Mulugeta Haile

Huazhen Fang, Ning Tian, Yebin Wang et al.

This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date, one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective, which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics (e.g., mean and covariance) conditioned on the system's measurement data. This article offers a systematic introduction of the Bayesian state estimation framework and reviews various Kalman filtering (KF) techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including the Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation forward to more complicated problems such as simultaneous state and parameter/input estimation.

Ling-Wei Kong, Yang Weng, Bryan Glaz et al.

We articulate the design imperatives for machine-learning based digital twins for nonlinear dynamical systems subject to external driving, which can be used to monitor the ``health'' of the target system and anticipate its future collapse. We demonstrate that, with single or parallel reservoir computing configurations, the digital twins are capable of challenging forecasting and monitoring tasks. Employing prototypical systems from climate, optics and ecology, we show that the digital twins can extrapolate the dynamics of the target system to certain parameter regimes never experienced before, make continual forecasting/monitoring with sparse real-time updates under non-stationary external driving, infer hidden variables and accurately predict their dynamical evolution, adapt to different forms of external driving, and extrapolate the global bifurcation behaviors to systems of some different sizes. These features make our digital twins appealing in significant applications such as monitoring the health of critical systems and forecasting their potential collapse induced by environmental changes.

Zheng-Meng Zhai, Mohammadamin Moradi, Ling-Wei Kong et al.

Nonlinear tracking control enabling a dynamical system to track a desired trajectory is fundamental to robotics, serving a wide range of civil and defense applications. In control engineering, designing tracking control requires complete knowledge of the system model and equations. We develop a model-free, machine-learning framework to control a two-arm robotic manipulator using only partially observed states, where the controller is realized by reservoir computing. Stochastic input is exploited for training, which consists of the observed partial state vector as the first and its immediate future as the second component so that the neural machine regards the latter as the future state of the former. In the testing (deployment) phase, the immediate-future component is replaced by the desired observational vector from the reference trajectory. We demonstrate the effectiveness of the control framework using a variety of periodic and chaotic signals, and establish its robustness against measurement noise, disturbances, and uncertainties.

Zheng-Meng Zhai, Mohammadamin Moradi, Bryan Glaz et al.

Complex and nonlinear dynamical systems often involve parameters that change with time, accurate tracking of which is essential to tasks such as state estimation, prediction, and control. Existing machine-learning methods require full state observation of the underlying system and tacitly assume adiabatic changes in the parameter. Formulating an inverse problem and exploiting reservoir computing, we develop a model-free and fully data-driven framework to accurately track time-varying parameters from partial state observation in real time. In particular, with training data from a subset of the dynamical variables of the system for a small number of known parameter values, the framework is able to accurately predict the parameter variations in time. Low- and high-dimensional, Markovian and non-Markovian nonlinear dynamical systems are used to demonstrate the power of the machine-learning based parameter-tracking framework. Pertinent issues affecting the tracking performance are addressed.

Gregory Barber, Mulugeta A. Haile, Tzikang Chen

Given an unknown dynamic system such as a coupled harmonic oscillator with $n$ springs and point masses. We are often interested in gaining insights into its physical parameters, i.e. stiffnesses and masses, by observing trajectories of motion. How do we achieve this from video frames or time-series data and without the knowledge of the dynamics model? We present a neural framework for estimating physical parameters in a manner consistent with the underlying physics. The neural framework uses a deep latent variable model to disentangle the system physical parameters from canonical coordinate observations. It then returns a Hamiltonian parameterization that generalizes well with respect to the discovered physical parameters. We tested our framework with simple harmonic oscillators, $n=1$, and noisy observations and show that it discovers the underlying system parameters and generalizes well with respect to these discovered parameters. Our model also extrapolates the dynamics of the system beyond the training interval and outperforms a non-physically constrained baseline model. Our source code and datasets can be found at this URL: https://github.com/gbarber94/ConSciNet.

Shirin Panahi, Ling-Wei Kong, Bryan Glaz et al.

For anticipating critical transitions in complex dynamical systems, the recent approach of parameter-driven reservoir computing requires explicit knowledge of the bifurcation parameter. We articulate a framework combining a variational autoencoder (VAE) and reservoir computing to address this challenge. In particular, the driving factor is detected from time series using the VAE in an unsupervised-learning fashion and the extracted information is then used as the parameter input to the reservoir computer for anticipating the critical transition. We demonstrate the power of the unsupervised learning scheme using prototypical dynamical systems including the spatiotemporal Kuramoto-Sivashinsky system. The scheme can also be extended to scenarios where the target system is driven by several independent parameters or with partial state observations.

Shirin Panahi, Ling-Wei Kong, Mohammadamin Moradi et al.

Anticipating a tipping point, a transition from one stable steady state to another, is a problem of broad relevance due to the ubiquity of the phenomenon in diverse fields. The steady-state nature of the dynamics about a tipping point makes its prediction significantly more challenging than predicting other types of critical transitions from oscillatory or chaotic dynamics. Exploiting the benefits of noise, we develop a general data-driven and machine-learning approach to predicting potential future tipping in nonautonomous dynamical systems and validate the framework using examples from different fields. As an application, we address the problem of predicting the potential collapse of the Atlantic Meridional Overturning Circulation (AMOC), possibly driven by climate-induced changes in the freshwater input to the North Atlantic. Our predictions based on synthetic and currently available empirical data place a potential collapse window spanning from 2040 to 2065, in consistency with the results in the current literature.

