arXiv subjects

J. Nathan Kutz

Publications and source records attributed to J. Nathan Kutz.

At least 109 records · Page 6Linked to original sources

arXiv2022

Universal Dynamics of Damped-Driven Systems: The Logistic Map as a Normal Form for Energy Balance

Damped-driven systems are ubiquitous in engineering and science. Despite the diversity of physical processes observed in a broad range of applications, the underlying instabilities observed in practice have a universal characterization which is determined by the overall gain and loss curves of a given system. The universal behavior of damped-driven systems can be understood from a geometrical description of the energy balance with a minimal number of assumptions. The assumptions on the energy dynamics are as follows: the energy increases monotonically as a function of increasing gain, and the losses become increasingly larger with increasing energy, i.e. there are many routes for dissipation in the system for large input energy. The intersection of the gain and loss curves define an energy balanced solution. By constructing an iterative map between the loss and gain curves, the dynamics can be shown to be homeomorphic to the logistic map, which exhibits a period doubling cascade to chaos. Indeed, the loss and gain curves allow for a geometrical description of the dynamics through a simple Verhulst diagram (cobweb plot). Thus irrespective of the physics and its complexities, this simple geometrical description dictates the universal set of logistic map instabilities that arise in complex damped-driven systems. More broadly, damped-driven systems are a class of non-equilibrium pattern forming systems which have a canonical set of instabilities that are manifest in practice.

nlin.AO↗

arXiv2022

Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants

Recent progress in autoencoder-based sparse identification of nonlinear dynamics (SINDy) under $\ell_1$ constraints allows joint discoveries of governing equations and latent coordinate systems from spatio-temporal data, including simulated video frames. However, it is challenging for $\ell_1$-based sparse inference to perform correct identification for real data due to the noisy measurements and often limited sample sizes. To address the data-driven discovery of physics in the low-data and high-noise regimes, we propose Bayesian SINDy autoencoders, which incorporate a hierarchical Bayesian sparsifying prior: Spike-and-slab Gaussian Lasso. Bayesian SINDy autoencoder enables the joint discovery of governing equations and coordinate systems with a theoretically guaranteed uncertainty estimate. To resolve the challenging computational tractability of the Bayesian hierarchical setting, we adapt an adaptive empirical Bayesian method with Stochatic gradient Langevin dynamics (SGLD) which gives a computationally tractable way of Bayesian posterior sampling within our framework. Bayesian SINDy autoencoder achieves better physics discovery with lower data and fewer training epochs, along with valid uncertainty quantification suggested by the experimental studies. The Bayesian SINDy autoencoder can be applied to real video data, with accurate physics discovery which correctly identifies the governing equation and provides a close estimate for standard physics constants like gravity $g$, for example, in videos of a pendulum.

cs.LG↗

arXiv2022

Walking droplets as a damped-driven system

We consider the dynamics of a droplet on a vibrating fluid bath. This hydrodynamic quantum analog system is shown to elicit the canonical behavior of damped-driven systems, including a period doubling route to chaos. By approximating the system as a compositional map between the gain and loss dynamics, the underlying nonlinear dynamics can be shown to be driven by energy balances in the systems. The gain-loss iterative mapping is similar to a normal form encoding for the pattern forming instabilities generated in such spatially-extended system. Similar to mode-locked lasers and rotating detonation engines, the underlying bifurcations persist for general forms of the loss and gain, both of which admit explicit representations in our approximation. Moreover, the resulting geometrical description of the particle-wave interaction completely characterizes the instabilities observed in experiments.

math.DS↗

arXiv2022

Pilot-Wave Dynamics: Using Dynamic Mode Decomposition to characterize Bifurcations, Routes to Chaos and Emergent Statistics

We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry, and apply the {\em Dynamic mode decomposition} (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. DMD provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatio-temporal data. The DMD characterization of the wave field yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Moreover, it provides a low-rank characterization of the bifurcation structure of the pilot wave physics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics.

