Sparse identification of nonlinear dynamics for model predictive control in the low-data limit

Nonlinear parametric models of viscoelastic fluid flows

Reduced-order models (ROMs) have been widely adopted in fluid mechanics, particularly in the context of Newtonian fluid flows. These models offer the ability to predict complex dynamics, such as instabilities and oscillations, at a considerably reduced computational cost. In contrast, the reduced-order modelling of non-Newtonian viscoelastic fluid flows remains relatively unexplored. This work leverages the sparse identification of nonlinear dynamics (SINDy) algorithm to develop interpretable ROMs for viscoelastic flows. In particular, we explore a benchmark oscillatory viscoelastic flow on the four-roll mill geometry using the classical Oldroyd-B fluid. This flow exemplifies many canonical challenges associated with non-Newtonian flows, including transitions, asymmetries, instabilities, and bifurcations arising from the interplay of viscous and elastic forces, all of which require expensive computations in order to resolve the fast timescales and long transients characteristic of such flows. First, we demonstrate the effectiveness of our data-driven surrogate model to predict the transient evolution and accurately reconstruct the spatial flow field for fixed flow parameters. We then develop a fully parametric, nonlinear model capable of capturing the dynamic variations as a function of the Weissenberg number. While the training data are predominantly concentrated on a limit cycle regime for moderate W i , we show that the parametrized model can be used to extrapolate, accurately predicting the dominant dynamics in the case of high Weissenberg numbers. The proposed methodology represents an initial step in applying machine learning and reduced-order modelling techniques to viscoelastic flows.

Nonlinear model predictive control of a conductance-based neuron model via data-driven forecasting

Objective. Precise control of neural systems is essential to experimental investigations of how the brain controls behavior and holds the potential for therapeutic manipulations to correct aberrant network states. Model predictive control, which employs a dynamical model of the system to find optimal control inputs, has promise for dealing with the nonlinear dynamics, high levels of exogenous noise, and limited information about unmeasured states and parameters that are common in a wide range of neural systems. However, the challenge still remains of selecting the right model, constraining its parameters, and synchronizing to the neural system.Approach. As a proof of principle, we used recent advances in data-driven forecasting to construct a nonlinear machine-learning model of a Hodgkin-Huxley type neuron when only the membrane voltage is observable and there are an unknown number of intrinsic currents.Main Results. We show that this approach is able to learn the dynamics of different neuron types and can be used with model predictive control (MPC) to force the neuron to engage in arbitrary, researcher-defined spiking behaviors.Significance.To the best of our knowledge, this is the first application of nonlinear MPC of a conductance-based model where there is only realistically limited information about unobservable states and parameters.

Model predictive complex system control from observational and interventional data

Complex systems, characterized by intricate interactions among numerous entities, give rise to emergent behaviors whose data-driven modeling and control are of utmost significance, especially when there is abundant observational data but the intervention cost is high. Traditional methods rely on precise dynamical models or require extensive intervention data, often falling short in real-world applications. To bridge this gap, we consider a specific setting of the complex systems control problem: how to control complex systems through a few online interactions on some intervenable nodes when abundant observational data from natural evolution is available. We introduce a two-stage model predictive complex system control framework, comprising an offline pre-training phase that leverages rich observational data to capture spontaneous evolutionary dynamics and an online fine-tuning phase that uses a variant of model predictive control to implement intervention actions. To address the high-dimensional nature of the state-action space in complex systems, we propose a novel approach employing action-extended graph neural networks to model the Markov decision process of complex systems and design a hierarchical action space for learning intervention actions. This approach performs well in three complex system control environments: Boids, Kuramoto, and Susceptible-Infectious-Susceptible (SIS) metapopulation. It offers accelerated convergence, robust generalization, and reduced intervention costs compared to the baseline algorithm. This work provides valuable insights into controlling complex systems with high-dimensional state-action spaces and limited intervention data, presenting promising applications for real-world challenges.

Dynamical system identification, model selection, and model uncertainty quantification by Bayesian inference

This study presents a Bayesian maximum a posteriori (MAP) framework for dynamical system identification from time-series data. This is shown to be equivalent to a generalized Tikhonov regularization, providing a rational justification for the choice of the residual and regularization terms, respectively, from the negative logarithms of the likelihood and prior distributions. In addition to the estimation of model coefficients, the Bayesian interpretation gives access to the full apparatus for Bayesian inference, including the ranking of models, the quantification of model uncertainties, and the estimation of unknown (nuisance) hyperparameters. Two Bayesian algorithms, joint MAP and variational Bayesian approximation, are compared to the least absolute shrinkage and selection operator (LASSO), ridge regression, and the sparse identification of nonlinear dynamics (SINDy) algorithms for sparse regression by application to several dynamical systems with added Gaussian or Laplace noise. For multivariate Gaussian likelihood and prior distributions, the Bayesian formulation gives Gaussian posterior and evidence distributions, in which the numerator terms can be expressed in terms of the Mahalanobis distance or "Gaussian norm" ||y-y^||M-12=(y-y^)⊤M-1(y-y^), where y is a vector variable, y^ is its estimator, and M is the covariance matrix. The posterior Gaussian norm is shown to provide a robust metric for quantitative model selection for the different systems and noise models examined.

