arXiv subjects

J. Nathan Kutz

Publications and source records attributed to J. Nathan Kutz.

At least 163 records · Page 9Linked to original sources

arXiv2020

Bracketing brackets with bras and kets

Brackets are an essential component in aircraft manufacture and design, joining parts together, supporting weight, holding wires, and strengthening joints. Hundreds or thousands of unique brackets are used in every aircraft, but manufacturing a large number of distinct brackets is inefficient and expensive. Fortunately, many so-called "different" brackets are in fact very similar or even identical to each other. In this manuscript, we present a data-driven framework for constructing a comparatively small group of representative brackets from a large catalog of current brackets, based on hierarchical clustering of bracket data. We find that for a modern commercial aircraft, the full set of brackets can be reduced by 30\% while still describing half of the test set sufficiently accurately. This approach is based on designing an inner product that quantifies a multi-objective similarity between two brackets, which are the "bra" and the "ket" of the inner product. Although we demonstrate this algorithm to reduce the number of brackets in aerospace manufacturing, it may be generally applied to any large-scale component standardization effort.

cs.LG↗

arXiv2020

Modeling Thermodynamic Trends of Rotating Detonation Engines

The formation of a number of co- and counter-rotating coherent combustion wave fronts is the hallmark feature of the Rotating Detonation Engine (RDE). The engineering implications of wave topology are not well understood nor quantified, especially with respect to parametric changes in combustor geometry, propellant chemistry, and injection and mixing schemes. In this article, a modeling framework that relates the time and spacial scales of the RDE to engineering performance metrics is developed and presented. The model is built under assumptions of backpressure-insensitivity and nominally choked gaseous propellant injection. The Euler equations of inviscid, compressible fluid flow in one dimension are adapted to model the combustion wave dynamics along the circumference of an annular-type rotating detonation engine. These adaptations provide the necessary mass and energy input and output channels to shape the traveling wave fronts and decaying tails. The associated unit processes of injection, mixing, combustion, and exhaust are all assigned representative time scales necessary for successful wave propagation. We find that the separation, or lack of, these time scales are responsible for the behavior of the system, including wave co- and counter-propagation and bifurcations between these regimes and wave counts. Furthermore, as there is no imposition of wave topology, the model output is used to estimate the net available mechanical work output and thermodynamic efficiency from the closed trajectories through pressure-volume and temperature-entropy spaces. These metrics are investigated with respect to variation in the characteristic scales for the RDE unit physical processes.

physics.flu-dyn↗

arXiv2020

Sparse Identification of Slow Timescale Dynamics

Multiscale phenomena that evolve on multiple distinct timescales are prevalent throughout the sciences. It is often the case that the governing equations of the persistent and approximately periodic fast scales are prescribed, while the emergent slow scale evolution is unknown. Yet the course-grained, slow scale dynamics is often of greatest interest in practice. In this work we present an accurate and efficient method for extracting the slow timescale dynamics from signals exhibiting multiple timescales that are amenable to averaging. The method relies on tracking the signal at evenly-spaced intervals with length given by the period of the fast timescale, which is discovered using clustering techniques in conjunction with the dynamic mode decomposition. Sparse regression techniques are then used to discover a mapping which describes iterations from one data point to the next. We show that for sufficiently disparate timescales this discovered mapping can be used to discover the continuous-time slow dynamics, thus providing a novel tool for extracting dynamics on multiple timescales.

nlin.CD↗

arXiv2020

A unified sparse optimization framework to learn parsimonious physics-informed models from data

Machine learning (ML) is redefining what is possible in data-intensive fields of science and engineering. However, applying ML to problems in the physical sciences comes with a unique set of challenges: scientists want physically interpretable models that can (i) generalize to predict previously unobserved behaviors, (ii) provide effective forecasting predictions (extrapolation), and (iii) be certifiable. Autonomous systems will necessarily interact with changing and uncertain environments, motivating the need for models that can accurately extrapolate based on physical principles (e.g. Newton's universal second law for classical mechanics, $F=ma$). Standard ML approaches have shown impressive performance for predicting dynamics in an interpolatory regime, but the resulting models often lack interpretability and fail to generalize. We introduce a unified sparse optimization framework that learns governing dynamical systems models from data, selecting relevant terms in the dynamics from a library of possible functions. The resulting models are parsimonious, have physical interpretations, and can generalize to new parameter regimes. Our framework allows the use of non-convex sparsity promoting regularization functions and can be adapted to address key challenges in scientific problems and data sets, including outliers, parametric dependencies, and physical constraints. We show that the approach discovers parsimonious dynamical models on several example systems. This flexible approach can be tailored to the unique challenges associated with a wide range of applications and data sets, providing a powerful ML-based framework for learning governing models for physical systems from data.