physics.flu-dyn↗

arXiv2022

Model predictive control for robust quantum state preparation

A critical engineering challenge in quantum technology is the accurate control of quantum dynamics. Model-based methods for optimal control have been shown to be highly effective when theory and experiment closely match. Consequently, realizing high-fidelity quantum processes with model-based control requires careful device characterization. In quantum processors based on cold atoms, the Hamiltonian can be well-characterized. For superconducting qubits operating at milli-Kelvin temperatures, the Hamiltonian is not as well-characterized. Unaccounted for physics (i.e., mode discrepancy), coherent disturbances, and increased noise compromise traditional model-based control. This work introduces model predictive control (MPC) for quantum control applications. MPC is a closed-loop optimization framework that (i) inherits a natural degree of disturbance rejection by incorporating measurement feedback, (ii) utilizes finite-horizon model-based optimizations to control complex multi-input, multi-output dynamical systems under state and input constraints, and (iii) is flexible enough to develop synergistically alongside other modern control strategies. We show how MPC can be used to generate practical optimized control sequences in representative examples of quantum state preparation. Specifically, we demonstrate for a qubit, a weakly-anharmonic qubit, and a system undergoing crosstalk, that MPC can realize successful model-based control even when the model is inadequate. These examples showcase why MPC is an important addition to the quantum engineering control suite.

quant-ph↗

arXiv2022

Robust, High-Rate Trajectory Tracking on Insect-Scale Soft-Actuated Aerial Robots with Deep-Learned Tube MPC

Accurate and agile trajectory tracking in sub-gram Micro Aerial Vehicles (MAVs) is challenging, as the small scale of the robot induces large model uncertainties, demanding robust feedback controllers, while the fast dynamics and computational constraints prevent the deployment of computationally expensive strategies. In this work, we present an approach for agile and computationally efficient trajectory tracking on the MIT SoftFly, a sub-gram MAV (0.7 grams). Our strategy employs a cascaded control scheme, where an adaptive attitude controller is combined with a neural network policy trained to imitate a trajectory tracking robust tube model predictive controller (RTMPC). The neural network policy is obtained using our recent work, which enables the policy to preserve the robustness of RTMPC, but at a fraction of its computational cost. We experimentally evaluate our approach, achieving position Root Mean Square Errors lower than 1.8 cm even in the more challenging maneuvers, obtaining a 60% reduction in maximum position error compared to our previous work, and demonstrating robustness to large external disturbances

cs.RO↗

arXiv2022

Koopman-theoretic Approach for Identification of Exogenous Anomalies in Nonstationary Time-series Data

In many scenarios, it is necessary to monitor a complex system via a time-series of observations and determine when anomalous exogenous events have occurred so that relevant actions can be taken. Determining whether current observations are abnormal is challenging. It requires learning an extrapolative probabilistic model of the dynamics from historical data, and using a limited number of current observations to make a classification. We leverage recent advances in long-term probabilistic forecasting, namely {\em Deep Probabilistic Koopman}, to build a general method for classifying anomalies in multi-dimensional time-series data. We also show how to utilize models with domain knowledge of the dynamics to reduce type I and type II error. We demonstrate our proposed method on the important real-world task of global atmospheric pollution monitoring, integrating it with NASA's Global Earth System Model. The system successfully detects localized anomalies in air quality due to events such as COVID-19 lockdowns and wildfires.

cs.LG↗

arXiv2022

Solving nonlinear ordinary differential equations using the invariant manifolds and Koopman eigenfunctions

Nonlinear ordinary differential equations can rarely be solved analytically. Koopman operator theory provides a way to solve nonlinear systems by mapping nonlinear dynamics to a linear space using eigenfunctions. Unfortunately, finding such eigenfunctions is difficult. We introduce a method for constructing eigenfunctions from a nonlinear ODE's invariant manifolds. This method, when successful, allows us to find analytical solutions for constant coefficient nonlinear systems. Previous data-driven methods have used Koopman theory to construct local Koopman eigenfunction approximations valid in different regions of phase space; our method finds analytic Koopman eigenfunctions that are exact and globally valid. We demonstrate our Koopman method of solving nonlinear systems on 1-dimensional and 2-dimensional ODEs. The nonlinear examples considered have simple expressions for their invariant manifolds which produce tractable analytical solutions. Thus our method allows for the construction of analytical solutions for previously unsolved ordinary differential equations. It also highlights the connection between invariant manifolds and eigenfunctions in nonlinear ordinary differential equations and presents avenues for extending this method to solve more nonlinear systems.

