arXiv subjects

J. Nathan Kutz

Publications and source records attributed to J. Nathan Kutz.

At least 181 records · Page 10Linked to original sources

arXiv2020

Multi-scale Physics of Rotating Detonation Engines: Autosolitons and Modulational Instabilities

We develop a theoretical framework that predicts and fully characterizes the diverse experimental observations of the nonlinear, combustion wave propagation in a rotating detonation engine (RDE), including the nucleation and formation of combustion pulses, the soliton-like interactions between these combustion fronts, and the fundamental, underlying Hopf bifurcation to time-periodic modulation of the waves. In this framework, the mode-locked structures are classified as autosolitons, or stably-propagating nonlinear waves where the local physics of nonlinearity, dispersion, gain, and dissipation exactly balance. We find that the global dominant balance physics in the RDE combustion chamber are dissipative and multi-scale in nature, with local fast scale (nano- to microseconds) combustion balances generating the fundamental mode-locked autosoliton state, while slow scale (milliseconds) gain-loss balances determine the instabilities and structure of the total number of autosolitons. In this manner, the global multi-scale balance physics give rise to the stable structures - not exclusively the frontal dynamics prescribed by classical detonation theory. Experimental observations and numerical models of the RDE combustion chamber are in strong qualitative agreement with no parameter tuning. Moreover, numerical continuation (computational bifurcation tracking) of the RDE analog system establishes that a Hopf bifurcation of the steadily propagating pulse train leads to the fundamental instability of the RDE, or time-periodic modulation of the waves. Along branches of Hopf orbits in parameter space exist a continuum of wave-pair interactions that exhibit solitonic interactions of varying strength.

physics.flu-dyn↗

arXiv2020

Randomized CP Tensor Decomposition

The CANDECOMP/PARAFAC (CP) tensor decomposition is a popular dimensionality-reduction method for multiway data. Dimensionality reduction is often sought after since many high-dimensional tensors have low intrinsic rank relative to the dimension of the ambient measurement space. However, the emergence of `big data' poses significant computational challenges for computing this fundamental tensor decomposition. By leveraging modern randomized algorithms, we demonstrate that coherent structures can be learned from a smaller representation of the tensor in a fraction of the time. Thus, this simple but powerful algorithm enables one to compute the approximate CP decomposition even for massive tensors. The approximation error can thereby be controlled via oversampling and the computation of power iterations. In addition to theoretical results, several empirical results demonstrate the performance of the proposed algorithm.

math.NA↗

arXiv2020

Time-Delay Observables for Koopman: Theory and Applications

Nonlinear dynamical systems are ubiquitous in science and engineering, yet analysis and prediction of these systems remains a challenge. Koopman operator theory circumvents some of these issues by considering the dynamics in the space of observable functions on the state, in which the dynamics are intrinsically linear and thus amenable to standard techniques from numerical analysis and linear algebra. However, practical issues remain with this approach, as the space of observables is infinite-dimensional and selecting a subspace of functions in which to accurately represent the system is a nontrivial task. In this work we consider time-delay observables to represent nonlinear dynamics in the Koopman operator framework. We prove the surprising result that Koopman operators for different systems admit universal (system-independent) representations in these coordinates, and give analytic expressions for these representations. In addition, we show that for certain systems a restricted class of these observables form an optimal finite-dimensional basis for representing the Koopman operator, and that the analytic representation of the Koopman operator in these coordinates coincides with results computed by the dynamic mode decomposition. We provide numerical examples to complement our results. In addition to being theoretically interesting, these results have implications for a number of linearization algorithms for dynamical systems.

math.NA↗

arXiv2020

Nonlinear control in the nematode C. elegans

Recent whole-brain calcium imaging recordings of the nematode C. elegans have demonstrated that neural activity is dominated by dynamics on a low-dimensional manifold that can be clustered according to behavioral states. Despite progress in modeling the dynamics with linear or locally linear models, it remains unclear how a single network of neurons can produce the observed features. In particular, there are multiple clusters, or fixed points, observed in the data which cannot be characterized by a single linear model. We propose a nonlinear control model which is global and parameterized by only four free parameters that match the features displayed by the low-dimensional C. elegans neural activity. In addition to reproducing the average probability distribution of the data, long and short time-scale changes in transition statistics can be characterized via changes in a single parameter. Some of these macro-scale transitions have experimental correlates to single neuro-modulators that seem to act as biological controls, allowing this model to generate testable hypotheses about the effect of these neuro-modulators on the global dynamics. The theory provides an elegant characterization of the neuron population dynamics in C. elegans. Moreover, the mathematical structure of the nonlinear control framework provides a paradigm that can be generalized to more complex systems with an arbitrary number of behavioral states.