Data-driven acoustic control of a spherical bubble using a Koopman linear quadratic regulator

Koopman operator theory has gained interest as a framework for transforming nonlinear dynamics on the state space into linear dynamics on abstract function spaces, which preserves the underlying nonlinear dynamics of the system. These spaces can be approximated through data-driven methodologies, which enables the application of classical linear control strategies to nonlinear systems. Here, a Koopman linear quadratic regulator (KLQR) was used to acoustically control the nonlinear dynamics of a single spherical bubble, as described by the well-known Rayleigh-Plesset equation, with several objectives: (1) simple harmonic oscillation at amplitudes large enough to incite nonlinearities, (2) stabilization of the bubble at a nonequilibrium radius, and (3) periodic and quasiperiodic oscillation with multiple frequency components of arbitrary amplitude. The results demonstrate that the KLQR controller can effectively drive a spherical bubble to radially oscillate according to prescribed trajectories using both broadband and single-frequency acoustic driving. This approach has several advantages over previous efforts to acoustically control bubbles, including the ability to track arbitrary trajectories, robustness, and the use of linear control methods, which do not depend on initial guesses.

From biological data to oscillator models using SINDy

Periodic changes in the concentration or activity of different molecules regulate vital cellular processes such as cell division and circadian rhythms. Developing mathematical models is essential to better understand the mechanisms underlying these oscillations. Recent data-driven methods like SINDy have fundamentally changed model identification, yet their application to experimental biological data remains limited. This study investigates SINDy's constraints by directly applying it to biological oscillatory data. We identify insufficient resolution, noise, dimensionality, and limited prior knowledge as primary limitations. Using various generic oscillator models of different complexity and/or dimensionality, we systematically analyze these factors. We then propose a comprehensive guide for inferring models from biological data, addressing these challenges step by step. Our approach is validated using glycolytic oscillation data from yeast.

Higher-order Granger reservoir computing: simultaneously achieving scalable complex structures inference and accurate dynamics prediction

Recently, machine learning methods, including reservoir computing (RC), have been tremendously successful in predicting complex dynamics in many fields. However, a present challenge lies in pushing for the limit of prediction accuracy while maintaining the low complexity of the model. Here, we design a data-driven, model-free framework named higher-order Granger reservoir computing (HoGRC), which owns two major missions: The first is to infer the higher-order structures incorporating the idea of Granger causality with the RC, and, simultaneously, the second is to realize multi-step prediction by feeding the time series and the inferred higher-order information into HoGRC. We demonstrate the efficacy and robustness of the HoGRC using several representative systems, including the classical chaotic systems, the network dynamical systems, and the UK power grid system. In the era of machine learning and complex systems, we anticipate a broad application of the HoGRC framework in structure inference and dynamics prediction.

Error-Triggered Adaptive Sparse Identification for Predictive Control and Its Application to Multiple Operating Conditions Processes

With the digital transformation of process manufacturing, identifying the system model from process data and then applying to predictive control has become the most dominant approach in process control. However, the controlled plant often operates under changing operating conditions. What is more, there are often unknown operating conditions such as first appearance operating conditions, which make traditional predictive control methods based on identified model difficult to adapt to changing operating conditions. Moreover, the control accuracy is low during operating condition switching. To solve these problems, this article proposes an error-triggered adaptive sparse identification for predictive control (ETASI4PC) method. Specifically, an initial model is established based on sparse identification. Then, a prediction error-triggered mechanism is proposed to monitor operating condition changes in real time. Next, the previously identified model is updated with the fewest modifications by identifying parameter change, structural change, and combination of changes in the dynamical equations, thus achieving precise control to multiple operating conditions. Considering the problem of low control accuracy during the operating condition switching, a novel elastic feedback correction strategy is proposed to significantly improve the control accuracy in the transition period and ensure accurate control under full operating conditions. To verify the superiority of the proposed method, a numerical simulation case and a continuous stirred tank reactor (CSTR) case are designed. Compared with some state-of-the-art methods, the proposed method can rapidly adapt to frequent changes in operating conditions, and it can achieve real-time control effects even for unknown operating conditions such as first appearance operating conditions.

Data-driven discovery of sparse dynamical model of cardiovascular system for model predictive control

Cardiovascular diseases remain the leading cause of death globally. In recent years, vagal nerve stimulation (VNS) has shown promising results in the treatment of a number of cardiovascular diseases. In this approach, mild electrical pulses are sent to the brain via the vagus nerve. This open-loop neurostimulation, however, leads to various side effects due to physiological and inter-patient variability and therefore a closed-loop delivery strategy of electrical pulses that accounts for this variability is desired. In this context, we envision data-driven sparse dynamical model parameterized by patient-specific data as appropriate for use in closed loop controller design. In this work, we build a dynamical model for mean arterial pressure and heart rate using the method sparse identification of nonlinear dynamics (SINDy). As a proxy for real datasets or measurements from a patient, we simulate a mechanistic model from the literature and then discover a data-driven model for predicting mean arterial pressure and heart rate in response to neural stimulus. This discovered model is then used to design a controller to be implemented in closed-loop via model predictive control. We observe that this data-driven model is interpretable, consistent with experiments, provides insights on the sensitivity of different stimulation locations and simplifies the formulation of the optimal control problem. Noting the set-point tracking performance of this closed-loop model-based controller that uses this discovered model, we conclude that the model is adequate in capturing the dynamics of a highly nonlinear cardiovascular system for the purpose of optimal predictive controller design.

