Sparse spatio temporal reconstruction with Closable Kernel Space
Keywords: Kernel Learning, Koopman Theory, Spatio-Temporal Reconstruction, Reproducing Kernel Hilbert space, Dynamical Systems
TL;DR: We propose a novel kernel based method for the sparse reconstruction of spatio temporal data.
Abstract: Quantifying spatio-temporal (ST) measures of dynamical systems is a crucial problem with wide ranging applications in climate modeling, epidemiology, physical processes to name a few.
We are interested in the same but motivated by a rather practical scenario where sparse information is collected non-uniformly.
To reconstruct the underlying dynamical system under such constraints, we propose a novel algorithm for learning the Koopman operator via a Reproducing Kernel Hilbert Space (RKHS) based on the Laplacian Kernel Extended Dynamic Mode Decomposition (Lap-KeDMD).
We further show that our kernel space resolves a fundamental issue that is required for a faithful reconstruction of the Koopman operator of the underlying ST data by proving its closability.
We demonstrate our method on standard benchmark cases --
Burger's Equation, fluid flow across cylinder and Duffing Oscillator.
We then reconstruct the Koopman operator for a real ST Seattle traffic flow data that is collected non-uniformly. Necessary comparisons are made between the current state of the art kernel methods corresponding to Gaussian Radial Basis Function (GRBF) Kernel.
Such empirical comparisons leads us to conclude that Lap-KeDMD remarkably outperforms as compared to that of aforementioned counter-part thereby, making the Laplacian Kernel a robust choice for such ST quantification.
Supplementary Material: zip
Primary Area: learning on time series and dynamical systems
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Submission Number: 2219
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