q-bio.NC↗

arXiv2020

Optimal Sensor and Actuator Selection using Balanced Model Reduction

Optimal sensor and actuator selection is a central challenge in high-dimensional estimation and control. Nearly all subsequent control decisions are affected by these sensor/actuator locations, and optimal placement amounts to an intractable brute-force search among the combinatorial possibilities. In this work, we exploit balanced model reduction and greedy optimization to efficiently determine sensor and actuator selections that optimize observability and controllability. In particular, we determine locations that optimize scalar measures of observability and controllability via greedy matrix QR pivoting on the dominant modes of the direct and adjoint balancing transformations. Pivoting runtime scales linearly with the state dimension, making this method tractable for high-dimensional systems. The results are demonstrated on the linearized Ginzburg-Landau system, for which our algorithm approximates known optimal placements computed using costly gradient descent methods.

math.OC↗

arXiv2019

Money on the Table: Statistical information ignored by Softmax can improve classifier accuracy

Softmax is a standard final layer used in Neural Nets (NNs) to summarize information encoded in the trained NN and return a prediction. However, Softmax leverages only a subset of the class-specific structure encoded in the trained model and ignores potentially valuable information: During training, models encode an array $D$ of class response distributions, where $D_{ij}$ is the distribution of the $j^{th}$ pre-Softmax readout neuron's responses to the $i^{th}$ class. Given a test sample, Softmax implicitly uses only the row of this array $D$ that corresponds to the readout neurons' responses to the sample's true class. Leveraging more of this array $D$ can improve classifier accuracy, because the likelihoods of two competing classes can be encoded in other rows of $D$. To explore this potential resource, we develop a hybrid classifier (Softmax-Pooling Hybrid, $SPH$) that uses Softmax on high-scoring samples, but on low-scoring samples uses a log-likelihood method that pools the information from the full array $D$. We apply $SPH$ to models trained on a vectorized MNIST dataset to varying levels of accuracy. $SPH$ replaces only the final Softmax layer in the trained NN, at test time only. All training is the same as for Softmax. Because the pooling classifier performs better than Softmax on low-scoring samples, $SPH$ reduces test set error by 6% to 23%, using the exact same trained model, whatever the baseline Softmax accuracy. This reduction in error reflects hidden capacity of the trained NN that is left unused by Softmax.

cs.LG↗

arXiv2019

Randomized Dynamic Mode Decomposition

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of `big data'. The idea is to derive a small matrix from the high-dimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension. For this work we assume that the dynamics of the problem under consideration is evolving on a low-dimensional subspace that is well characterized by a fast decaying singular value spectrum.

math.NA↗

arXiv2019

Randomized Matrix Decompositions using R

Matrix decompositions are fundamental tools in the area of applied mathematics, statistical computing, and machine learning. In particular, low-rank matrix decompositions are vital, and widely used for data analysis, dimensionality reduction, and data compression. Massive datasets, however, pose a computational challenge for traditional algorithms, placing significant constraints on both memory and processing power. Recently, the powerful concept of randomness has been introduced as a strategy to ease the computational load. The essential idea of probabilistic algorithms is to employ some amount of randomness in order to derive a smaller matrix from a high-dimensional data matrix. The smaller matrix is then used to compute the desired low-rank approximation. Such algorithms are shown to be computationally efficient for approximating matrices with low-rank structure. We present the \proglang{R} package rsvd, and provide a tutorial introduction to randomized matrix decompositions. Specifically, randomized routines for the singular value decomposition, (robust) principal component analysis, interpolative decomposition, and CUR decomposition are discussed. Several examples demonstrate the routines, and show the computational advantage over other methods implemented in R.