A Levenberg-Marquardt Algorithm for Sparse Identification of Dynamical Systems

Low complexity of a system model is essential for its use in real-time applications. However, sparse identification methods commonly have stringent requirements that exclude them from being applied in an industrial setting. In this article, we introduce a flexible method for the sparse identification of dynamical systems described by ordinary differential equations. Our method relieves many of the requirements imposed by other methods that relate to the structure of the model and the dataset, such as fixed sampling rates, full state measurements, and linearity of the model. The Levenberg-Marquardt algorithm is used to solve the identification problem. We show that the Levenberg-Marquardt algorithm can be written in a form that enables parallel computing, which greatly diminishes the time required to solve the identification problem. An efficient backward elimination strategy is presented to construct a lean system model.

Kernel-based learning framework for discovering the governing equations of stochastic jump-diffusion processes directly from data

Discovering the underlying mathematical-physical equations of complex systems directly from observational data has been a challenging inversion problem. We propose a data-driven framework for identifying dynamical information in stochastic diffusion or stochastic jump-diffusion systems. The probability density function is utilized to relate the Kramers-Moyal expansion to the governing equations, and the kernel density estimation method, improved by the Fourier transform idea, is used to extract the Kramers-Moyal coefficients from the time series of the state variables of the system. These coefficients provide the data expression of the governing equations of the system. Then a data-driven sparse identification algorithm is used to reconstruct the underlying dynamic equations. The proposed framework does not rely on prior assumptions, and all results are obtained directly from the data. In addition, we demonstrate its validity and accuracy using illustrative one- and two-dimensional examples.

SPADE4: Sparsity and Delay Embedding Based Forecasting of Epidemics

Predicting the evolution of diseases is challenging, especially when the data availability is scarce and incomplete. The most popular tools for modelling and predicting infectious disease epidemics are compartmental models. They stratify the population into compartments according to health status and model the dynamics of these compartments using dynamical systems. However, these predefined systems may not capture the true dynamics of the epidemic due to the complexity of the disease transmission and human interactions. In order to overcome this drawback, we propose Sparsity and Delay Embedding based Forecasting (SPADE4) for predicting epidemics. SPADE4 predicts the future trajectory of an observable variable without the knowledge of the other variables or the underlying system. We use random features model with sparse regression to handle the data scarcity issue and employ Takens' delay embedding theorem to capture the nature of the underlying system from the observed variable. We show that our approach outperforms compartmental models when applied to both simulated and real data.

Data driven model discovery and interpretation for CAR T-cell killing using sparse identification and latent variables

In the development of cell-based cancer therapies, quantitative mathematical models of cellular interactions are instrumental in understanding treatment efficacy. Efforts to validate and interpret mathematical models of cancer cell growth and death hinge first on proposing a precise mathematical model, then analyzing experimental data in the context of the chosen model. In this work, we present the first application of the sparse identification of non-linear dynamics (SINDy) algorithm to a real biological system in order discover cell-cell interaction dynamics in in vitro experimental data, using chimeric antigen receptor (CAR) T-cells and patient-derived glioblastoma cells. By combining the techniques of latent variable analysis and SINDy, we infer key aspects of the interaction dynamics of CAR T-cell populations and cancer. Importantly, we show how the model terms can be interpreted biologically in relation to different CAR T-cell functional responses, single or double CAR T-cell-cancer cell binding models, and density-dependent growth dynamics in either of the CAR T-cell or cancer cell populations. We show how this data-driven model-discovery based approach provides unique insight into CAR T-cell dynamics when compared to an established model-first approach. These results demonstrate the potential for SINDy to improve the implementation and efficacy of CAR T-cell therapy in the clinic through an improved understanding of CAR T-cell dynamics.

Design, Control, and Experimental Evaluation of A Novel Robotic Glove System for Patients with Brachial Plexus Injuries

This paper presents the development of an exoskeleton glove system for people who suffer from brachial plexus injuries, aiming to assist their lost grasping functionality. The robotic system consists of a portable glove system and an embedded controller. The glove system consists of Linear Series Elastic Actuators (LSEA), Rotary Series Elastic Actuators (RSEA), and optimized finger linkages to provide imitated human motion to each finger and a coupled motion of the hand. The design principles and optimization strategies were investigated to balance functionality, portability, and stability. The model-based force control strategy compensated with a backlash model and model-free force control strategy are presented and compared. Results show that our proposed model-free control method achieves the goal of accurate force control. Finally, experiments were conducted with the prototype of the developed integrated exoskeleton glove system. Results from 3 subjects with 150 trials show that our proposed exoskeleton glove system has the potential to be used as a rehabilitation device for patients.

Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches

Micro-electro-mechanical-systems are complex structures, often involving nonlinearites of geometric and multiphysics nature, that are used as sensors and actuators in countless applications. Starting from full-order representations, we apply deep learning techniques to generate accurate, efficient, and real-time reduced order models to be used for the simulation and optimization of higher-level complex systems. We extensively test the reliability of the proposed procedures on micromirrors, arches, and gyroscopes, as well as displaying intricate dynamical evolutions such as internal resonances. In particular, we discuss the accuracy of the deep learning technique and its ability to replicate and converge to the invariant manifolds predicted using the recently developed direct parametrization approach that allows the extraction of the nonlinear normal modes of large finite element models. Finally, by addressing an electromechanical gyroscope, we show that the non-intrusive deep learning approach generalizes easily to complex multiphysics problems.

Deep learning delay coordinate dynamics for chaotic attractors from partial observable data

A common problem in time-series analysis is to predict dynamics with only scalar or partial observations of the underlying dynamical system. For data on a smooth compact manifold, Takens' theorem proves a time-delayed embedding of the partial state is diffeomorphic to the attractor, although for chaotic and highly nonlinear systems, learning these delay coordinate mappings is challenging. We utilize deep artificial neural networks (ANNs) to learn discrete time maps and continuous time flows of the partial state. Given training data for the full state, we also learn a reconstruction map. Thus, predictions of a time series can be made from the current state and several previous observations with embedding parameters determined from time-series analysis. The state space for time evolution is of comparable dimension to reduced order manifold models. These are advantages over recurrent neural network models, which require a high-dimensional internal state or additional memory terms and hyperparameters. We demonstrate the capacity of deep ANNs to predict chaotic behavior from a scalar observation on a manifold of dimension three via the Lorenz system. We also consider multivariate observations on the Kuramoto-Sivashinsky equation, where the observation dimension required for accurately reproducing dynamics increases with the manifold dimension via the spatial extent of the system.

A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge

Discovering a meaningful symbolic expression that explains experimental data is a fundamental challenge in many scientific fields. We present a novel, open-source computational framework called Scientist-Machine Equation Detector (SciMED), which integrates scientific discipline wisdom in a scientist-in-the-loop approach, with state-of-the-art symbolic regression (SR) methods. SciMED combines a wrapper selection method, that is based on a genetic algorithm, with automatic machine learning and two levels of SR methods. We test SciMED on five configurations of a settling sphere, with and without aerodynamic non-linear drag force, and with excessive noise in the measurements. We show that SciMED is sufficiently robust to discover the correct physically meaningful symbolic expressions from the data, and demonstrate how the integration of domain knowledge enhances its performance. Our results indicate better performance on these tasks than the state-of-the-art SR software packages , even in cases where no knowledge is integrated. Moreover, we demonstrate how SciMED can alert the user about possible missing features, unlike the majority of current SR systems.

Implementation of Kalman Filtering with Spiking Neural Networks

A Kalman filter can be used to fill space-state reconstruction dynamics based on knowledge of a system and partial measurements. However, its performance relies on accurate modeling of the system dynamics and a proper characterization of the uncertainties, which can be hard to obtain in real-life scenarios. In this work, we explore how the values of a Kalman gain matrix can be estimated by using spiking neural networks through a combination of biologically plausible neuron models with spike-time-dependent plasticity learning algorithms. The performance of proposed neural architecture is verified with simulations of some representative nonlinear systems, which show promising results. This approach traces a path for its implementation in neuromorphic analog hardware that can learn and reconstruct partial and changing dynamics of a system without the massive power consumption that is typically needed in a Von Neumann-based computer architecture.

An improved sparse identification of nonlinear dynamics with Akaike information criterion and group sparsity

A crucial challenge encountered in diverse areas of engineering applications involves speculating the governing equations based upon partial observations. On this basis, a variant of the sparse identification of nonlinear dynamics (SINDy) algorithm is developed. First, the Akaike information criterion (AIC) is integrated to enforce model selection by hierarchically ranking the most informative model from several manageable candidate models. This integration avoids restricting the number of candidate models, which is a disadvantage of the traditional methods for model selection. The subsequent procedure expands the structure of dynamics from ordinary differential equations (ODEs) to partial differential equations (PDEs), while group sparsity is employed to identify the nonconstant coefficients of partial differential equations. Of practical consideration within an integrated frame is data processing, which tends to treat noise separate from signals and tends to parametrize the noise probability distribution. In particular, the coefficients of a species of canonical ODEs and PDEs, such as the Van der Pol, Rössler, Burgers' and Kuramoto-Sivashinsky equations, can be identified correctly with the introduction of noise. Furthermore, except for normal noise, the proposed approach is able to capture the distribution of uniform noise. In accordance with the results of the experiments, the computational speed is markedly advanced and possesses robustness.

Online Weak-form Sparse Identification of Partial Differential Equations

This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in the sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the ℓ 0 -pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions.