math.DS↗

arXiv2022

Data-driven discovery of governing equations for coarse-grained heterogeneous network dynamics

We leverage data-driven model discovery methods to determine the governing equations for the emergent behavior of heterogeneous networked dynamical systems. Specifically, we consider networks of coupled nonlinear oscillators whose collective behaviour approaches a limit cycle. Stable limit-cycles are of interest in many biological applications as they model self-sustained oscillations (e.g. heart beats, chemical oscillations, neurons firing, circadian rhythm). For systems that display relaxation oscillations, our method automatically detects boundary (time) layer structures in the dynamics, fitting inner and outer solutions and matching them in a data-driven manner. We demonstrate the method on well-studied systems: the Rayleigh Oscillator and the Van der Pol Oscillator. We then apply the mathematical framework to discovering low-dimensional dynamics in networks of semi-synchronized Kuramoto, Rayleigh, Rossler, and Fitzhugh-Nagumo oscillators, as well as heterogeneous combinations thereof. We also provide a numerical exploration of the dimension of collective network dynamics as a function of several network parameters, showing that the discovery of coarse-grained variables and dynamics can be accomplished with the proposed architecture.

math.DS↗

arXiv2022

Effcient magnetometer sensor array selection for signal reconstruction and brain source localization

Magnetoencephalography (MEG) is a noninvasive method for measuring magnetic flux signals caused by brain activity using sensor arrays located on or above the scalp. A common strategy for monitoring brain activity is to place sensors on a nearly uniform grid, or sensor array, around the head. By increasing the total number of sensors, higher spatial-frequency components of brain activity can be resolved as dictated by Nyquist sampling theory. Currently, there are few principled mathematical architectures for sensor placement aside from Nyquist considerations. However, global brain activity often exhibits low-dimensional patterns of spatio-temporal dynamics. The low-dimensional global patterns can be computed from the singular value decomposition and can be leveraged to select a small number of sensors optimized for reconstructing brain signals and localizing brain sources. Moreover, a smaller number of sensors which are systematically chosen can outperform the entire sensor array when considering noisy measurements. We propose a greedy selection algorithm based upon the QR decomposition that is computationally efficient to implement for MEG. We demonstrate the performance of the sensor selection algorithm for the tasks of signal reconstruction and localization. The performance is dependent upon source localization, with shallow sources easily identified and reconstructed, and deep sources more difficult to locate. Our findings suggest that principled methods for sensor selection can improve MEG capabilities and potentially add cost savings for monitoring brain-wide activity.

physics.med-ph↗

arXiv2022

The Experimental Multi-Arm Pendulum on a Cart: A Benchmark System for Chaos, Learning, and Control

The single, double, and triple pendulum has served as an illustrative experimental benchmark system for scientists to study dynamical behavior for more than four centuries. The pendulum system exhibits a wide range of interesting behaviors, from simple harmonic motion in the single pendulum to chaotic dynamics in multi-arm pendulums. Under forcing, even the single pendulum may exhibit chaos, providing a simple example of a damped-driven system. All multi-armed pendulums are characterized by the existence of index-one saddle points, which mediate the transport of trajectories in the system, providing a simple mechanical analog of various complex transport phenomena, from biolocomotion to transport within the solar system. Further, pendulum systems have long been used to design and test both linear and nonlinear control strategies, with the addition of more arms making the problem more challenging. In this work, we provide extensive designs for the construction and operation of a high-performance, multi-link pendulum on a cart system. Although many experimental setups have been built to study the behavior of pendulum systems, such an extensive documentation on the design, construction, and operation is missing from the literature. The resulting experimental system is highly flexible, enabling a wide range of benchmark problems in dynamical systems modeling, system identification and learning, and control. To promote reproducible research, we have made our entire system open-source, including 3D CAD drawings, basic tutorial code, and data. Moreover, we discuss the possibility of extending our system capability to be operated remotely to enable researchers all around the world to use it, thus increasing access.