physics.comp-ph↗

arXiv2020

Inferring Causal Networks of Dynamical Systems through Transient Dynamics and Perturbation

Inferring causal relations from time series measurements is an ill-posed mathematical problem, where typically an infinite number of potential solutions can reproduce the given data. We explore in depth a strategy to disambiguate between possible underlying causal networks by perturbing the network, where the actuations are either targeted or applied at random. The resulting transient dynamics provide the critical information necessary to infer causality. Two methods are shown to provide accurate causal reconstructions: Granger causality (GC) with perturbations, and our proposed perturbation cascade inference (PCI). Perturbed GC is capable of inferring smaller networks under low coupling strength regimes. Our proposed PCI method demonstrated consistently strong performance in inferring causal relations for small (2-5 node) and large (10-20 node) networks, with both linear and nonlinear dynamics. Thus the ability to apply a large and diverse set of perturbations/actuations to the network is critical for successfully and accurately determining causal relations and disambiguating between various viable networks.

math.DS↗

arXiv2020

Nonlinear control of networked dynamical systems

We develop a principled mathematical framework for controlling nonlinear, networked dynamical systems. Our method integrates dimensionality reduction, bifurcation theory and emerging model discovery tools to find low-dimensional subspaces where feed-forward control can be used to manipulate a system to a desired outcome. The method leverages the fact that many high-dimensional networked systems have many fixed points, allowing for the computation of control signals that will move the system between any pair of fixed points. The sparse identification of nonlinear dynamics (SINDy) algorithm is used to fit a nonlinear dynamical system to the evolution on the dominant, low-rank subspace. This then allows us to use bifurcation theory to find collections of constant control signals that will produce the desired objective path for a prescribed outcome. Specifically, we can destabilize a given fixed point while making the target fixed point an attractor. The discovered control signals can be easily projected back to the original high-dimensional state and control space. We illustrate our nonlinear control procedure on established bistable, low-dimensional biological systems, showing how control signals are found that generate switches between the fixed points. We then demonstrate our control procedure for high-dimensional systems on random high-dimensional networks and Hopfield memory networks.

math.DS↗

arXiv2020

Physics-informed machine learning for sensor fault detection with flight test data

We develop data-driven algorithms to fully automate sensor fault detection in systems governed by underlying physics. The proposed machine learning method uses a time series of typical behavior to approximate the evolution of measurements of interest by a linear time-invariant system. Given additional data from related sensors, a Kalman observer is used to maintain a separate real-time estimate of the measurement of interest. Sustained deviation between the measurements and the estimate is used to detect anomalous behavior. A decision tree, informed by integrating other sensor measurement values, is used to determine the amount of deviation required to identify a sensor fault. We validate the method by applying it to three test systems exhibiting various types of sensor faults: commercial flight test data, an unsteady aerodynamics model with dynamic stall, and a model for longitudinal flight dynamics forced by atmospheric turbulence. In the latter two cases we test fault detection for several prototypical failure modes. The combination of a learned dynamical model with the automated decision tree accurately detects sensor faults in each case.

eess.SY↗

arXiv2020

Deep reinforcement learning for optical systems: A case study of mode-locked lasers

We demonstrate that deep reinforcement learning (deep RL) provides a highly effective strategy for the control and self-tuning of optical systems. Deep RL integrates the two leading machine learning architectures of deep neural networks and reinforcement learning to produce robust and stable learning for control. Deep RL is ideally suited for optical systems as the tuning and control relies on interactions with its environment with a goal-oriented objective to achieve optimal immediate or delayed rewards. This allows the optical system to recognize bi-stable structures and navigate, via trajectory planning, to optimally performing solutions, the first such algorithm demonstrated to do so in optical systems. We specifically demonstrate the deep RL architecture on a mode-locked laser, where robust self-tuning and control can be established through access of the deep RL agent to its waveplates and polarizers. We further integrate transfer learning to help the deep RL agent rapidly learn new parameter regimes and generalize its control authority. Additionally, the deep RL learning can be easily integrated with other control paradigms to provide a broad framework to control any optical system.

eess.SP↗

arXiv2020

SINDy-BVP: Sparse Identification of Nonlinear Dynamics for Boundary Value Problems

