Journal
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume 477, Issue 2254, Pages -Publisher
ROYAL SOC
DOI: 10.1098/rspa.2021.0097
Keywords
dynamic mode decomposition; time-delay coordinates; Frenet-Serret; Koopman operator; Hankel matrix
Categories
Funding
- Army Research Office [W911NF-17-1-0306]
- Air Force Office of Scientific Research [FA9550-17-1-0329]
- Air Force Research Laboratory [FA8651-16-1-0003]
- National Science Foundation [1514556]
- Alfred P. Sloan Foundation
- Washington Research Foundation
- Direct For Computer & Info Scie & Enginr
- Div Of Information & Intelligent Systems [1514556] Funding Source: National Science Foundation
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In this paper, a theoretical connection between HAVOK and the Frenet-Serret frame from differential geometry is established, and an improved algorithm is developed to identify more stable and accurate models. The study demonstrates that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in the Frenet-Serret frame, and modifying the algorithm to promote this antisymmetric structure improves modeling accuracy even with noisy, low-data limits.
Time-delay embedding and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal; for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet-Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in the Frenet-Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modelling procedure on data from several nonlinear synthetic and real-world examples.
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