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Published work

55 published item(s)

preprint2026arXiv

CTF4Nuclear: Common Task Framework for Nuclear Fission and Fusion Models

The demand for clean energy is ever increasing, with new nuclear technologies presenting a complementary solution to renewable energies. However, designing and operating these systems is exceptionally difficult, given the complexity of the physical phenomena that interact to form the system dynamics. While high-fidelity simulations help to understand the non-linear, multi-physics interactions within a reactor, they are computationally expensive and rarely suitable for real-time applications. Furthermore, model-based approaches are inherently sensitive to simplifying assumptions required to derive their governing equations and parameters, leading to inevitable discrepancies with real-world measurements. In contrast, Machine Learning (ML) methods have the potential to generate reliable surrogate models which may be able to quickly predict the system's behaviour. However, the number of data-driven methods that can potentially be used for this task is large and diverse. In a safety-critical setting such as nuclear engineering, a fair comparison of different ML methods, and a clear understanding of their advantages and limitations, is of paramount importance. To address this, we introduce a Common Task Framework (CTF) for ML in nuclear engineering, building upon previous efforts in dynamical systems and seismology. This CTF considers a curated set of datasets from different nuclear and nuclear-adjacent systems. The CTF evaluates the performance of a method on 12 established metrics, alongside a new paradigm focused on system monitoring from sparse measurements only. We illustrate the framework by benchmarking standard ML baselines against these datasets, revealing current method limitations. Our vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets, raising the bar for rigour and reproducibility in scientific ML for the nuclear industry.

preprint2026arXiv

Data-driven methods to discover stable linear models of the helicity injectors on HIT-SIU

Accurate and efficient circuit models are necessary to control the power electronic circuits found on plasma physics experiments. Tuning and controlling the behavior of these circuits is inextricably linked to plasma performance. Linear models are greatly preferred for control applications due to their well-established performance guarantees, but they typically fail to capture nonlinear dynamics and changes in experimental parameters. Data-driven system identification can help mitigate these shortcomings by learning interpretable and accurate reduced-order models of a complex system, in this case the injector circuits of the Helicity Injected Torus - Steady Inductive Upgrade (HIT-SIU) experiment. Specifically, the Bagging Optimized Dynamic Mode Decomposition (BOP-DMD), is leveraged to learn stable, reduced order models of the interaction between the spheromak plasma formed in the confinement volume, and the injector circuits of the device. BOP-DMD is trained and evaluated on an analytic model of the vacuum dynamics of the injector circuits of HIT-SIU, as well as an analytic linear reduced-order model for the injector dynamics when a plasma is present. BOP-DMD is then fit on experimental data, both on shots with and without a plasma in the confinement volume. In doing so, we demonstrate the capability of data-driven methods to produce stable, linear models for control and uncertainty quantification in plasma experiments.

preprint2026arXiv

Terrain-Adaptive Mobile 3D Printing with Hierarchical Control

Mobile 3D printing on unstructured terrain remains challenging due to the conflict between platform mobility and deposition precision. Existing gantry-based systems achieve high accuracy but lack mobility, while mobile platforms struggle to maintain print quality on uneven ground. We present a framework that tightly integrates AI-driven disturbance prediction with multi-modal sensor fusion and hierarchical hardware control, forming a closed-loop perception-learning-actuation system. The AI module learns terrain-to-perturbation mappings from IMU, vision, and depth sensors, enabling proactive compensation rather than reactive correction. This intelligence is embedded into a three-layer control architecture: path planning, predictive chassis-manipulator coordination, and precision hardware execution. Through outdoor experiments on terrain with slopes and surface irregularities, we demonstrate sub-centimeter printing accuracy while maintaining full platform mobility. This AI-hardware integration establishes a practical foundation for autonomous construction in unstructured environments.