We develop a data-driven model discovery and system identification technique for spatially-dependent boundary value problems (BVPs). Specifically, we leverage the sparse identification of nonlinear dynamics (SINDy) algorithm and group sparse regression techniques with a set of forcing functions and corresponding state variable measurements to yield a parsimonious model of the system. The approach models forced systems governed by linear or nonlinear operators of the form $L[u(x)] = f(x)$ on a prescribed domain $x \in [a, b]$. We demonstrate the approach on a range of example systems, including Sturm-Liouville operators, beam theory (elasticity), and a class of nonlinear BVPs. The generated data-driven model is used to infer both the operator and/or spatially-dependent parameters that describe the heterogenous, physical quantities of the system. Our SINDy-BVP framework will enables the characterization of a broad range of systems, including for instance, the discovery of anisotropic materials with heterogeneous variability.

cs.CE↗

arXiv2020

Spatio-Temporal Mode-locking in Quadratic Nonlinear Media

A new theoretical model is developed to characterize spatio-temporal mode-locking (ML) in quadratic nonlinear media. The model is based on the two-dimensional nonlinear Schrödinger equation with coupling to a mean term (NLSM) and constructed as an extension of the master mode-locking model. It is numerically demonstrated that there exists steady state soliton solutions of the ML-NLSM mode-locking model that are astigmatic in nature. A full stability analysis and bifurcation study is performed for the ML-NLSM model and it is manifest that spatio-temporal mode-locking of the astigmatic steady-state solutions is possible in quadratic nonlinear media.

physics.optics↗

arXiv2020

Dimensionality Reduction and Reduced Order Modeling for Traveling Wave Physics

We develop an unsupervised machine learning algorithm for the automated discovery and identification of traveling waves in spatio-temporal systems governed by partial differential equations (PDEs). Our method uses sparse regression and subspace clustering to robustly identify translational invariances that can be leveraged to build improved reduced order models (ROMs). Invariances, whether translational or rotational, are well known to compromise the ability of ROMs to produce accurate and/or low-rank representations of the spatio-temporal dynamics. However, by discovering translations in a principled way, data can be shifted into a coordinate systems where quality, low-dimensional ROMs can be constructed. This approach can be used on either numerical or experimental data with or without knowledge of the governing equations. We demonstrate our method on a variety of PDEs of increasing difficulty, taken from the field of fluid dynamics, showing the efficacy and robustness of the proposed approach.

physics.comp-ph↗

arXiv2020

Multi-fidelity sensor selection: Greedy algorithms to place cheap and expensive sensors with cost constraints

We develop greedy algorithms to approximate the optimal solution to the multi-fidelity sensor selection problem, which is a cost constrained optimization problem prescribing the placement and number of cheap (low signal-to-noise) and expensive (high signal-to-noise) sensors in an environment or state space. Specifically, we evaluate the composition of cheap and expensive sensors, along with their placement, required to achieve accurate reconstruction of a high-dimensional state. We use the column-pivoted QR decomposition to obtain preliminary sensor positions. How many of each type of sensor to use is highly dependent upon the sensor noise levels, sensor costs, overall cost budget, and the singular value spectrum of the data measured. Such nuances allow us to provide sensor selection recommendations based on computational results for asymptotic regions of parameter space. We also present a systematic exploration of the effects of the number of modes and sensors on reconstruction error when using one type of sensor. Our extensive exploration of multi-fidelity sensor composition as a function of data characteristics is the first of its kind to provide guidelines towards optimal multi-fidelity sensor selection.

eess.SP↗

arXiv2020

PySINDy: A Python package for the Sparse Identification of Nonlinear Dynamics from Data

PySINDy is a Python package for the discovery of governing dynamical systems models from data. In particular, PySINDy provides tools for applying the sparse identification of nonlinear dynamics (SINDy) (Brunton et al. 2016) approach to model discovery. In this work we provide a brief description of the mathematical underpinnings of SINDy, an overview and demonstration of the features implemented in PySINDy (with code examples), practical advice for users, and a list of potential extensions to PySINDy. Software is available at https://github.com/dynamicslab/pysindy.

math.DS↗

arXiv2020

Discovery of Physics from Data: Universal Laws and Discrepancies

Machine learning (ML) and artificial intelligence (AI) algorithms are now being used to automate the discovery of physics principles and governing equations from measurement data alone. However, positing a universal physical law from data is challenging without simultaneously proposing an accompanying discrepancy model to account for the inevitable mismatch between theory and measurements. By revisiting the classic problem of modeling falling objects of different size and mass, we highlight a number of nuanced issues that must be addressed by modern data-driven methods for automated physics discovery. Specifically, we show that measurement noise and complex secondary physical mechanisms, like unsteady fluid drag forces, can obscure the underlying law of gravitation, leading to an erroneous model. We use the sparse identification of nonlinear dynamics (SINDy) method to identify governing equations for real-world measurement data and simulated trajectories. Incorporating into SINDy the assumption that each falling object is governed by a similar physical law is shown to improve the robustness of the learned models, but discrepancies between the predictions and observations persist due to subtleties in drag dynamics. This work highlights the fact that the naive application of ML/AI will generally be insufficient to infer universal physical laws without further modification.

