arXiv subjects

Christopher W. Curtis

Publications and source records attributed to Christopher W. Curtis.

13 recordsLinked to original sources

arXiv2026

Weak-form Extended Dynamic Mode Decomposition

In this work, we develop a weak-form version of Extended Dynamic Mode Decomposition that we call WEDMD. We establish a number of analytic results about the method and show explicitly how the weak form is able to mitigate the impacts of noise in linear stochastic differential equations. In nonlinear systems, we likewise show how the method is able to generate excellent approximations to Koopman modes without making recourse to sparsity promoting approaches. This excellent approximation property allows for meaningful forecasting even in the presence of noisy data.

math.DS↗

arXiv2026

Forecasting Local Ionospheric Parameters Using Transformers

We present a novel method for forecasting key ionospheric parameters using transformer-based neural networks. The model provides accurate forecasts and uncertainty quantification of the F2-layer peak plasma frequency (foF2), the F2-layer peak density height (hmF2), and total electron content (TEC) for a given geographic location. It includes a number of exogenous variables, including F10.7cm solar flux and disturbance storm time (Dst). We demonstrate how transformers can be trained in a data assimilation-like fashion that uses these exogenous variables along with naive predictions from climatology to generate 24-hour forecasts with nonparametric uncertainty bounds. We call this method the Local Ionospheric Forecast Transformer (LIFT). We demonstrate that the trained model can generalize to new geographic locations and time periods not seen during training, and we compare its performance to that of the International Reference Ionosphere (IRI) using CCIR coefficients.

physics.space-ph↗

arXiv2026

Enhancing Dynamic-Mode Decomposition via Multi-Scale Analysis

Through the use of wavelet based Besov norms, we compute nontrivial multiscale nonlinear features of a given data set so as to enhance the standard Dynamic-Mode Decomposition algorithm. Thus we are able to build sophisticated observables which enhance algorithm performance without placing undue computational burdens on the user.

math.DS↗

arXiv2026

Empirical Discovery of Multi-Scale Transfer of Information in Dynamical Systems

In this work, we quantify the time scales and information flow associated with multiscale energy transfer in a weakly turbulent system. This is done through a greedy optimization algorithm which finds the maximum conditional-mutual information across lagged embeddings of time series localized by wavenumber. For our chosen weakly turbulent system, the algorithm finds asymmetries in the information flow across wavenumbers, reflecting what are typically described as forward and inverse cascades. However, our approach goes beyond typical heuristic arguments and provides quantitative insight into the intricate multi-wave mixing dynamics necessary to maintain the steady statistical state characterizing weak turbulence. Our work then provides a novel, detailed, and fully nonlinear statistical analysis of a weakly turbulent system. The flexibility of our approach points to broader applicability in real-world data coming from chaotic or turbulent dynamical systems.

nlin.CD↗

arXiv2024

Entropic Regression DMD (ERDMD) Discovers Informative Sparse and Nonuniformly Time Delayed Models

In this work, we present a method which determines optimal multi-step dynamic mode decomposition (DMD) models via entropic regression, which is a nonlinear information flow detection algorithm. Motivated by the higher-order DMD (HODMD) method of \cite{clainche}, and the entropic regression (ER) technique for network detection and model construction found in \cite{bollt, bollt2}, we develop a method that we call ERDMD that produces high fidelity time-delay DMD models that allow for nonuniform time space, and the time spacing is discovered by consider most informativity based on ER. These models are shown to be highly efficient and robust. We test our method over several data sets generated by chaotic attractors and show that we are able to build excellent reconstructions using relatively minimal models. We likewise are able to better identify multiscale features via our models which enhances the utility of dynamic mode decomposition.

stat.ML↗

arXiv2023

Machine Learning Enhanced Hankel Dynamic-Mode Decomposition

While the acquisition of time series has become more straightforward, developing dynamical models from time series is still a challenging and evolving problem domain. Within the last several years, to address this problem, there has been a merging of machine learning tools with what is called the dynamic mode decomposition (DMD). This general approach has been shown to be an especially promising avenue for accurate model development. Building on this prior body of work, we develop a deep learning DMD based method which makes use of the fundamental insight of Takens' Embedding Theorem to build an adaptive learning scheme that better approximates higher dimensional and chaotic dynamics. We call this method the Deep Learning Hankel DMD (DLHDMD). We likewise explore how our method learns mappings which tend, after successful training, to significantly change the mutual information between dimensions in the dynamics. This appears to be a key feature in enhancing the DMD overall, and it should help provide further insight for developing other deep learning methods for time series analysis and model generation.

cs.LG↗

arXiv2022

Scale-Separated Dynamic Mode Decomposition and Ionospheric Forecasting

We present a method for forecasting the foF2 and hmF2 parameters using modal decompositions of ionospheric electron density profile time series. Our method is based on the Dynamic Mode Decomposition (DMD), which provides a means of determining spatiotemporal modes from measurements alone. DMD models are easily updated as new data is recorded and do not require any physics to inform the dynamics. However, in the case of ionospheric profiles, we find a wide range of oscillations, including some far above the diurnal frequency. Therefore, we propose nontrivial extensions to DMD using wavelet decompositions. We call this method the Scale-Separated Dynamic Mode Decomposition (SSDMD) since the wavelets isolate fluctuations at different time scales in the data into separated components. We show that this method provides a stable reconstruction of the peak plasma density and can be used to predict the state of foF2 and hmF2 at future time steps. We demonstrate the SSDMD method on data sets covering periods of high and low solar activity as well as low, mid, and high latitude locations.