preprint2025arXiv

Coarse graining and reduced order models for plume ejection dynamics

Monitoring the atmospheric dispersion of pollutants is increasingly critical for environmental impact assessments. High-fidelity computational models are often employed to simulate plume dynamics, guiding decision-making and prioritizing resource deployment. However, such models can be prohibitively expensive to simulate, as they require resolving turbulent flows at fine spatial and temporal resolutions. Moreover, there are at least two distinct dynamical regimes of interest in the plume: (i) the initial ejection of the plume where turbulent mixing is generated by the shear-driven Kelvin-Helmholtz instability, and (ii) the ensuing turbulent diffusion and advection which is often modeled by the Gaussian plume model. We address the challenge of modeling the initial plume generation. Specifically, we propose a data-driven framework that identifies a reduced-order analytical model for plume dynamics -- directly from video data. We extract a time series of plume center and edge points from video snapshots and evaluate different regressions based to their extrapolation performance to generate a time series of coefficients that characterize the plume's overall direction and spread. We regress to a sinusoidal model inspired by the Kelvin-Helmholtz instability for the edge points in order to identify the plume's dispersion and vorticity. Overall, this reduced-order modeling framework provides a data-driven and lightweight approach to capture the dominant features of the initial nonlinear point-source plume dynamics, agnostic to plume type and starting only from video. The resulting model is a pre-cursor to standard models such as the Gaussian plume model and has the potential to enable rapid assessment and evaluation of critical environmental hazards, such as methane leaks, chemical spills, and pollutant dispersal from smokestacks.

preprint2023arXiv

Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data

High-dimensional spatio-temporal dynamics can often be encoded in a low-dimensional subspace. Engineering applications for modeling, characterization, design, and control of such large-scale systems often rely on dimensionality reduction to make solutions computationally tractable in real-time. Common existing paradigms for dimensionality reduction include linear methods, such as the singular value decomposition (SVD), and nonlinear methods, such as variants of convolutional autoencoders (CAE). However, these encoding techniques lack the ability to efficiently represent the complexity associated with spatio-temporal data, which often requires variable geometry, non-uniform grid resolution, adaptive meshing, and/or parametric dependencies. To resolve these practical engineering challenges, we propose a general framework called Neural Implicit Flow (NIF) that enables a mesh-agnostic, low-rank representation of large-scale, parametric, spatial-temporal data. NIF consists of two modified multilayer perceptrons (MLPs): (i) ShapeNet, which isolates and represents the spatial complexity, and (ii) ParameterNet, which accounts for any other input complexity, including parametric dependencies, time, and sensor measurements. We demonstrate the utility of NIF for parametric surrogate modeling, enabling the interpretable representation and compression of complex spatio-temporal dynamics, efficient many-spatial-query tasks, and improved generalization performance for sparse reconstruction.

preprint2022arXiv

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.

preprint2022arXiv

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.

preprint2022arXiv

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.

preprint2022arXiv

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.

preprint2022arXiv

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.

preprint2022arXiv

Dynamic Mode Decomposition for Aero-Optic Wavefront Characterization

Aero-optical beam control relies on the development of low-latency forecasting techniques to quickly predict wavefronts aberrated by the Turbulent Boundary Layer (TBL) around an airborne optical system, and its study applies to a multi-domain need from astronomy to microscopy for high-fidelity laser propagation. We leverage the forecasting capabilities of the Dynamic Mode Decomposition (DMD) -- an equation-free, data-driven method for identifying coherent flow structures and their associated spatiotemporal dynamics -- in order to estimate future state wavefront phase aberrations to feed into an adaptive optic (AO) control loop. We specifically leverage the optimized DMD (opt-DMD) algorithm on a subset of the Airborne Aero-Optics Laboratory Transonic (AAOL-T) experimental dataset, characterizing aberrated wavefront dynamics for 23 beam propagation directions via the spatiotemporal decomposition underlying DMD. Critically, we show that opt-DMD produces an optimally de-biased eigenvalue spectrum with imaginary eigenvalues, allowing for arbitrarily long forecasting to produce a robust future-state prediction, while exact DMD loses structural information due to modal decay rates.

preprint2022arXiv

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.

preprint2022arXiv

Principal component trajectories for modeling spectrally-continuous dynamics as forced linear systems