stat.CO↗

arXiv2019

Deep Learning Models for Global Coordinate Transformations that Linearize PDEs

We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a nonlinear PDE into a linear PDE. Our architecture is motivated by the linearizing transformations provided by the Cole-Hopf transform for Burgers equation and the inverse scattering transform for completely integrable PDEs. By leveraging a residual network architecture, a near-identity transformation can be exploited to encode intrinsic coordinates in which the dynamics are linear. The resulting dynamics are given by a Koopman operator matrix $\mathbf{K}$. The decoder allows us to transform back to the original coordinates as well. Multiple time step prediction can be performed by repeated multiplication by the matrix $\mathbf{K}$ in the intrinsic coordinates. We demonstrate our method on a number of examples, including the heat equation and Burgers equation, as well as the substantially more challenging Kuramoto-Sivashinsky equation, showing that our method provides a robust architecture for discovering interpretable, linearizing transforms for nonlinear PDEs.

math.DS↗

arXiv2019

Mode-Locked Rotating Detonation Waves: Experiments and a Model Equation

Direct observation of a Rotating Detonation Engine combustion chamber has enabled the extraction of the kinematics of its detonation waves. These records exhibit a rich set of instabilities and bifurcations arising from the interaction of coherent wave fronts and global gain dynamics. We develop a model of the observed dynamics by recasting the Majda detonation analog as an autowave. The solution fronts become attractors of the engine; i.e., mode-locked rotating detonation waves. We find that detonative energy release competes with dissipation and gain recovery to produce the observed dynamics and a bifurcation structure common to driven-dissipative systems, such as mode-locked lasers.

nlin.PS↗

arXiv2019

Learning Discrepancy Models From Experimental Data

First principles modeling of physical systems has led to significant technological advances across all branches of science. For nonlinear systems, however, small modeling errors can lead to significant deviations from the true, measured behavior. Even in mechanical systems, where the equations are assumed to be well-known, there are often model discrepancies corresponding to nonlinear friction, wind resistance, etc. Discovering models for these discrepancies remains an open challenge for many complex systems. In this work, we use the sparse identification of nonlinear dynamics (SINDy) algorithm to discover a model for the discrepancy between a simplified model and measurement data. In particular, we assume that the model mismatch can be sparsely represented in a library of candidate model terms. We demonstrate the efficacy of our approach on several examples including experimental data from a double pendulum on a cart. We further design and implement a feed-forward controller in simulations, showing improvement with a discrepancy model.

cs.LG↗

arXiv2019

Frequency comb generation at 800nm in waveguide array quantum well diode lasers

A traveling wave model for a semiconductor diode laser based on quantum wells is presented as well as a comprehensive theoretical model of the lasing dynamics produced by the intensity discrimination of the nonlinear mode-coupling in a waveguide array. By leveraging a recently developed model for the detailed semiconductor gain dynamics, the temporal shaping effects of the nonlinear mode-coupling induced by the waveguide arrays can be characterized. Specifically, the enhanced nonlinear pulse shaping provided by the waveguides are capable of generating stable frequency combs wavelength of 800 nm in a GaAs device, a parameter regime not feasible for stable combline generation using a single waveguide. Extensive numerical simulations showed that stable waveform generation could be achieved and optimized by an appropriate choice of the linear waveguide coupling coefficient, quantum well depth, and the input currents to the first and second waveguides. The model provides a first demonstration that a compact, efficient and robust on-chip comb source can be produced in GaAs.

physics.optics↗

arXiv2019

Stable Numerical Schemes for Nonlinear Dispersive Equations with Counter-Propagation and Gain Dynamics

We develop a stable and efficient numerical scheme for modeling the optical field evolution in a nonlinear dispersive cavity with counter propagating waves and complex, semiconductor physics gain dynamics that are expensive to evaluate. Our stability analysis is characterized by a von-Neumann analysis which shows that many standard numerical schemes are unstable due to competing physical effects in the propagation equations. We show that the combination of a predictor-corrector scheme with an operator-splitting not only results in a stable scheme, but provides a highly efficient, single-stage evaluation of the gain dynamics. Given that the gain dynamics is the rate-limiting step of the algorithm, our method circumvents the numerical instability induced by the other cavity physics when evaluating the gain in an efficient manner. We demonstrate the stability and efficiency of the algorithm on a diode laser model which includes three waveguides and semiconductor gain dynamics. The laser is able to produce a repeating temporal waveform and stable optical comblines, thus demonstrating that frequency combs generation may be possible in chip scale, diode lasers.

