STOCHASTIC ANALYSIS AND NONLINEAR DYNAMICS Dynamically data-driven morphing of reduced order models and the prediction of transients Joint NSF/AFOSR EAGER on Dynamic Data Systems Themis Sapsis Massachusetts Institute of Technology Department of Mechanical Engineering ABS Career Development Assistant Professor Yannis Kevrekidis Princeton University Department of Chemical Engineering Smith Professor of Engineering 1
Complex systems with inherently transient dynamics Intermittent phenomena in CFD/GFD Objective Dynamically, data-driven prediction and filtering Challenges Extreme events in nonlinear waves Very high dimensionality (both physical and intrinsic) Inherently time-dependent features (rare events, non-stationary statistics) Model error (neglected dynamics, unknown parameters) Transient responses in networks Data (observation errors, sparse data) Approach Physics-constrained, data-driven modeling
Optimally time dependent modes Develop an approach that will adaptively select the modes associated with transient instabilities Setup Dynamical system: Linearized dynamics around a trajectory: We introduce the following minimization principle: Optimally time dependent modes: H. Babaee & T. Sapsis, A minimization principle for the description of modes associated with finite-time instabilities, Proceedings of the Royal Society A (2016) In Press.
Optimally time dependent modes Theorem 1. The minimization principle defined within the basis elements that satisfy the orthonormality constraint is equivalent with the set of evolution equations Theorem 2. Let L be a steady and diagonalizable operator that represents the linearization of an autonomous dynamical system. Then i) The OTD modes equations have equilibrium states that consist of all the r- dimensional subspaces in the span of r distinct eigenvectors of L. ii) From all the equilibrium states there is only one that is a stable solution for equation. This is given by the subspace spanned by the eigenvectors of L associated with the r eigenvalues having the largest real part. H. Babaee & T. Sapsis, A minimization principle for the description of modes associated with finite-time instabilities, Proceedings of the Royal Society A (2016) In Press.
Optimally time dependent modes A simple 3D example with 1 OTD mode H. Babaee & T. Sapsis, A minimization principle for the description of modes associated with finite-time instabilities, Proceedings of the Royal Society A (2016) In Press.
Physics-constrained, data-driven modeling 3D unstable jet in cross flow 1st OTD mode during the initial transient 4 OTD modes in the statistical steady state Growth rate of the OTD modes
Dynamic Data-Driven reduced-order dynamics OTD basis adaptively captures the most important directions of phase space. Simple Galerkin projection of the governing equations contains important truncation error and model error. Can we utilize available data streams to stochastically reconstruct the reducedorder vector field within the reduced-order OTD subspace? Approach Given the OTD basis u i (t), i = 1,,r up to the current time instant t we project the available data points z #, z #, j = 1,, D. This gives the projected data points for the evolution vector within the OTD subspace: y # = z #, u and y # = z #, u We then use Gaussian Process (GP) Regression to reconstruct the reduced-order vector field: f y = f y; u Quantifies the mean vector field as well as the truncation & model error
Dynamic Data-Driven reduced-order dynamics Demonstration over a fixed-in-time subspace Demonstration over a fixed-in-time subspace Z. Y. Wan & T. Sapsis, Reduced-space Gaussian Process Regression Forecast for Nonlinear Dynamical Systems, 2016 (Submitted).
Summary Dynamical Equations System state OTD equations Data Projection in the OTD subspace Update of the system state only along OTD directions Machine-learn the reduced-order dynamics DO subspace aligns with the mostunstable directions in the neighborhood of the solution Probabilistic Estimate for the current state 9 Machine learned manifold H. Babaee & T. Sapsis, A minimization principle for the description of modes associated with finite-time instabilities, Proceedings of the Royal Society A (2016) In Press. Z. Y. Wan & T. Sapsis, Reduced-space Gaussian Process Regression Forecast for Nonlinear Dynamical Systems, 2016 (Submitted).