Data-driven discovery of the governing equations of dynamical systems via moving horizon optimization

Discovering the governing laws underpinning physical and chemical phenomena entirely from data is a key step towards understanding and ultimately controlling systems in science and engineering. Noisy measurements and complex, highly nonlinear underlying dynamics hinder the identification of such governing laws. In this work, we introduce a machine learning framework rooted in moving horizon nonlinear optimization for identifying governing equations in the form of ordinary differential equations from noisy experimental data sets. Our approach evaluates sequential subsets of measurement data, and exploits statistical arguments to learn truly parsimonious governing equations from a large dictionary of basis functions. The proposed framework reduces gradient approximation errors by implicitly embedding an advanced numerical discretization scheme, which improves robustness to noise as well as to model stiffness. Canonical nonlinear dynamical system examples are used to demonstrate that our approach can accurately recover parsimonious governing laws under increasing levels of measurement noise, and outperform state of the art frameworks in the literature. Further, we consider a non-isothermal chemical reactor example to demonstrate that the proposed framework can cope with basis functions that have nonlinear (unknown) parameterizations.

Data-Driven Predictive Control of Interconnected Systems Using the Koopman Operator

Interconnected systems are widespread in modern technological systems. Designing a reliable control strategy requires modeling and analysis of the system, which can be a complicated, or even impossible, task in some cases. However, current technological developments in data sensing, processing, and storage make data-driven control techniques an appealing alternative solution. In this work, a design methodology of a decentralized control strategy is developed for interconnected systems based only on local and interconnection time series. Then, the optimization problem associated with the predictive control design is defined. Finally, an extension to interconnected systems coupled through their input signals is discussed. Simulations of two coupled Duffing oscillators, a bipedal locomotion model, and a four water tank system show the effectiveness of the approach.

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 the 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 datasets 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 to 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.

Automatic differentiation to simultaneously identify nonlinear dynamics and extract noise probability distributions from data

The sparse identification of nonlinear dynamics (SINDy) is a regression framework for the discovery of parsimonious dynamic models and governing equations from time-series data. As with all system identification methods, noisy measurements compromise the accuracy and robustness of the model discovery procedure. In this work we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al (2019 J. Computat. Phys. 396 483–506) for simultaneously (1) denoising the data, (2) learning and parametrizing the noise probability distribution, and (3) identifying the underlying parsimonious dynamical system responsible for generating the time-series data. Thus within an integrated optimization framework, noise can be separated from signal, resulting in an architecture that is approximately twice as robust to noise as state-of-the-art methods, handling as much as 40% noise on a given time-series signal and explicitly parametrizing the noise probability distribution. We demonstrate this approach on several numerical examples, from Lotka-Volterra models to the spatio-temporal Lorenz 96 model. Further, we show the method can learn a diversity of probability distributions for the measurement noise, including Gaussian, uniform, Gamma, and Rayleigh distributions.

Deeptime: a Python library for machine learning dynamical models from time series data

Generation and analysis of time-series data is relevant to many quantitative fields ranging from economics to fluid mechanics. In the physical sciences, structures such as metastable and coherent sets, slow relaxation processes, collective variables, dominant transition pathways or manifolds and channels of probability flow can be of great importance for understanding and characterizing the kinetic, thermodynamic and mechanistic properties of the system. Deeptime is a general purpose Python library offering various tools to estimate dynamical models based on time-series data including conventional linear learning methods, such as Markov state models (MSMs), Hidden Markov Models and Koopman models, as well as kernel and deep learning approaches such as VAMPnets and deep MSMs. The library is largely compatible with scikit-learn, having a range of Estimator classes for these different models, but in contrast to scikit-learn also provides deep Model classes, e.g. in the case of an MSM, which provide a multitude of analysis methods to compute interesting thermodynamic, kinetic and dynamical quantities, such as free energies, relaxation times and transition paths. The library is designed for ease of use but also easily maintainable and extensible code. In this paper we introduce the main features and structure of the deeptime software. Deeptime can be found under https://deeptime-ml.github.io/.

Finite-horizon, energy-efficient trajectories in unsteady flows

Intelligent mobile sensors, such as uninhabited aerial or underwater vehicles, are becoming prevalent in environmental sensing and monitoring applications. These active sensing platforms operate in unsteady fluid flows, including windy urban environments, hurricanes and ocean currents. Often constrained in their actuation capabilities, the dynamics of these mobile sensors depend strongly on the background flow, making their deployment and control particularly challenging. Therefore, efficient trajectory planning with partial knowledge about the background flow is essential for teams of mobile sensors to adaptively sense and monitor their environments. In this work, we investigate the use of finite-horizon model predictive control (MPC) for the energy-efficient trajectory planning of an active mobile sensor in an unsteady fluid flow field. We uncover connections between trajectories optimized over a finite-time horizon and finite-time Lyapunov exponents of the background flow, confirming that energy-efficient trajectories exploit invariant coherent structures in the flow. We demonstrate our findings on the unsteady double gyre vector field, which is a canonical model for chaotic mixing in the ocean. We present an exhaustive search through critical MPC parameters including the prediction horizon, maximum sensor actuation, and relative penalty on the accumulated state error and actuation effort. We find that even relatively short prediction horizons can often yield energy-efficient trajectories. We also explore these connections on a three-dimensional flow and ocean flow data from the Gulf of Mexico. These results are promising for the adaptive planning of energy-efficient trajectories for swarms of mobile sensors in distributed sensing and monitoring.

