(53)(54)(55)(56).…”

Deep learning of turbulent scalar mixing

“…The proposed framework replaces the guessing work often involved in such model development with a data-driven approach that uncovers the closure from data in a systematic fashion. Our approach draws inspiration from the early and contemporary contributions in deep learning for partial differential equations [14][15][16][17][18][19][20] and data-driven modeling strategies [21][22][23], and in particular relies on recent developments in physics-informed deep learning [24] and deep hidden physics models [25].…”