physics.space-ph↗

arXiv2022

Deep Learning Enhanced Dynamic Mode Decomposition

Koopman operator theory shows how nonlinear dynamical systems can be represented as an infinite-dimensional, linear operator acting on a Hilbert space of observables of the system. However, determining the relevant modes and eigenvalues of this infinite-dimensional operator can be difficult. The extended dynamic mode decomposition (EDMD) is one such method for generating approximations to Koopman spectra and modes, but the EDMD method faces its own set of challenges due to the need of user defined observables. To address this issue, we explore the use of autoencoder networks to simultaneously find optimal families of observables which also generate both accurate embeddings of the flow into a space of observables and submersions of the observables back into flow coordinates. This network results in a global transformation of the flow and affords future state prediction via the EDMD and the decoder network. We call this method the deep learning dynamic mode decomposition (DLDMD). The method is tested on canonical nonlinear data sets and is shown to produce results that outperform a standard DMD approach and enable data-driven prediction where the standard DMD fails.

cs.LG↗

arXiv2021

Detection of Functional Communities in Networks of Randomly Coupled Oscillators Using the Dynamic-Mode Decomposition

Dynamic-mode decomposition (DMD) is a versatile framework for model-free analysis of time series that are generated by dynamical systems. We develop a DMD-based algorithm to investigate the formation of "functional communities" in networks of coupled, heterogeneous Kuramoto oscillators. In these functional communities, the oscillators in the network have similar dynamics. We consider two common random-graph models (Watts--Strogatz networks and Barabási--Albert networks) with different amounts of heterogeneities among the oscillators. In our computations, we find that membership in a community reflects the extent to which there is establishment and sustainment of locking between oscillators. We construct forest graphs that illustrate the complex ways in which the heterogeneous oscillators associate and disassociate with each other.

nlin.AO↗

arXiv2020

The Dynamic-Mode Decomposition and Optimal Prediction

The Dynamic-Mode Decomposition (DMD) is a well established data-driven method of finding temporally evolving linear-mode decompositions of nonlinear time series. Traditionally, this method presumes that all relevant dimensions are sampled through measurement. To address dynamical systems in which the data may be incomplete or represent only partial observation of a more complex system, we extend the DMD algorithm by including a Mori-Zwanzig Decomposition to derive memory kernels that capture the averaged dynamics of the unresolved variables as projected onto the resolved dimensions. From this, we then derive what we call the Memory-Dependent Dynamic Mode Decomposition (MDDMD). Through numerical examples, the MDDMD method is shown to produce reasonable approximations of the ensemble-averaged dynamics of the full system given a single time series measurement of the resolved variables.

math.DS↗

arXiv2019

Particle trajectories in nonlinear Schrodinger models

The nonlinear Schrodinger equation is well known as a universal equation in the study of wave motion. In the context of wave motion at the free surface of an incompressible fluid, the equation accurately predicts the evolution of modulated wave trains with low to moderate wave steepness. While there is an abundance of studies investigating the reconstruction of the surface profile $η$, and the fidelity of such profiles provided by the nonlinear Schrodinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid. In the current work, it is shown that the velocity potential $ϕ$ can be reconstructed in a similar way as the free-surface profile. This observation opens up a range of potential applications since the nonlinear Schrodinger equation features fairly simple closed-form solutions and can be solved numerically with comparatively little effort. In particular, it is shown that particle trajectories in the fluid can be described with relative ease not only in the context of the nonlinear Schrodinger equation, but also in higher-order models such as the Dysthe equation, and in models incorporating certain types of viscous effects.

physics.flu-dyn↗

arXiv2012

On Nonlocal Gross-Pitaevskii Equations with Periodic Potentials

The Gross-Pitaevskii equation is a widely used model in physics, in particular in the context of Bose-Einstein condensates. However, it only takes into account local interactions between particles. This paper demonstrates the validity of using a nonlocal formulation as a generalization of the local model. In particular, the paper demonstrates that the solution of the nonlocal model approaches in norm the solution of the local model as the nonlocal model approaches the local model. The nonlocality and potential used for the Gross-Pitaevskii equation are quite general, thus this paper shows that one can easily add nonlocal effects to interesting classes of Bose-Einstein condensate models. Based on a particular choice of potential for the nonlocal Gross-Pitaevskii equation, we establish the orbital stability of a class of parameter-dependent solutions to the nonlocal problem for certain parameter regimes. Numerical results corroborate the analytical stability results and lead to predictions about the stability of the class of solutions for parameter values outside of the purview of the theory established in this paper.

math.AP↗

arXiv2012

Propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion

The propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion coefficients is studied. Using coordinate changes, WKB approximations, and multiple scales analysis, we provide an analytic framework that describes propagation of the front up to the minimum of the diffusion coefficient. We also present results showing the behavior of the front after it passes the minimum. In each case, we show that standard traveling coordinate frames do not properly describe front propagation. Lastly, we provide numerical simulations to support our analysis and to show, that around the minimum, the motion of the front is arrested on asymptotically significant timescales.

math.AP↗