nlin.CD↗

arXiv2022

Transitions between peace and systemic war as bifurcations in a signed network dynamical system

We investigate structural features and processes associated with the onset of systemic conflict using an approach which integrates complex systems theory with network modeling and analysis. We present a signed network model of cooperation and conflict dynamics in the context of international relations between states. The model evolves ties between nodes under the influence of a structural balance force and a dyad-specific force. Model simulations exhibit a sharp bifurcation from peace to systemic war as structural balance pressures increase, a bistable regime in which both peace and war stable equilibria exist, and a hysteretic reverse bifurcation from war to peace. We show how the analytical expression we derive for the peace-to-war bifurcation condition implies that polarized network structure increases susceptibility to systemic war. We develop a framework for identifying patterns of relationship perturbations that are most destabilizing and apply it to the network of European great powers before World War I. We also show that the model exhibits critical slowing down, in which perturbations to the peace equilibrium take longer to decay as the system draws closer to the bifurcation. We discuss how our results relate to international relations theories on the causes and catalysts of systemic war.

cs.SI↗

arXiv2022

A new approach for determining optimal placement of PM2.5 air quality sensors: case study for the contiguous United States

Considerable financial resources are allocated for measuring ambient air pollution in the United States, yet the locations for these monitoring sites may not be optimized to capture the full extent of current pollution variability. Prior research on best sensor placement for monitoring fine particulate matter (PM2.5) pollution is scarce: most studies do not span areas larger than a medium-sized city or examine timescales longer than one week. Here we present a pilot study using multiresolution modal decomposition (mrDMD) to identify the optimal placement of PM2.5 sensors from 2000-2016 over the contiguous United States. This novel approach incorporates the variation of PM2.5 on timescales ranging from one day to over a decade to capture air pollution variability. We find that the mrDMD algorithm identifies high-priority sensor locations in the western United States, but a significantly lower density of sensors than expected along the eastern coast, where a large number of EPA PM2.5 monitors currently reside. Specifically, 53% of mrDMD optimized sensor locations are west of the 100th meridian, compared to only 32% in the current EPA network. The mrDMD sensor locations can capture PM2.5 from wildfires and high pollution events, with particularly high skill in the West. These results suggest significant gaps in the current EPA monitoring network in the San Joaquin Valley in California, northern California, and in the Pacific Northwest (Idaho, and Eastern Washington and Oregon). Our framework diagnoses where to place air quality sensors so that they can best monitor smoke from wildfires. Our framework may also be applied to urban areas for equitable placement of PM2.5 monitors.

stat.AP↗

arXiv2022

Data-driven sensor placement with shallow decoder networks

Sensor placement is an important and ubiquitous problem across the engineering and physical sciences for tasks such as reconstruction, forecasting and control. Surprisingly, there are few principled mathematical techniques developed to date for optimizing sensor locations, with the leading sensor placement algorithms often based upon the discovery of linear, low-rank sub-spaces and the QR algorithm. QR is a computationally efficient greedy search algorithm which selects sensor locations from candidate positions with maximal variance exhibited in a training data set. More recently, neural networks, specifically shallow decoder networks (SDNs), have been shown to be very successful in mapping sensor measurements to the original high-dimensional state space. SDNs outperform linear subspace representations in reconstruction accuracy, noise tolerance, and robustness to sensor locations. However, SDNs lack principled mathematical techniques for determining sensor placement. In this work, we develop two algorithms for optimizing sensor locations for use with SDNs: one which is a linear selection algorithm based upon QR (Q-SDN), and one which is a nonlinear selection algorithm based upon neural network pruning (P-SDN). Such sensor placement algorithms promise to enhance the already impressive reconstruction capabilities of SDNs. We demonstrate our sensor selection algorithms on two example data sets from fluid dynamics. Moreover, we provide a detailed comparison between our linear (Q-SDN) and nonlinear (P-SDN) algorithms with traditional linear embedding techniques (proper orthogonal decomposition) and QR greedy selection. We show that QR selection with SDNs enhances performance. QR even out-performs our nonlinear selection method that uses magnitude-based pruning. Thus, the combination of a greedy linear selection (QR) with nonlinear encoding (SDN) provides a synergistic combination.