cs.LG↗

arXiv2020

Unsupervised learning of control signals and their encodings in $\textit{C. elegans}$ whole-brain recordings

Recent whole brain imaging experiments on $\textit{C. elegans}$ has revealed that the neural population dynamics encode motor commands and stereotyped transitions between behaviors on low dimensional manifolds. Efforts to characterize the dynamics on this manifold have used piecewise linear models to describe the entire state space, but it is unknown how a single, global dynamical model can generate the observed dynamics. Here, we propose a control framework to achieve such a global model of the dynamics, whereby underlying linear dynamics is actuated by sparse control signals. This method learns the control signals in an unsupervised way from data, then uses $\textit{ Dynamic Mode Decomposition with control}$ (DMDc) to create the first global, linear dynamical system that can reconstruct whole-brain imaging data. These control signals are shown to be implicated in transitions between behaviors. In addition, we analyze the time-delay encoding of these control signals, showing that these transitions can be predicted from neurons previously implicated in behavioral transitions, but also additional neurons previously unidentified. Moreover, our decomposition method allows one to understand the observed nonlinear global dynamics instead as linear dynamics with control. The proposed mathematical framework is generic and can be generalized to other neurosensory systems, potentially revealing transitions and their encodings in a completely unsupervised way.

q-bio.QM↗

arXiv2020

Multiresolution Convolutional Autoencoders

We propose a multi-resolution convolutional autoencoder (MrCAE) architecture that integrates and leverages three highly successful mathematical architectures: (i) multigrid methods, (ii) convolutional autoencoders and (iii) transfer learning. The method provides an adaptive, hierarchical architecture that capitalizes on a progressive training approach for multiscale spatio-temporal data. This framework allows for inputs across multiple scales: starting from a compact (small number of weights) network architecture and low-resolution data, our network progressively deepens and widens itself in a principled manner to encode new information in the higher resolution data based on its current performance of reconstruction. Basic transfer learning techniques are applied to ensure information learned from previous training steps can be rapidly transferred to the larger network. As a result, the network can dynamically capture different scaled features at different depths of the network. The performance gains of this adaptive multiscale architecture are illustrated through a sequence of numerical experiments on synthetic examples and real-world spatial-temporal data.

cs.LG↗

arXiv2020

Poincaré Maps for Multiscale Physics Discovery and Nonlinear Floquet Theory

Poincaré maps are an integral aspect to our understanding and analysis of nonlinear dynamical systems. Despite this fact, the construction of these maps remains elusive and is primarily left to simple motivating examples. In this manuscript we propose a method of data-driven discovery of Poincaré maps based upon sparse regression techniques, specifically the sparse identification of nonlinear dynamics (SINDy) algorithm. This work can be used to determine the dynamics on and near invariant manifolds of a given dynamical system, as well as provide long-time forecasting of the coarse-grained dynamics of multiscale systems. Moreover, the method provides a mathematical formalism for determining nonlinear Floquet theory for the stability of nonlinear periodic orbits. The methods are applied to a range of examples including both ordinary and partial differential equations that exhibit periodic, quasi-periodic, and chaotic behavior.

math.DS↗

arXiv2020

Sensor Selection With Cost Constraints for Dynamically Relevant Bases

We consider cost-constrained sparse sensor selection for full-state reconstruction, applying a well-known greedy algorithm to dynamical systems for which the usual singular value decomposition (SVD) basis may not be available or preferred. We apply the cost-modified, column-pivoted QR decomposition to a physically relevant basis -- the pivots correspond to sensor locations, and these locations are penalized with a heterogeneous cost function. In considering different bases, we are able to account for the dynamics of the particular system, yielding sensor arrays that are nearly Pareto optimal in sensor cost and performance in the chosen basis. This flexibility extends our framework to include actuation and dynamic estimation, and to select sensors without training data. We provide three examples from the physical and engineering sciences and evaluate sensor selection in three dynamically relevant bases: truncated balanced modes for control systems, dynamic mode decomposition (DMD) modes, and a basis of analytic modes. We find that these bases all yield effective sensor arrays and reconstructions for their respective systems. When possible, we compare to results using an SVD basis and evaluate tradeoffs between methods.

math.OC↗