Delay embeddings of time series data have emerged as a promising coordinate basis for data-driven estimation of the Koopman operator, which seeks a linear representation for observed nonlinear dynamics. Recent work has demonstrated the efficacy of Dynamic Mode Decomposition (DMD) for obtaining finite-dimensional Koopman approximations in delay coordinates. In this paper we demonstrate how nonlinear dynamics with sparse Fourier spectra can be (i) represented by a superposition of principal component trajectories (PCT) and (ii) modeled by DMD in this coordinate space. For continuous or mixed (discrete and continuous) spectra, DMD can be augmented with an external forcing term. We present a method for learning linear control models in delay coordinates while simultaneously discovering the corresponding exogeneous forcing signal in a fully unsupervised manner. This extends the existing DMD with control (DMDc) algorithm to cases where a control signal is not known a priori. We provide examples to validate the learned forcing against a known ground truth and illustrate their statistical similarity. Finally we offer a demonstration of this method applied to real-world power grid load data to show its utility for diagnostics and interpretation on systems in which somewhat periodic behavior is strongly forced by unknown and unmeasurable environmental variables.

preprint2022arXiv

Pruning deep neural networks generates a sparse, bio-inspired nonlinear controller for insect flight

Insect flight is a strongly nonlinear and actuated dynamical system. As such, strategies for understanding its control have typically relied on either model-based methods or linearizations thereof. Here we develop a framework that combines model predictive control on an established flight dynamics model and deep neural networks (DNN) to create an efficient method for solving the inverse problem of flight control. We turn to natural systems for inspiration since they inherently demonstrate network pruning with the consequence of yielding more efficient networks for a specific set of tasks. This bio-inspired approach allows us to leverage network pruning to optimally sparsify a DNN architecture in order to perform flight tasks with as few neural connections as possible, however, there are limits to sparsification. Specifically, as the number of connections falls below a critical threshold, flight performance drops considerably. We develop sparsification paradigms and explore their limits for control tasks. Monte Carlo simulations also quantify the statistical distribution of network weights during pruning given initial random weights of the DNNs. We demonstrate that on average, the network can be pruned to retain approximately 7% of the original network weights, with statistical distributions quantified at each layer of the network. Overall, this work shows that sparsely connected DNNs are capable of predicting the forces required to follow flight trajectories. Additionally, sparsification has sharp performance limits.

preprint2022arXiv

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.

preprint2022arXiv

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.

preprint2022arXiv

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.

preprint2022arXiv

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.

preprint2021arXiv

A toolkit for data-driven discovery of governing equations in high-noise regimes

We consider the data-driven discovery of governing equations from time-series data in the limit of high noise. The algorithms developed describe an extensive toolkit of methods for circumventing the deleterious effects of noise in the context of the sparse identification of nonlinear dynamics (SINDy) framework. We offer two primary contributions, both focused on noisy data acquired from a system x' = f(x). First, we propose, for use in high-noise settings, an extensive toolkit of critically enabling extensions for the SINDy regression method, to progressively cull functionals from an over-complete library and yield a set of sparse equations that regress to the derivate x'. These innovations can extract sparse governing equations and coefficients from high-noise time-series data (e.g. 300% added noise). For example, it discovers the correct sparse libraries in the Lorenz system, with median coefficient estimate errors equal to 1% - 3% (for 50% noise), 6% - 8% (for 100% noise); and 23% - 25% (for 300% noise). The enabling modules in the toolkit are combined into a single method, but the individual modules can be tactically applied in other equation discovery methods (SINDy or not) to improve results on high-noise data. Second, we propose a technique, applicable to any model discovery method based on x' = f(x), to assess the accuracy of a discovered model in the context of non-unique solutions due to noisy data. Currently, this non-uniqueness can obscure a discovered model's accuracy and thus a discovery method's effectiveness. We describe a technique that uses linear dependencies among functionals to transform a discovered model into an equivalent form that is closest to the true model, enabling more accurate assessment of a discovered model's accuracy.