math.DS↗

arXiv2022

Dimensionally Consistent Learning with Buckingham Pi

In the absence of governing equations, dimensional analysis is a robust technique for extracting insights and finding symmetries in physical systems. Given measurement variables and parameters, the Buckingham Pi theorem provides a procedure for finding a set of dimensionless groups that spans the solution space, although this set is not unique. We propose an automated approach using the symmetric and self-similar structure of available measurement data to discover the dimensionless groups that best collapse this data to a lower dimensional space according to an optimal fit. We develop three data-driven techniques that use the Buckingham Pi theorem as a constraint: (i) a constrained optimization problem with a non-parametric input-output fitting function, (ii) a deep learning algorithm (BuckiNet) that projects the input parameter space to a lower dimension in the first layer, and (iii) a technique based on sparse identification of nonlinear dynamics (SINDy) to discover dimensionless equations whose coefficients parameterize the dynamics. We explore the accuracy, robustness and computational complexity of these methods as applied to three example problems: a bead on a rotating hoop, a laminar boundary layer, and Rayleigh-Bénard convection.

cs.LG↗

arXiv2022

PySINDy: A comprehensive Python package for robust sparse system identification

Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy, we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differential equations (PDEs), and implicit differential equations. Robust formulations, including the integral form of SINDy and ensembling techniques, are also implemented to improve performance for real-world data. Finally, we provide a range of new optimization algorithms, including several sparse regression techniques and algorithms to enforce and promote inequality constraints and stability. Together, these updates enable entirely new SINDy model discovery capabilities that have not been reported in the literature, such as constrained PDE identification and ensembling with different sparse regression optimizers.

eess.SY↗

arXiv2022

Quantifying yeast colony morphologies with feature engineering from time-lapse photography

Baker's yeast (Saccharomyces cerevisiae) is a model organism for studying the morphology that emerges at the scale of multi-cell colonies. To look at how morphology develops, we collect a dataset of time-lapse photographs of the growth of different strains of S. cerevisiae. We discuss the general statistical challenges that arise when using time-lapse photographs to extract time-dependent features. In particular, we show how texture-based feature engineering and representative clustering can be successfully applied to categorize the development of yeast colony morphology using our dataset. The local binary pattern (LBP) from image processing is used to score the surface texture of colonies. This texture score develops along a smooth trajectory during growth. The path taken depends on how the morphology emerges. A hierarchical clustering of the colonies is performed according to their texture development trajectories. The clustering method is designed for practical interpretability; it obtains the best representative colony image for any hierarchical sub-cluster.

q-bio.QM↗

arXiv2022

Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders

A central challenge in data-driven model discovery is the presence of hidden, or latent, variables that are not directly measured but are dynamically important. Takens' theorem provides conditions for when it is possible to augment these partial measurements with time delayed information, resulting in an attractor that is diffeomorphic to that of the original full-state system. However, the coordinate transformation back to the original attractor is typically unknown, and learning the dynamics in the embedding space has remained an open challenge for decades. Here, we design a custom deep autoencoder network to learn a coordinate transformation from the delay embedded space into a new space where it is possible to represent the dynamics in a sparse, closed form. We demonstrate this approach on the Lorenz, Rössler, and Lotka-Volterra systems, learning dynamics from a single measurement variable. As a challenging example, we learn a Lorenz analogue from a single scalar variable extracted from a video of a chaotic waterwheel experiment. The resulting modeling framework combines deep learning to uncover effective coordinates and the sparse identification of nonlinear dynamics (SINDy) for interpretable modeling. Thus, we show that it is possible to simultaneously learn a closed-form model and the associated coordinate system for partially observed dynamics.

cs.LG↗