Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userKernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)
358
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Research in modern data-driven dynamical systems is typically focused on the three key challenges of high dimensionality, unknown dynamics, and nonlinearity. The dynamic mode decomposition (DMD) has emerged as a cornerstone for modeling high-dimensional systems from data. However, the quality of the linear DMD model is known to be fragile with respect to strong nonlinearity, which contaminates the model estimate. In contrast, sparse identification of nonlinear dynamics (SINDy) learns fully nonlinear models, disambiguating the linear and nonlinear effects, but is restricted to low-dimensional systems. In this work, we present a kernel method that learns interpretable data-driven models for high-dimensional, nonlinear systems. Our method performs kernel regression on a sparse dictionary of samples that appreciably contribute to the underlying dynamics. We show that this kernel method efficiently handles high-dimensional data and is flexible enough to incorporate partial knowledge of system physics. It is possible to accurately recover the linear model contribution with this approach, disambiguating the effects of the implicitly defined nonlinear terms, resulting in a DMD-like model that is robust to strongly nonlinear dynamics. We demonstrate our approach on data from a wide range of nonlinear ordinary and partial differential equations that arise in the physical sciences. This framework can be used for many practical engineering tasks such as model order reduction, diagnostics, prediction, control, and discovery of governing laws.
rate research
Read More
Koopman mode analysis has provided a framework for analysis of nonlinear phenomena across a plethora of fields. Its numerical implementation via Dynamic Mode Decomposition (DMD) has been extensively deployed and improved upon over the last decade. We address the problems of mean subtraction and DMD mode selection in the context of finite dimensional Koopman invariant subspaces. Preprocessing of data by subtraction of the temporal mean of a time series has been a point of contention in companion matrix-based DMD. This stems from the potential of said preprocessing to render DMD equivalent to temporal DFT. We prove that this equivalence is impossible when the order of the DMD-based representation of the dynamics exceeds the dimension of the system. Moreover, this parity of DMD and DFT is mostly indicative of an inadequacy of data, in the sense that the number of snapshots taken is not enough to represent the true dynamics of the system. We then vindicate the practice of pruning DMD eigenvalues based on the norm of the respective modes. Once a minimum number of time delays has been taken, DMD eigenvalues corresponding to DMD modes with low norm are shown to be spurious, and hence must be discarded. When dealing with mean-subtracted data, the above criterion for detecting synthetic eigenvalues can be applied after additional pre-processing. This takes the form of an eigenvalue constraint on Companion DMD, or yet another time delay.
The 2D second-mode is a potent instability in hypersonic boundary layers (HBLs). We study its linear and nonlinear evolution, followed by its role in transition and eventual breakdown of the HBL into a fully turbulent state. Linear stability theory (LST) is utilized to identify the second-mode wave through FS-synchronization, which is then recreated in linearly and nonlinearly forced 2D direct numerical simulations (DNSs). The nonlinear DNS shows saturation of the fundamental frequency, and the resulting superharmonics induce tightly braided ``rope-like patterns near the generalized inflection point (GIP). The instability exhibits a second region of growth constituted by the fundamental frequency, downstream of the primary envelope, which is absent in the linear scenario. Subsequent 3D DNS identifies this region to be crucial in amplifying oblique instabilities riding on the 2D second-mode ``rollers. This results in lambda vortices below the GIP, which are detached from the ``rollers in the inner boundary layer. Streamwise vortex-stretching results in a localized peak in length-scales inside the HBL, eventually forming haripin vortices. Spectral analyses track the transformation of harmonic peaks into a turbulent spectrum, and appearance of oblique modes at the fundamental frequency, which suggests that fundamental resonance is the most dominant mechanism of transition. Bispectrum reveals coupled nonlinear interactions between the fundamental and its superharmonics leading to spectral broadening, and traces of subharmonic resonance as well.
Dynamic mode decomposition (DMD) is utilised to identify the intrinsic signals arising from planetary interiors. Focusing on an axisymmetric quasi-geostrophic magnetohydrodynamic (MHD) wave -called torsional Alfv{e}n waves (TW) - we examine the utility of DMD in two types of MHD direct numerical simulations: Boussinesq magnetoconvection and anelastic convection-driven dynamos in rapidly rotating spherical shells, which model the dynamics in Earths core and in Jupiter, respectively. We demonstrate that DMD is capable of distinguishing internal modes and boundary/interface-related modes from the timeseries of the internal velocity. Those internal modes may be realised as free TW, in terms of eigenvalues and eigenfunctions of their normal mode solutions. Meanwhile it turns out that, in order to account for the details, the global TW eigenvalue problems in spherical shells need to be further addressed.
The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical variational algorithm which offers the potential to handle combinatorial optimization problems. Introducing constraints in such combinatorial optimization problems poses a major challenge in the extensions of QAOA to support relevant larger scale problems. In this paper, we introduce a quantum machine learning approach to learn the mixer Hamiltonian that is required to hard constrain the search subspace. We show that this method can be used for encoding any general form of constraints. By using a form of an adaptable ansatz, one can directly plug the learnt unitary into the QAOA framework. This procedure gives the flexibility to control the depth of the circuit at the cost of accuracy of enforcing the constraint, thus having immediate application in the Noisy Intermediate Scale Quantum (NISQ) era. We also develop an intuitive metric that uses Wasserstein distance to assess the performance of general approximate optimization algorithms with/without constrains. Finally using this metric, we evaluate the performance of the proposed algorithm.
We employ the framework of the Koopman operator and dynamic mode decomposition to devise a computationally cheap and easily implementable method to detect transient dynamics and regime changes in time series. We argue that typically transient dynamics experiences the full state space dimension with subsequent fast relaxation towards the attractor. In equilibrium, on the other hand, the dynamics evolves on a slower time scale on a lower dimensional attractor. The reconstruction error of a dynamic mode decomposition is used to monitor the inability of the time series to resolve the fast relaxation towards the attractor as well as the effective dimension of the dynamics. We illustrate our method by detecting transient dynamics in the Kuramoto-Sivashinsky equation. We further apply our method to atmospheric reanalysis data; our diagnostics detects the transition from a predominantly negative North Atlantic Oscillation (NAO) to a predominantly positive NAO around 1970, as well as the recently found regime change in the Southern Hemisphere atmospheric circulation around 1970.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