preprint2021arXiv

Data-driven discovery of Koopman eigenfunctions for control

Data-driven transformations that reformulate nonlinear systems in a linear framework have the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics, and its eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly. Previous studies have used finite-dimensional approximations of the Koopman operator for model-predictive control approaches. In this work, we illustrate a fundamental closure issue of this approach and argue that it is beneficial to first validate eigenfunctions and then construct reduced-order models in these validated eigenfunctions. These coordinates form a Koopman-invariant subspace by design and, thus, have improved predictive power. We show then how the control can be formulated directly in these intrinsic coordinates and discuss potential benefits and caveats of this perspective. The resulting control architecture is termed Koopman Reduced Order Nonlinear Identification and Control (KRONIC). It is demonstrated that these eigenfunctions can be approximated with data-driven regression and power series expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. Validating discovered eigenfunctions is crucial and we show that lightly damped eigenfunctions may be faithfully extracted from EDMD or an implicit formulation. These lightly damped eigenfunctions are particularly relevant for control, as they correspond to nearly conserved quantities that are associated with persistent dynamics, such as the Hamiltonian. KRONIC is then demonstrated on a number of relevant examples, including 1) a nonlinear system with a known linear embedding, 2) a variety of Hamiltonian systems, and 3) a high-dimensional double-gyre model for ocean mixing.

preprint2021arXiv

Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control

Sparse model identification enables the discovery of nonlinear dynamical systems purely from data; however, this approach is sensitive to noise, especially in the low-data limit. In this work, we leverage the statistical approach of bootstrap aggregating (bagging) to robustify the sparse identification of nonlinear dynamics (SINDy) algorithm. First, an ensemble of SINDy models is identified from subsets of limited and noisy data. The aggregate model statistics are then used to produce inclusion probabilities of the candidate functions, which enables uncertainty quantification and probabilistic forecasts. We apply this ensemble-SINDy (E-SINDy) algorithm to several synthetic and real-world data sets and demonstrate substantial improvements to the accuracy and robustness of model discovery from extremely noisy and limited data. For example, E-SINDy uncovers partial differential equations models from data with more than twice as much measurement noise as has been previously reported. Similarly, E-SINDy learns the Lotka Volterra dynamics from remarkably limited data of yearly lynx and hare pelts collected from 1900-1920. E-SINDy is computationally efficient, with similar scaling as standard SINDy. Finally, we show that ensemble statistics from E-SINDy can be exploited for active learning and improved model predictive control.

preprint2021arXiv

PySensors: A Python Package for Sparse Sensor Placement

PySensors is a Python package for selecting and placing a sparse set of sensors for classification and reconstruction tasks. Specifically, PySensors implements algorithms for data-driven sparse sensor placement optimization for reconstruction (SSPOR) and sparse sensor placement optimization for classification (SSPOC). In this work we provide a brief description of the mathematical algorithms and theory for sparse sensor optimization, along with an overview and demonstration of the features implemented in PySensors (with code examples). We also include practical advice for user and a list of potential extensions to PySensors. Software is available at https://github.com/dynamicslab/pysensors.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

Built to Last: Functional and structural mechanisms in the moth olfactory network mitigate effects of neural injury

Most organisms suffer neuronal damage throughout their lives, which can impair performance of core behaviors. Their neural circuits need to maintain function despite injury, which in particular requires preserving key system outputs. In this work, we explore whether and how certain structural and functional neuronal network motifs act as injury mitigation mechanisms. Specifically, we examine how (i) Hebbian learning, (ii) high levels of noise, and (iii) parallel inhibitory and excitatory connections contribute to the robustness of the olfactory system in the Manduca sexta moth. We simulate injuries on a detailed computational model of the moth olfactory network calibrated to in vivo data. The injuries are modeled on focal axonal swellings, a ubiquitous form of axonal pathology observed in traumatic brain injuries and other brain disorders. Axonal swellings effectively compromise spike train propagation along the axon, reducing the effective neural firing rate delivered to downstream neurons. All three of the network motifs examined significantly mitigate the effects of injury on readout neurons, either by reducing injury's impact on readout neuron responses or by restoring these responses to pre-injury levels. These motifs may thus be partially explained by their value as adaptive mechanisms to minimize the functional effects of neural injury. More generally, robustness to injury is a vital design principle to consider when analyzing neural systems.