Uncertainty Analysis and Experimental Validation of Identifying the Governing Equation of an Oscillator Using Sparse Regression

In recent years, the rapid growth of computing technology has enabled identifying mathematical models for vibration systems using measurement data instead of domain knowledge. Within this category, the method Sparse Identification of Nonlinear Dynamical Systems (SINDy) shows potential for interpretable identification. Therefore, in this work, a procedure of system identification based on the SINDy framework is developed and validated on a single-mass oscillator. To estimate the parameters in the SINDy model, two sparse regression methods are discussed. Compared with the Least Squares method with Sequential Threshold (LSST), which is the original estimation method from SINDy, the Least Squares method Post-LASSO (LSPL) shows better performance in numerical Monte Carlo Simulations (MCSs) of a single-mass oscillator in terms of sparseness, convergence, identified eigenfrequency, and coefficient of determination. Furthermore, the developed method SINDy-LSPL was successfully implemented with real measurement data of a single-mass oscillator with known theoretical parameters. The identified parameters using a sweep signal as excitation are more consistent and accurate than those identified using impulse excitation. In both cases, there exists a dependency of the identified parameter on the excitation amplitude that should be investigated in further research.

Challenges in dynamic mode decomposition

Dynamic mode decomposition (DMD) is a powerful tool for extracting spatial and temporal patterns from multi-dimensional time series, and it has been used successfully in a wide range of fields, including fluid mechanics, robotics and neuroscience. Two of the main challenges remaining in DMD research are noise sensitivity and issues related to Krylov space closure when modelling nonlinear systems. Here, we investigate the combination of noise and nonlinearity in a controlled setting, by studying a class of systems with linear latent dynamics which are observed via multinomial observables. Our numerical models include system and measurement noise. We explore the influences of dataset metrics, the spectrum of the latent dynamics, the normality of the system matrix and the geometry of the dynamics. Our results show that even for these very mildly nonlinear conditions, DMD methods often fail to recover the spectrum and can have poor predictive ability. Our work is motivated by our experience modelling multilegged robot data, where we have encountered great difficulty in reconstructing time series for oscillatory systems with intermediate transients, which decay only slightly faster than a period.

Data-enabled predictive control for quadcopters

We study the application of a data-enabled predictive control (DeePC) algorithm for position control of real-world nano-quadcopters. The DeePC algorithm is a finite-horizon, optimal control method that uses input/output measurements from the system to predict future trajectories without the need for system identification or state estimation. The algorithm predicts future trajectories of the quadcopter by linearly combining previously measured trajectories (motion primitives). We illustrate the necessity of a regularized variant of the DeePC algorithm to handle the nonlinear nature of the real-world quadcopter dynamics with noisy measurements. Simulation-based analysis is used to gain insights into the effects of regularization, and experimental results validate that these insights carry over to the real-world quadcopter. Moreover, we demonstrate the reliability of the DeePC algorithm by collecting a new set of input/output measurements for every real-world experiment performed. The performance of the DeePC algorithm is compared to Model Predictive Control based on a first-principles model of the quadcopter. The results are demonstrated with a video of successful trajectory tracking of the real-world quadcopter.

Physics-informed learning of governing equations from scarce data

Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel approach called physics-informed neural network with sparse regression to discover governing partial differential equations from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this discovery approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, automatic differentiation and sparse regression to approximate the solution of system variables, compute essential derivatives, as well as identify the key derivative terms and parameters that form the structure and explicit expression of the equations. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of partial differential equation systems with different levels of data scarcity and noise accounting for different initial/boundary conditions. The resulting computational framework shows the potential for closed-form model discovery in practical applications where large and accurate datasets are intractable to capture.

Bayesian uncertainty quantification for data-driven equation learning

Equation learning aims to infer differential equation models from data. While a number of studies have shown that differential equation models can be successfully identified when the data are sufficiently detailed and corrupted with relatively small amounts of noise, the relationship between observation noise and uncertainty in the learned differential equation models remains unexplored. We demonstrate that for noisy datasets there exists great variation in both the structure of the learned differential equation models and their parameter values. We explore how to exploit multiple datasets to quantify uncertainty in the learned models, and at the same time draw mechanistic conclusions about the target differential equations. We showcase our results using simulation data from a relatively straightforward agent-based model (ABM) which has a well-characterized partial differential equation description that provides highly accurate predictions of averaged ABM behaviours in relevant regions of parameter space. Our approach combines equation learning methods with Bayesian inference approaches so that a quantification of uncertainty can be given by the posterior parameter distribution of the learned model.

Identifying indicator species in ecological habitats using Deep Optimal Feature Learning

Much of the current research on supervised modelling is focused on maximizing outcome prediction accuracy. However, in engineering disciplines, an arguably more important goal is that of feature extraction, the identification of relevant features associated with the various outcomes. For instance, in microbial communities, the identification of keystone species can often lead to improved prediction of future behavioral shifts. This paper proposes a novel feature extractor based on Deep Learning, which is largely agnostic to underlying assumptions regarding the training data. Starting from a collection of microbial species abundance counts, the Deep Learning model first trains itself to classify the selected distinct habitats. It then identifies indicator species associated with the habitats. The results are then compared and contrasted with those obtained by traditional statistical techniques. The indicator species are similar when compared at top taxonomic levels such as Domain and Phylum, despite visible differences in lower levels such as Class and Order. More importantly, when our estimated indicators are used to predict final habitat labels using simpler models (such as Support Vector Machines and traditional Artificial Neural Networks), the prediction accuracy is improved. Overall, this study serves as a preliminary step that bridges modern, black-box Machine Learning models with traditional, domain expertise-rich techniques.

