arXiv subjects

J Nathan Kutz

Publications and source records attributed to J Nathan Kutz.

4 recordsLinked to original sources

arXiv2024

Separation of periodic orbits in the delay embedded space of chaotic attractors

This work explores the intersection of time-delay embeddings, periodic orbit theory, and symbolic dynamics. Time-delay embeddings have been effectively applied to chaotic time series data, offering a principled method to reconstruct relevant information of the full attractor from partial time series observations. In this study, we investigate the structure of the unstable periodic orbits of an attractor using time-delay embeddings. First, we embed time-series data from a periodic orbit into a higher-dimensional space through the construction of a Hankel matrix, formed by arranging time-shifted copies of the data. We then examine the influence of the width and height of the Hankel matrix on the geometry of unstable periodic orbits in the delay-embedded space. The right singular vectors of the Hankel matrix provide a basis for embedding the periodic orbits. We observe that increasing the length of the delay (e.g., the height of the Hankel matrix) leads to a clear separation of the periodic orbits into distinct clusters within the embedded space. Our analysis characterizes these separated clusters and provides a mathematical framework to determine the relative position of individual unstable periodic orbits in the embedded space. Additionally, we present a modified formula to derive the symbolic representation of distinct periodic orbits for a specified sequence length, extending the Polyá-Redfield enumeration theorem.

nlin.CD↗

arXiv2020

Insect cyborgs: Bio-mimetic feature generators improve machine learning accuracy on limited data

Machine learning (ML) classifiers always benefit from more informative input features. We seek to auto-generate stronger feature sets in order to address the difficulty that ML methods often experience given limited training data. A wide range of biological neural nets (BNNs) excel at fast learning, implying that they are adept at extracting informative features. We can thus look to BNNs for tools to improve ML performance in this low-data regime. The insect olfactory network learns new odors very rapidly, by means of three key elements: A competitive inhibition layer; a high-dimensional sparse plastic layer; and Hebbian updates of synaptic weights. In this work, we deployed MothNet, a computational model of the insect olfactory network, as an automatic feature generator: Attached as a front-end pre-processor, its Readout Neurons provided new features, derived from the original features, for use by standard ML classifiers. We found that these "insect cyborgs", i.e. classifiers that are part-insect model and part-ML method, had significantly better performance than baseline ML methods alone on a vectorized MNIST dataset. The MothNet feature generator also substantially out-performed other feature generating methods such as PCA, PLS, and NNs, as well as pre-training to initialize NN weights. Cyborgs improved relative test set accuracy by an average of 6% to 33% depending on baseline ML accuracy, while relative reduction in test set error exceeded 50% for higher baseline accuracy ML models. These results indicate the potential value of BNN-inspired feature generators in the ML context.

cs.ET↗

arXiv2018

Model selection for hybrid dynamical systems via sparse regression

Hybrid systems are traditionally difficult to identify and analyze using classical dynamical systems theory. Moreover, recently developed model identification methodologies largely focus on identifying a single set of governing equations solely from measurement data. In this article, we develop a new methodology, Hybrid-Sparse Identification of Nonlinear Dynamics (Hybrid-SINDy), which identifies separate nonlinear dynamical regimes, employs information theory to manage uncertainty, and characterizes switching behavior. Specifically, we utilize the nonlinear geometry of data collected from a complex system to construct a set of coordinates based on measurement data and augmented variables. Clustering the data in these measurement-based coordinates enables the identification of nonlinear hybrid systems. This methodology broadly empowers nonlinear system identification without constraining the data locally in time and has direct connections to hybrid systems theory. We demonstrate the success of this method on numerical examples including a mass-spring hopping model and an infectious disease model. Characterizing complex systems that switch between dynamic behaviors is integral to overcoming modern challenges such as eradication of infectious diseases, the design of efficient legged robots, and the protection of cyber infrastructures.

math.DS↗

arXiv2016

Sparsity enabled cluster reduced-order models for control

Characterizing and controlling nonlinear, multi-scale phenomena play important roles in science and engineering. Cluster-based reduced-order modeling (CROM) was introduced to exploit the underlying low-dimensional dynamics of complex systems. CROM builds a data-driven discretization of the Perron-Frobenius operator, resulting in a probabilistic model for ensembles of trajectories. A key advantage of CROM is that it embeds nonlinear dynamics in a linear framework, and uncertainty can be managed with data assimilation. CROM is typically computed on high-dimensional data, however, access to and computations on this full-state data limit the online implementation of CROM for prediction and control. Here, we address this key challenge by identifying a small subset of critical measurements to learn an efficient CROM, referred to as sparsity-enabled CROM. In particular, we leverage compressive measurements to faithfully embed the cluster geometry and preserve the probabilistic dynamics. Further, we show how to identify fewer optimized sensor locations tailored to a specific problem that outperform random measurements. Both of these sparsity-enabled sensing strategies significantly reduce the burden of data acquisition and processing for low-latency in-time estimation and control. We illustrate this unsupervised learning approach on three different high-dimensional nonlinear dynamical systems from fluids with increasing complexity, with one application in flow control. Sparsity-enabled CROM is a critical facilitator for real-time implementation on high-dimensional systems where full-state information may be inaccessible.

physics.data-an↗