preprint2020arXiv

Data-Driven Aerospace Engineering: Reframing the Industry with Machine Learning

Data science, and machine learning in particular, is rapidly transforming the scientific and industrial landscapes. The aerospace industry is poised to capitalize on big data and machine learning, which excels at solving the types of multi-objective, constrained optimization problems that arise in aircraft design and manufacturing. Indeed, emerging methods in machine learning may be thought of as data-driven optimization techniques that are ideal for high-dimensional, non-convex, and constrained, multi-objective optimization problems, and that improve with increasing volumes of data. In this review, we will explore the opportunities and challenges of integrating data-driven science and engineering into the aerospace industry. Importantly, we will focus on the critical need for interpretable, generalizeable, explainable, and certifiable machine learning techniques for safety-critical applications. This review will include a retrospective, an assessment of the current state-of-the-art, and a roadmap looking forward. Recent algorithmic and technological trends will be explored in the context of critical challenges in aerospace design, manufacturing, verification, validation, and services. In addition, we will explore this landscape through several case studies in the aerospace industry. This document is the result of close collaboration between UW and Boeing to summarize past efforts and outline future opportunities.

preprint2020arXiv

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.

preprint2020arXiv

DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems

Boundary value problems (BVPs) play a central role in the mathematical analysis of constrained physical systems subjected to external forces. Consequently, BVPs frequently emerge in nearly every engineering discipline and span problem domains including fluid mechanics, electromagnetics, quantum mechanics, and elasticity. The fundamental solution, or Green's function, is a leading method for solving linear BVPs that enables facile computation of new solutions to systems under any external forcing. However, fundamental Green's function solutions for nonlinear BVPs are not feasible since linear superposition no longer holds. In this work, we propose a flexible deep learning approach to solve nonlinear BVPs using a dual-autoencoder architecture. The autoencoders discover an invertible coordinate transform that linearizes the nonlinear BVP and identifies both a linear operator $L$ and Green's function $G$ which can be used to solve new nonlinear BVPs. We find that the method succeeds on a variety of nonlinear systems including nonlinear Helmholtz and Sturm--Liouville problems, nonlinear elasticity, and a 2D nonlinear Poisson equation. The method merges the strengths of the universal approximation capabilities of deep learning with the physics knowledge of Green's functions to yield a flexible tool for identifying fundamental solutions to a variety of nonlinear systems.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers

Nonlinear differential equations rarely admit closed-form solutions, thus requiring numerical time-stepping algorithms to approximate solutions. Further, many systems characterized by multiscale physics exhibit dynamics over a vast range of timescales, making numerical integration computationally expensive due to numerical stiffness. In this work, we develop a hierarchy of deep neural network time-steppers to approximate the flow map of the dynamical system over a disparate range of time-scales. The resulting model is purely data-driven and leverages features of the multiscale dynamics, enabling numerical integration and forecasting that is both accurate and highly efficient. Moreover, similar ideas can be used to couple neural network-based models with classical numerical time-steppers. Our multiscale hierarchical time-stepping scheme provides important advantages over current time-stepping algorithms, including (i) circumventing numerical stiffness due to disparate time-scales, (ii) improved accuracy in comparison with leading neural-network architectures, (iii) efficiency in long-time simulation/forecasting due to explicit training of slow time-scale dynamics, and (iv) a flexible framework that is parallelizable and may be integrated with standard numerical time-stepping algorithms. The method is demonstrated on a wide range of nonlinear dynamical systems, including the Van der Pol oscillator, the Lorenz system, the Kuramoto-Sivashinsky equation, and fluid flow pass a cylinder; audio and video signals are also explored. On the sequence generation examples, we benchmark our algorithm against state-of-the-art methods, such as LSTM, reservoir computing, and clockwork RNN. Despite the structural simplicity of our method, it outperforms competing methods on numerical integration.