Control of stochastic quantum dynamics by differentiable programming

Control of the stochastic dynamics of a quantum system is indispensable in fields such as quantum information processing and metrology. However, there is no general ready-made approach to the design of efficient control strategies. Here, we propose a framework for the automated design of control schemes based on differentiable programming. We apply this approach to the state preparation and stabilization of a qubit subjected to homodyne detection. To this end, we formulate the control task as an optimization problem where the loss function quantifies the distance from the target state, and we employ neural networks (NNs) as controllers. The system’s time evolution is governed by a stochastic differential equation (SDE). To implement efficient training, we backpropagate the gradient information from the loss function through the SDE solver using adjoint sensitivity methods. As a first example, we feed the quantum state to the controller and focus on different methods of obtaining gradients. As a second example, we directly feed the homodyne detection signal to the controller. The instantaneous value of the homodyne current contains only very limited information on the actual state of the system, masked by unavoidable photon-number fluctuations. Despite the resulting poor signal-to-noise ratio, we can train our controller to prepare and stabilize the qubit to a target state with a mean fidelity of around 85%. We also compare the solutions found by the NN to a hand-crafted control strategy.

Sparse nonlinear models of chaotic electroconvection

Convection is a fundamental fluid transport phenomenon, where the large-scale motion of a fluid is driven, for example, by a thermal gradient or an electric potential. Modelling convection has given rise to the development of chaos theory and the reduced-order modelling of multiphysics systems; however, these models have been limited to relatively simple thermal convection phenomena. In this work, we develop a reduced-order model for chaotic electroconvection at high electric Rayleigh number. The chaos in this system is related to the standard Lorenz model obtained from Rayleigh-Benard convection, although our system is driven by a more complex three-way coupling between the fluid, the charge density, and the electric field. Coherent structures are extracted from temporally and spatially resolved charge density fields via proper orthogonal decomposition (POD). A nonlinear model is then developed for the chaotic time evolution of these coherent structures using the sparse identification of nonlinear dynamics (SINDy) algorithm, constrained to preserve the symmetries observed in the original system. The resulting model exhibits the dominant chaotic dynamics of the original high-dimensional system, capturing the essential nonlinear interactions with a simple reduced-order model.

Discovering dynamic models of COVID-19 transmission

Existing models about the dynamics of COVID‐19 transmission often assume the mechanism of virus transmission and the form of the differential equations. These assumptions are hard to verify. Due to the biases of country‐level data, it is inaccurate to construct the global dynamic of COVID‐19. This research aims to provide a robust data‐driven global model of the transmission dynamics. We apply sparse identification of nonlinear dynamics (SINDy) to model the dynamics of COVID‐19 global transmission. One advantage is that we can discover the nonlinear dynamics from data without assumptions in the form of the governing equations. To overcome the problem of biased country‐level data on the number of reported cases, we propose a robust global model of the dynamics by using maximin aggregation. Real data analysis shows the efficiency of our model.

Trigonometric Embeddings in Polynomial Extended Mode Decomposition—Experimental Application to an Inverted Pendulum

The extended dynamic mode decomposition algorithm is a tool for accurately approximating the point spectrum of the Koopman operator. This algorithm provides an approximate linear expansion of non-linear discrete-time systems, which can be useful for system analysis and controller design. The accuracy of this algorithm depends heavily on the availability of a set of basis functions that provide the ability to capture the nonlinear dynamics of the underlying system. Recently, the use of orthogonal polynomials, along with reduction techniques for the dimension and maximum order of the polynomial basis, have been successfully used to approximate nonlinear systems with the additional benefit of using smaller datasets. This paper expands the current methods for selecting the set of observables for nonlinear systems with periodic behavior, which is prone to a representation in terms of trigonometric functions. The benefit of working with orthogonal polynomials is preserved by embedding the trigonometric functions into the orthogonal basis. The algorithm is illustrated with the data-driven modelling of an inverted pendulum in simulation and real-life experiments.

Using First Principles for Deep Learning and Model-Based Control of Soft Robots

Model-based optimal control of soft robots may enable compliant, underdamped platforms to operate in a repeatable fashion and effectively accomplish tasks that are otherwise impossible for soft robots. Unfortunately, developing accurate analytical dynamic models for soft robots is time-consuming, difficult, and error-prone. Deep learning presents an alternative modeling approach that only requires a time history of system inputs and system states, which can be easily measured or estimated. However, fully relying on empirical or learned models involves collecting large amounts of representative data from a soft robot in order to model the complex state space–a task which may not be feasible in many situations. Furthermore, the exclusive use of empirical models for model-based control can be dangerous if the model does not generalize well. To address these challenges, we propose a hybrid modeling approach that combines machine learning methods with an existing first-principles model in order to improve overall performance for a sampling-based non-linear model predictive controller. We validate this approach on a soft robot platform and demonstrate that performance improves by 52% on average when employing the combined model.

