Chapter 1: Dynamic Mode Decomposition: An Introduction

Kutz, J. Nathan; Brunton, Steven L.; Brunton, Bingni W.; Proctor, Joshua L.

doi:10.1137/1.9781611974508.ch1

Cited by 3 publications

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“… Compute the single value decomposition (SVD) of the data matrix Perform a low-rank truncation of the data. Obtain U r , Σ r , V r by only considering the first r columns of U and V , and the first r rows and columns of Σ where , . Compute Ã , the r × r projection of the full data operator A into a reduced dimensionality space: Compute eigenvalues μ i and eigenvectors w i of Ã , where Ã w i = λ i w i . Every DMD mode ϕ i with a nonzero eigenvalue μ i can be written as To extract the characteristic time constants τ i describing the evolution of spectra, we need to convert the discrete-time eigenvalues λ i to continuous eigenvalues ω i using ω i = ln (λ i )/Δ t , where Δ t is the sampling time interval. , The exponential decay/growth rate τ i and oscillation frequency ω′ i are given by ω i = τ i + i ω′ i . The DMD modes associated with each time scale are plotted to visualize the spectral features described by each time scale. Finally, if desired, the reconstruction of 2D spectra at any given time can be written as …”

Section: Methodsmentioning

confidence: 99%

“…To extract the characteristic time constants τ i describing the evolution of spectra, we need to convert the discrete-time eigenvalues λ i to continuous eigenvalues ω i using ω i = ln (λ i )/Δ t , where Δ t is the sampling time interval. , The exponential decay/growth rate τ i and oscillation frequency ω′ i are given by ω i = τ i + i ω′ i . The DMD modes associated with each time scale are plotted to visualize the spectral features described by each time scale.…”

Section: Methodsmentioning

confidence: 99%

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Cutting through the Noise: Extracting Dynamics from Ultrafast Spectra Using Dynamic Mode Decomposition

Xu,

Baiz

2023

J. Phys. Chem. A

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Coherent multidimensional spectroscopy provides experimental access to molecular structure and subpicosecond dynamics in solution. Dynamics are typically inferred from the evolution of lineshapes over a function of waiting time. Numerous spectral analysis methods, such as center/nodal line slope, have been developed to extract these dynamics. However, the extracted dynamics can depend heavily on subjective choices, such as the region selected for CLS analysis or the chosen models. In this study, we introduce a novel approach to extracting dynamics from ultrafast twodimensional infrared (2D IR) spectra by using dynamic mode decomposition (DMD). As a data-driven method, DMD directly extracts spatiotemporal structures from the complex 2D IR spectra. We evaluated the performance of DMD in simulated and experimental spectra containing overlapped peaks. We show that DMD can retrieve the dynamics of overlapped transitions and cross peaks that are typically challenging to extract with traditional methods. In addition, we demonstrate that combining conditional generative adversarial neural networks with DMD can recover dynamics even at low signalto-noise ratios. DMD methods do not require preliminary assumptions and can be readily extended to other multidimensional spectroscopies.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Cutting through the Noise: Extracting Dynamics from Ultrafast Spectra Using Dynamic Mode Decomposition

Xu,

Baiz

2023

J. Phys. Chem. A

View full text Add to dashboard Cite

show abstract

“…To extract the characteristic time constants I describing the evolution of spectra, we need to convert the discrete-time eigenvalues & to continuous eigenvalues using = ln(& ) /∆ , where ∆ is the sampling time interval. 31,32 The exponential decay/growth rate I and oscillation frequency ′ are given by = I + O ′ . The DMD modes associated with each timescale are plotted to visualize the spectral features described by each timescale.…”

Section: Dmd Applications To 2d Ir Spectroscopymentioning

confidence: 99%

Cutting Through the Noise: Extracting Dynamics from Ultrafast Spectra using Dynamic Mode Decomposition

Xu,

Baiz

2023

Preprint

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Coherent multidimensional spectroscopy provides experimental access to molecular structure and sub-picosecond dynamics in solution. Dynamics are typically inferred from the evolution of lineshapes over was a function of waiting time. Numerous spectral analysis methods, such as center/nodal line slope, have been developed to extract these dynamics. However, the extracted dynamics can depend heavily on subjective choices such as the region selected for CLS analysis or the chosen models. In this study, we introduce a novel approach to extracting dynamics from ultrafast two-dimensional infrared (2D IR) spectra using dynamic mode decomposition (DMD). As a data-driven method, DMD directly extracts spatio-temporal structures from the complex 2D IR spectra. We evaluate the performance of DMD in synthetic and experimental spectra containing overlapped peaks. We show that, DMD can retrieve the dynamics of overlapped transitions and cross peaks that are typically challenging to extract with traditional methods. In addition, we demonstrate that combining conditional generative adversarial neural networks (cGANNs) with DMD can recover dynamics even at low signal-to-noise ratios. DMD methods do not require preliminary assumptions and can be readily extended to other multidimensional spectroscopies.

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Testing Methods of Pattern Extraction for Climate Data Using Synthetic Modes

Fulton

Hegerl

2021

Journal of Climate

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In this paper we develop a method to quantify the accuracy of different pattern extraction techniques for the additive space–time modes often assumed to be present in climate data. It has previously been shown that the standard technique of principal component analysis (PCA; also known as empirical orthogonal functions) may extract patterns that are not physically meaningful. Here we analyze two modern pattern extraction methods, namely dynamical mode decomposition (DMD) and slow feature analysis (SFA), in comparison with PCA. We develop a Monte Carlo method to generate synthetic additive modes that mimic the properties of climate modes described in the literature. The datasets composed of these generated modes do not satisfy the assumptions of any pattern extraction method presented. We find that both alternative methods significantly outperform PCA in extracting local and global modes in the synthetic data. These techniques had a higher mean accuracy across modes in 60 out of 60 mixed synthetic climates, with SFA slightly outperforming DMD. We show that in the majority of simple cases PCA extracts modes that are not significantly better than a random guess. Finally, when applied to real climate data these alternative techniques extract a more coherent and less noisy global warming signal, as well as an El Niño signal with a clearer spectral peak in the time series, and more a physically plausible spatial pattern.

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Chapter 1: Dynamic Mode Decomposition: An Introduction

Cited by 3 publications

References 0 publications

Cutting through the Noise: Extracting Dynamics from Ultrafast Spectra Using Dynamic Mode Decomposition

Cutting through the Noise: Extracting Dynamics from Ultrafast Spectra Using Dynamic Mode Decomposition

Cutting Through the Noise: Extracting Dynamics from Ultrafast Spectra using Dynamic Mode Decomposition

Testing Methods of Pattern Extraction for Climate Data Using Synthetic Modes

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