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Dynamic Mode Decomposition for Compressive System Identification
Dynamic mode decomposition has emerged as a leading technique to identify
spatiotemporal coherent structures from high-dimensional data, benefiting from
a strong connection to nonlinear dynamical systems via the Koopman operator. In
this work, we integrate and unify two recent innovations that extend DMD to
systems with actuation [Proctor et al., 2016] and systems with heavily
subsampled measurements [Brunton et al., 2015]. When combined, these methods
yield a novel framework for compressive system identification [code is publicly
available at: https://github.com/zhbai/cDMDc]. It is possible to identify a
low-order model from limited input-output data and reconstruct the associated
full-state dynamic modes with compressed sensing, adding interpretability to
the state of the reduced-order model. Moreover, when full-state data is
available, it is possible to dramatically accelerate downstream computations by
first compressing the data. We demonstrate this unified framework on two model
systems, investigating the effects of sensor noise, different types of
measurements (e.g., point sensors, Gaussian random projections, etc.),
compression ratios, and different choices of actuation (e.g., localized,
broadband, etc.). In the first example, we explore this architecture on a test
system with known low-rank dynamics and an artificially inflated state
dimension. The second example consists of a real-world engineering application
given by the fluid flow past a pitching airfoil at low Reynolds number. This
example provides a challenging and realistic test-case for the proposed method,
and results demonstrate that the dominant coherent structures are well
characterized despite actuation and heavily subsampled data
Efficient estimation of nearly sparse many-body quantum Hamiltonians
We develop an efficient and robust approach to Hamiltonian identification for
multipartite quantum systems based on the method of compressed sensing. This
work demonstrates that with only O(s log(d)) experimental configurations,
consisting of random local preparations and measurements, one can estimate the
Hamiltonian of a d-dimensional system, provided that the Hamiltonian is nearly
s-sparse in a known basis. We numerically simulate the performance of this
algorithm for three- and four-body interactions in spin-coupled quantum dots
and atoms in optical lattices. Furthermore, we apply the algorithm to
characterize Hamiltonian fine structure and unknown system-bath interactions.Comment: 8 pages, 2 figures. Title is changed. Detailed error analysis is
added. Figures are updated with additional clarifying discussion
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