Advances in monitoring and control of refolding kinetics combining PAT and modeling

Overexpression of recombinant proteins in Escherichia coli results in misfolded and non-active protein aggregates in the cytoplasm, so-called inclusion bodies (IB). In recent years, a change in the mindset regarding IBs could be observed: IBs are no longer considered an unwanted waste product, but a valid alternative to produce a product with high yield, purity, and stability in short process times. However, solubilization of IBs and subsequent refolding is necessary to obtain a correctly folded and active product. This protein refolding process is a crucial downstream unit operation-commonly done as a dilution in batch or fed-batch mode. Drawbacks of the state-of-the-art include the following: the large volume of buffers and capacities of refolding tanks, issues with uniform mixing, challenging analytics at low protein concentrations, reaction kinetics in non-usable aggregates, and generally low re-folding yields. There is no generic platform procedure available and a lack of robust control strategies. The introduction of Quality by Design (QbD) is the method-of-choice to provide a controlled and reproducible refolding environment. However, reliable online monitoring techniques to describe the refolding kinetics in real-time are scarce. In our view, only monitoring and control of re-folding kinetics can ensure a productive, scalable, and versatile platform technology for re-folding processes. For this review, we screened the current literature for a combination of online process analytical technology (PAT) and modeling techniques to ensure a controlled refolding process. Based on our research, we propose an integrated approach based on the idea that all aspects that cannot be monitored directly are estimated via digital twins and used in real-time for process control. KEY POINTS: • Monitoring and a thorough understanding of refolding kinetics are essential for model-based control of refolding processes. • The introduction of Quality by Design combining Process Analytical Technology and modeling ensures a robust platform for inclusion body refolding.

Model predictive control in pharmaceutical continuous manufacturing: A review from a user's perspective

Pharmaceutical continuous manufacturing is considered as an emerging technology by the regulatory agencies, which have defined a framework guided by an effective quality risk management. With the understanding of process dynamics and the appropriate control strategy, pharmaceutical continuous manufacturing is able to tackle the Quality-by-Design paradigm that paves the way to the future smart manufacturing described by Quality-by-Control. The introduction of soft sensors seems to be a helpful tool to reach smart manufacturing. In fact, soft sensors have the ability to keep the quality attributes of the final drug product as close as possible to their references set by regulatory agencies and to mitigate the undesired events by potentially discard out of specification products. Within this review, challenges related to implementing these technologies are discussed. Then, automation control strategies for pharmaceutical continuous manufacturing are presented and discussed: current control tools such as the proportional integral derivative controllers are compared to advanced control techniques like model predictive control, which holds promise to be an advanced automation concept for pharmaceutical continuous manufacturing. Finally, industrial applications of model predictive control in pharmaceutical continuous manufacturing are outlined. Simulations studies as well as real implementation on pharmaceutical plant are gathered from the control of one single operation unit such as the tablet press to the control of a full direct compaction line. Model predictive control is a key to enable the industrial revolution or Industry 4.0.

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.

Predictive Control of District Heating System Using Multi-Stage Nonlinear Approximation with Selective Memory

Innovative heating networks with a hybrid generation park can make an important contribution to the energy turnaround. By integrating heat from several heat generators and a high proportion of different renewable energies, they also have a high degree of flexibility. Optimizing the operation of such systems is a complex task due to the diversity of producers, the use of storage systems with stratified charging and continuous changes in system properties. Besides, it is necessary to consider conflicting economic and ecological targets. Operational optimization of district heating systems using nonlinear models is underrepresented in practice and science. Considering ecological and economic targets, the current work focuses on developing a procedure for an operational optimization, which ensures a continuous optimal operation of the heat and power generators of a local heating network. The approach presented uses machine learning methods, including Gaussian process regressions for a repeatedly updated multi-stage approximation of the nonlinear system behavior. For the formation of the approximation models, a selection algorithm is utilized to choose only essential and current process data. By using a global optimization algorithm, a multi-objective optimal setting of the controllable variables of the system can be found in feasible time. Implemented in the control system of a dynamic simulation, significant improvements of the target variables (operating costs, CO2 emissions) can be seen in comparison with a standard control system. The investigation of different scenarios illustrates the high relevance of the presented methodology.

Koopman analysis in oscillator synchronization

Synchronization is an important dynamical phenomenon in coupled nonlinear systems, which has been studied extensively in recent years. However, analysis focused on individual orbits seems hard to extend to complex systems, while a global statistical approach is overly cursory. Koopman operator technique seems to balance well the two approaches. In this paper we extend Koopman analysis to the study of synchronization of coupled oscillators by extracting important eigenvalues and eigenfunctions from the observed time series. A renormalization group analysis is designed to derive an analytic approximation of the eigenfunction in the case of weak coupling that dominates the oscillation. For moderate or strong couplings, numerical computation further confirms the importance of the average frequencies and the associated eigenfunctions. The synchronization transition points could be located with quite high accuracy by checking the correlation of neighboring eigenfunctions at different coupling strengths, which is readily applied to other nonlinear systems.