Gregory Barber, Mulugeta A. Haile, Tzikang Chen

Neural networks often require large amounts of data to generalize and can be ill-suited for modeling small and noisy experimental datasets. Standard network architectures trained on scarce and noisy data will return predictions that violate the underlying physics. In this paper, we present methods for embedding even--odd symmetries and conservation laws in neural networks and propose novel extensions and use cases for physical constraint embedded neural networks. We design an even--odd decomposition architecture for disentangling a neural network parameterized function into its even and odd components and demonstrate that it can accurately infer symmetries without prior knowledge. We highlight the noise resilient properties of physical constraint embedded neural networks and demonstrate their utility as physics-informed noise regulators. Here we employed a conservation of energy constraint embedded network as a physics-informed noise regulator for a symbolic regression task. We showed that our approach returns a symbolic representation of the neural network parameterized function that aligns well with the underlying physics while outperforming a baseline symbolic regression approach.

Chen-Di Han, Bryan Glaz, Mulugeta Haile et al.

Interacting spin networks are fundamental to quantum computing. Data-based tomography of time-independent spin networks has been achieved, but an open challenge is to ascertain the structures of time-dependent spin networks using time series measurements taken locally from a small subset of the spins. Physically, the dynamical evolution of a spin network under time-dependent driving or perturbation is described by the Heisenberg equation of motion. Motivated by this basic fact, we articulate a physics-enhanced machine learning framework whose core is Heisenberg neural networks. In particular, we develop a deep learning algorithm according to some physics motivated loss function based on the Heisenberg equation, which "forces" the neural network to follow the quantum evolution of the spin variables. We demonstrate that, from local measurements, not only the local Hamiltonian can be recovered but the Hamiltonian reflecting the interacting structure of the whole system can also be faithfully reconstructed. We test our Heisenberg neural machine on spin networks of a variety of structures. In the extreme case where measurements are taken from only one spin, the achieved tomography fidelity values can reach about 90%. The developed machine learning framework is applicable to any time-dependent systems whose quantum dynamical evolution is governed by the Heisenberg equation of motion.

Chen-Di Han, Bryan Glaz, Mulugeta Haile et al.

The rapid growth of research in exploiting machine learning to predict chaotic systems has revived a recent interest in Hamiltonian Neural Networks (HNNs) with physical constraints defined by the Hamilton's equations of motion, which represent a major class of physics-enhanced neural networks. We introduce a class of HNNs capable of adaptable prediction of nonlinear physical systems: by training the neural network based on time series from a small number of bifurcation-parameter values of the target Hamiltonian system, the HNN can predict the dynamical states at other parameter values, where the network has not been exposed to any information about the system at these parameter values. The architecture of the HNN differs from the previous ones in that we incorporate an input parameter channel, rendering the HNN parameter--cognizant. We demonstrate, using paradigmatic Hamiltonian systems, that training the HNN using time series from as few as four parameter values bestows the neural machine with the ability to predict the state of the target system in an entire parameter interval. Utilizing the ensemble maximum Lyapunov exponent and the alignment index as indicators, we show that our parameter-cognizant HNN can successfully predict the route of transition to chaos. Physics-enhanced machine learning is a forefront area of research, and our adaptable HNNs provide an approach to understanding machine learning with broad applications.

Levi McClenny, Mulugeta Haile, Vahid Attari et al.

In many real-world applications of deep learning, estimation of a target may rely on various types of input data modes, such as audio-video, image-text, etc. This task can be further complicated by a lack of sufficient data. Here we propose a Deep Multimodal Transfer-Learned Regressor (DMTL-R) for multimodal learning of image and feature data in a deep regression architecture effective at predicting target parameters in data-poor domains. Our model is capable of fine-tuning a given set of pre-trained CNN weights on a small amount of training image data, while simultaneously conditioning on feature information from a complimentary data mode during network training, yielding more accurate single-target or multi-target regression than can be achieved using the images or the features alone. We present results using phase-field simulation microstructure images with an accompanying set of physical features, using pre-trained weights from various well-known CNN architectures, which demonstrate the efficacy of the proposed multimodal approach.

Huazhen Fang, Mulugeta A. Haile, Yebin Wang

Outliers can contaminate the measurement process of many nonlinear systems, which can be caused by sensor errors, model uncertainties, change in ambient environment, data loss or malicious cyber attacks. When the extended Kalman filter (EKF) is applied to such systems for state estimation, the outliers can seriously reduce the estimation accuracy. This paper proposes an innovation saturation mechanism to modify the EKF toward building robustness against outliers. This mechanism applies a saturation function to the innovation process that the EKF leverages to correct the state estimation. As such, when an outlier occurs, the distorting innovation is saturated and thus prevented from damaging the state estimation. The mechanism features an adaptive adjustment of the saturation bound. The design leads to the development robust EKF approaches for continuous- and discrete-time systems. They are proven to be capable of generating bounded-error estimation in the presence of bounded outlier disturbances. An application study about mobile robot localization is presented, with the numerical simulation showing the efficacy of the proposed design. Compared to existing methods, the proposed approaches can effectively reject outliers of various magnitudes, types and durations, at significant computational efficiency and without requiring measurement redundancy.