Sparse Convolution-based Markov Models for Nonlinear Fluid Flows

Abstract

Data-driven modeling for nonlinear fluid flows using sparse convolution-based mapping into a feature space where the dynamics are Markov linear is explored in this article. The underlying principle of low-order models for fluid systems is identifying convolutions to a feature space where the system evolution (a) is simpler and efficient to model and (b) the predictions can be reconstructed accurately through deconvolution. Such methods are useful when real-time models from sensor data are needed for online decision making. The Markov linear approximation is popular as it allows us to leverage the vast linear systems machinery. Examples include the Koopman operator approximation techniques and evolutionary kernel methods in machine learning. The success of these models in approximating nonlinear dynamical systems is tied to the effectiveness of the convolution map in accomplishing both (a) and (b) mentioned above. To assess this, we perform in-depth study of two classes of sparse convolution operators: (i) a pure data-driven POD-convolution that uses left singular vectors of the data snapshots - a staple of Koopman approximation methods and (ii) a sparse Gaussian Process (sGP) convolution that combines sparse sampling with a Gaussian kernel embedding an implicit feature map to an inner product reproducing kernel Hilbert space.

arXiv commentsubmitted to Computers and Fluids

References in corpus (8)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Model-Based Compressive Sensing
• Cluster-based reduced-order modelling of a mixing layer
• Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation
• A Kernel-Based Approach to Data-Driven Koopman Spectral Analysis
• Variational approach for learning Markov processes from time series data
• Online dynamic mode decomposition for time-varying systems

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