preprint2020arXiv

Machine Learning and Feature Engineering for Predicting Pulse Status during Chest Compressions

Objective: Current resuscitation protocols require pausing chest compressions during cardiopulmonary resuscitation (CPR) to check for a pulse. However, pausing CPR during a pulseless rhythm can worsen patient outcome. Our objective is to design an ECG-based algorithm that predicts pulse status during uninterrupted CPR and evaluate its performance. Methods: We evaluated 383 patients being treated for out-of-hospital cardiac arrest using defibrillator data. We collected paired and immediately adjacent ECG segments having an organized rhythm. Segments were collected during the 10s period of ongoing CPR prior to a pulse check, and 5s segments without CPR during the pulse check. ECG segments with or without a pulse were identified by the audio annotation of a paramedic's pulse check findings and recorded blood pressures. We developed an algorithm to predict the clinical pulse status based on the wavelet transform of the bandpass-filtered ECG, applying principle component analysis. We then trained a linear discriminant model using 3 principle component modes. Model performance was evaluated on test group segments with and without CPR using receiver operating curves and according to the initial arrest rhythm. Results: There were 230 patients (540 pulse checks) in the training set and 153 patients (372 pulse checks) in the test set. Overall 38% (351/912) of checks had a spontaneous pulse. The areas under the receiver operating characteristic curve (AUCs) for predicting pulse status with and without CPR on test data were 0.84 and 0.89, respectively. Conclusion: A novel ECG-based algorithm demonstrates potential to improve resuscitation by predicting presence of a spontaneous pulse without pausing CPR. Significance: Our algorithm predicts pulse status during uninterrupted CPR, allowing for CPR to proceed unimpeded by pauses to check for a pulse and potentially improving resuscitation performance.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

Numerical differentiation of noisy data: A unifying multi-objective optimization framework

Computing derivatives of noisy measurement data is ubiquitous in the physical, engineering, and biological sciences, and it is often a critical step in developing dynamic models or designing control. Unfortunately, the mathematical formulation of numerical differentiation is typically ill-posed, and researchers often resort to an \textit{ad hoc} process for choosing one of many computational methods and its parameters. In this work, we take a principled approach and propose a multi-objective optimization framework for choosing parameters that minimize a loss function to balance the faithfulness and smoothness of the derivative estimate. Our framework has three significant advantages. First, the task of selecting multiple parameters is reduced to choosing a single hyper-parameter. Second, where ground-truth data is unknown, we provide a heuristic for automatically selecting this hyper-parameter based on the power spectrum and temporal resolution of the data. Third, the optimal value of the hyper-parameter is consistent across different differentiation methods, thus our approach unifies vastly different numerical differentiation methods and facilitates unbiased comparison of their results. Finally, we provide an extensive open-source Python library \texttt{pynumdiff} to facilitate easy application to diverse datasets (https://github.com/florisvb/PyNumDiff).

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2020arXiv

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.

preprint2019arXiv

Data-driven discovery of coordinates and governing equations

The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam's razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom autoencoder to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system. We demonstrate this approach on several example high-dimensional dynamical systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. It is the first method of its kind to place the discovery of coordinates and models on an equal footing.

preprint2019arXiv

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.

preprint2019arXiv

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.

preprint2019arXiv

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.

preprint2017arXiv

Data-Driven Sparse Sensor Placement for Reconstruction

Optimal sensor placement is a central challenge in the design, prediction, estimation, and control of high-dimensional systems. High-dimensional states can often leverage a latent low-dimensional representation, and this inherent compressibility enables sparse sensing. This article explores optimized sensor placement for signal reconstruction based on a tailored library of features extracted from training data. Sparse point sensors are discovered using the singular value decomposition and QR pivoting, which are two ubiquitous matrix computations that underpin modern linear dimensionality reduction. Sparse sensing in a tailored basis is contrasted with compressed sensing, a universal signal recovery method in which an unknown signal is reconstructed via a sparse representation in a universal basis. Although compressed sensing can recover a wider class of signals, we demonstrate the benefits of exploiting known patterns in data with optimized sensing. In particular, drastic reductions in the required number of sensors and improved reconstruction are observed in examples ranging from facial images to fluid vorticity fields. Principled sensor placement may be critically enabling when sensors are costly and provides faster state estimation for low-latency, high-bandwidth control. MATLAB code is provided for all examples.