physics.optics↗

arXiv2019

Dynamic mode decomposition for multiscale nonlinear physics

We present a data-driven method for separating complex, multiscale systems into their constituent time-scale components using a recursive implementation of dynamic mode decomposition (DMD). Local linear models are built from windowed subsets of the data, and dominant time scales are discovered using spectral clustering on their eigenvalues. This approach produces time series data for each identified component, which sum to a faithful reconstruction of the input signal. It differs from most other methods in the field of multiresolution analysis (MRA) in that it 1) accounts for spatial and temporal coherencies simultaneously, making it more robust to scale overlap between components, and 2) yields a closed-form expression for local dynamics at each scale, which can be used for short-term prediction of any or all components. Our technique is an extension of multi-resolution dynamic mode decomposition (mrDMD), generalized to treat a broader variety of multiscale systems and more faithfully reconstruct their isolated components. In this paper we present an overview of our algorithm and its results on two example physical systems, and briefly discuss some advantages and potential forecasting applications for the technique.

eess.SY↗

arXiv2019

Centering Data Improves the Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) is a data-driven method that models high-dimensional time series as a sum of spatiotemporal modes, where the temporal modes are constrained by linear dynamics. For nonlinear dynamical systems exhibiting strongly coherent structures, DMD can be a useful approximation to extract dominant, interpretable modes. In many domains with large spatiotemporal data---including fluid dynamics, video processing, and finance---the dynamics of interest are often perturbations about fixed points or equilibria, which motivates the application of DMD to centered (i.e. mean-subtracted) data. In this work, we show that DMD with centered data is equivalent to incorporating an affine term in the dynamic model and is not equivalent to computing a discrete Fourier transform. Importantly, DMD with centering can always be used to compute eigenvalue spectra of the dynamics. However, in many cases DMD without centering cannot model the corresponding dynamics, most notably if the dynamics have full effective rank. Additionally, we generalize the notion of centering to extracting arbitrary, but known, fixed frequencies from the data. We corroborate these theoretical results numerically on three nonlinear examples: the Lorenz system, a surveillance video, and brain recordings. Since centering the data is simple and computationally efficient, we recommend it as a preprocessing step before DMD; furthermore, we suggest that it can be readily used in conjunction with many other popular implementations of the DMD algorithm.

math.DS↗

arXiv2019

Dynamic Mode Decomposition and Sparse Measurements for Characterization and Monitoring of Power System Disturbances

We introduce the dynamics mode decomposition for monitoring wide-area power grid networks from sparse measurement data. The mathematical framework fuses data from multiple sensors based on multivariate statistics, providing accurate full state estimation from limited measurements and generating data-driven forecasts for the state of the system. Our proposed data-driven strategy, which is based on energy metrics, can be used for the analysis of major disturbances in the network. The approach is tested and validated using time domain simulations in the IEEE 118 bus system under various disturbance scenarios and under different sparse observations of the system. In addition to state reconstruction, the minimal number of sensors required for monitoring disturbances can be evaluated. Visualization techniques are developed in order to aid in the analysis and characterization of the system after disturbance.

nlin.PS↗

arXiv2019

Deep Model Predictive Control with Online Learning for Complex Physical Systems

The control of complex systems is of critical importance in many branches of science, engineering, and industry. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high-dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems. We present a novel deep learning model predictive control (DeepMPC) framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system. The RNN is then embedded into a MPC framework to construct a feedback loop, and incoming sensor data is used to perform online updates to improve prediction accuracy. The results are validated using varying fluid flow examples of increasing complexity.

cs.LG↗

arXiv2019

Engineering Structural Robustness in Power Grid Networks Susceptible to Coherent Swing Instability

Networked power grid systems are susceptible to a phenomenon known as Coherent Swing Instability (CSI), in which a subset of machines in the grid lose synchrony with the rest of the network. We develop network level evaluation metrics to (i) identify community substructures in the power grid network, (ii) determine weak points in the network that are particularly sensitive to CSI, and (iii) produce an engineering approach for the addition of transmission lines to reduce the incidences of CSI in existing networks, or design new power grid networks that are robust to CSI by their network design. For simulations on a reduced model for the American Northeast power grid, where a block of buses representing the New England region exhibit a strong propensity for CSI, we show that modifying the network's connectivity structure can markedly improve the grid's resilience to CSI. Our analysis provides a versatile diagnostic tool for evaluating the efficacy of adding lines to a power grid which is known to be prone to CSI. This is a particularly relevant problem in large-scale power systems, where improving stability and robustness to interruptions by increasing overall network connectivity is not feasible due to financial and infrastructural constraints.

physics.soc-ph↗