preprint2017arXiv

Environment Identification in Flight using Sparse Approximation of Wing Strain

This paper addresses the problem of identifying different flow environments from sparse data collected by wing strain sensors. Insects regularly perform this feat using a sparse ensemble of noisy strain sensors on their wing. First, we obtain strain data from numerical simulation of a Manduca sexta hawkmoth wing undergoing different flow environments. Our data-driven method learns low-dimensional strain features originating from different aerodynamic environments using proper orthogonal decomposition (POD) modes in the frequency domain, and leverages sparse approximation to classify a set of strain frequency signatures using a dictionary of POD modes. This bio-inspired machine learning architecture for dictionary learning and sparse classification permits fewer costly physical strain sensors while being simultaneously robust to sensor noise. A measurement selection algorithm identifies frequencies that best discriminate the different aerodynamic environments in low-rank POD feature space. In this manner, sparse and noisy wing strain data can be exploited to robustly identify different aerodynamic environments encountered in flight, providing insight into the stereotyped placement of neurons that act as strain sensors on a Manduca sexta hawkmoth wing.

preprint2017arXiv

Optimized Sampling for Multiscale Dynamics

The characterization of intermittent, multiscale and transient dynamics using data-driven analysis remains an open challenge. We demonstrate an application of the Dynamic Mode Decomposition (DMD) with sparse sampling for the diagnostic analysis of multiscale physics. The DMD method is an ideal spatiotemporal matrix decomposition that correlates spatial features of computational or experimental data to periodic temporal behavior. DMD can be modified into a multiresolution analysis to separate complex dynamics into a hierarchy of multiresolution timescale components, where each level of the hierarchy divides dynamics into distinct background (slow) and foreground (fast) timescales. The multiresolution DMD is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank spatial modes and their temporal Fourier dynamics. Moreover, these multiresolution DMD modes can be used to determined sparse sampling locations which are nearly optimal for dynamic regime classification and full state reconstruction. Specifically, optimized sensors are efficiently chosen using QR column pivots of the DMD library, thus avoiding an NP-hard selection process. We demonstrate the efficacy of the method on several examples, including global sea-surface temperature data, and show that only a small number of sensors are needed for accurate global reconstructions and classification of El Niño events.

preprint2017arXiv

Predicting shim gaps in aircraft assembly with machine learning and sparse sensing

A modern aircraft may require on the order of thousands of custom shims to fill gaps between structural components in the airframe that arise due to manufacturing tolerances adding up across large structures. These shims are necessary to eliminate gaps, maintain structural performance, and minimize pull-down forces required to bring the aircraft into engineering nominal configuration for peak aerodynamic efficiency. Gap filling is a time-consuming process, involving either expensive by-hand inspection or computations on vast quantities of measurement data from increasingly sophisticated metrology equipment. Either case amounts to significant delays in production, with much of the time spent in the critical path of aircraft assembly. This work presents an alternative strategy for predictive shimming, based on machine learning and sparse sensing to first learn gap distributions from historical data, and then design optimized sparse sensing strategies to streamline data collection and processing. This new approach is based on the assumption that patterns exist in shim distributions across aircraft, which may be mined and used to reduce the burden of data collection and processing in future aircraft. Specifically, robust principal component analysis is used to extract low-dimensional patterns in the gap measurements while rejecting outliers. Next, optimized sparse sensors are obtained that are most informative about the dimensions of a new aircraft in these low-dimensional principal components. We demonstrate the success of the proposed approach, called PIXel Identification Despite Uncertainty in Sensor Technology (PIXI-DUST), on historical production data from 54 representative Boeing commercial aircraft. Our algorithm successfully predicts $99\%$ of shim gaps within the desired measurement tolerance using $3\%$ of the laser scan points typically required; all results are cross-validated.