Complex Nonlinear Dynamical Systems Research Articles

ModelFLOWs-app: Data-driven post-processing and reduced order modelling tools

This article presents an innovative open-source software named ModelFLOWs-app,1 written in Python, which has been created and tested to generate precise and robust hybrid reduced order models (ROMs) fully data-driven. By integrating modal decomposition and deep learning in diverse ways, the software uncovers the fundamental patterns in dynamic systems. This acquired knowledge is then employed to enrich the comprehension of the underlying physics, reconstruct databases from limited measurements, and forecast the progression of system dynamics. The hybrid ROMs produced by ModelFLOWs-app combine experimental and numerical databases, serving as highly accurate alternatives to numerical simulations. As a result, computational expenses are significantly reduced, and the models become powerful tools for optimization and control in various applications. The exceptional capability of ModelFLOWs-app in developing reliable data-driven hybrid ROMs has been demonstrated across a wide range of applications, making it a valuable resource for understanding complex nonlinear dynamical systems and providing insights in diverse domains. This article presents the mathematical background, as well as a review of some examples of applications. Program summaryProgram title: ModelFLOWs-appCPC Library link to program files:https://doi.org/10.17632/49tzcc8sf3.1Developer's repository link:github.com/modelflows/ModelFLOWs-appLicensing provisions: MIT licenseProgramming language: PythonSupplementary material: Tutorial, example datasets.Nature of problem: ModelFLOWs-app is an open-source Software for data post-processing, patterns identification and development of reduced order models using modal decomposition and deep learning architectures. ModelFLOWs-app provides its users with a user-friendly interface to efficiently identify patterns in data, reconstruct data by repairing or enhancing it, and make predictions based on the available data.Solution method: ModelFLOWs-app methodological framework is formed by two big modules: Module 1 uses modal decomposition methods, and Module 2 is formed by hybrid machine learning tools, which combine modal decomposition with deep learning architectures. Each module solves three different applications: (1) patterns identification, suitable to study the physics behind the data analysed; (2) data reconstruction, capable to reconstruct two- or three- dimensional databases from a set of selected points, using data from sensors, or repairing missing data; (3) data forecasting, which builds reduced order models (ROMs) to predict the spatio-temporal evolution of the signal analysed.Additional comments including restrictions and unusual features: ModelFLOWs-app offers an intuitive and user-friendly interface, making it accessible to users with varying levels of technical expertise. It is compatible with Windows, macOS, and Linux operating systems. Users can import their data in popular formats (.mat, .csv, .npy, .pkl, .h5), and the results can be exported in Matlab “.mat” or Numpy “.npy” for post-processing. The Deep Learning models have been developed using TensorFlow, and users have the option to search for the optimal hyperparameters for their database since hyperparameter optimization has been implemented for all models. See Refs. [1–10].

Explicit machine learning-based model predictive control of nonlinear processes via multi-parametric programming

Machine learning-based model predictive control (ML-MPC) has been developed to control nonlinear processes with unknown first-principles models. While ML models can capture nonlinear dynamics of complex systems, the complexity of ML models leads to increased computation time for real-time implementation of ML-MPC. To address this issue, in this work, we propose an explicit ML-MPC framework for nonlinear processes using multi-parametric programming. Specifically, a self-adaptive approximation algorithm is first developed to obtain a piecewise linear affine function that approximates the behaviors of ML models. Then, multi-parametric quadratic programming (mpQP) problems are formulated to generate the solution map for states in discretized state–space. Furthermore, to accelerate the implementation of explicit ML-MPC, a neighbor-first search algorithm is developed. Finally, an example of a chemical reactor is used to demonstrate the effectiveness of the explicit ML-MPC.

A recurrent neural network-based identification of complex nonlinear dynamical systems: a novel structure, stability analysis and a comparative study

A recurrent neural network-based identification of complex nonlinear dynamical systems: a novel structure, stability analysis and a comparative study

A novel deep reinforcement learning framework with BiLSTM-Attention networks for algorithmic trading

The financial market, as a complex nonlinear dynamic system frequently influenced by various factors, such as international investment capital, is very challenging to build trading strategies from the obtained market information. Deep Reinforcement Learning (DRL) combines the perceptual capability of deep learning and the control decision making capability of reinforcement learning to learn the mapping between financial market states and trading decisions by interacting with the environment. In this paper, an enhanced stock trading strategy, denominated efficient deep State-Action-Reward-State-Action (SARSA), is presented to tackle the algorithmic trading problem of determining optimal trading positions in the daily trading activities of the stock market. This algorithm is recognized for its properties of stable learning and convergence, attributes of critical significance within the financial domain, where stability assumes paramount importance, and excessive risk-taking should be averted. Furthermore, a novel deep network architecture called Bidirectional Long Short-Term Memory (BiLSTM)-Attention is introduced to address the challenge of accurately presenting the complex and volatile stock market. The BiLSTM-Attention greatly enhances the network’s capacity to recognize key features and patterns in stock market data, allowing agents to focus on the most relevant aspects of the data. Evaluations on DJI, SP500, GE, IXIC datasets from January 1, 2008 to December 31, 2022 show that our efficient deep SARSA algorithm outperforms a wide range of traditional strategies (B&H, S&H, MR, TF) and DRL-based strategies (TDQN, DQN-Vanilla). For example, on the IXIC dataset, the efficient deep SARSA strategy achieves an attractive Cumulative Return (CR) of 582.17% and a Sharpe Ratio (ShR) of 1.86, outperforming all other methods. These experimental results prove the performance of our method in enhancing stock trading strategies.

Performance assessment of hydropower generator unit with the nonlinear transmissibility function method

The hydropower generator unit (HGU) is a complex nonlinear dynamic system, and how to utilize the monitoring data to achieve the performance assessment of the HGU is a challenge at present. In this paper, based on the linear transmissibility function and the nonlinear output frequency response function, the nonlinear transmissibility function (NTF) method is proposed to characterize the variation rule of the HGU performance, and the effectiveness of the proposed method is confirmed by the monitoring data from a hydropower station located in Guangxi, China. Firstly, the feature parameters related to the operation state of the unit are extracted from the monitoring data of the HGU. Secondly, the HGU system is divided into three subsystems according to the installation position of the sensors collecting these feature parameters, and the NTFs of them are estimated from multiple time scales. Finally, the performance of the HGU is comprehensively assessed utilizing the Fe indicator and the correlation transmissibility damage indicator. The analysis results demonstrate that the NTF method can reveal the performance variation trend and detect the early fault of the HGU, which is of great significance to the stable operation of the HGU.

Sparse Bayesian machine learning for the interpretable identification of nonlinear structural dynamics: Towards the experimental data-driven discovery of a quasi zero stiffness device

Data-driven discovery of governing laws for complex nonlinear structural dynamic systems remains a challenging issue of paramount importance. This work addresses the above issue by leveraging the available noisy data and integrating sparse Bayesian machine learning (ML) techniques to discover the governing equations. The problem of discovery is re-cast as the automatic relevance determination of models (model selection) from the library of potential candidate basis terms and their coefficients are determined (parameter identification) using sparse Bayesian linear regression. Two sparsity promoting ML algorithms based on relevance vector machines have been employed. Both these approaches use Bayesian statistics and quantify the uncertainty associated with the model predictions. Results from four representative numerical examples of nonlinear structural dynamics illustrate excellent performance of both proposed approaches. The results have been validated with the true governing equations and time response data. Comparison has also been made with a recent and popular sparse discovery approach. Finally, the proposed framework is applied to real datasets that were generated from an in-house designed experimental setup of a quasi zero stiffness device and good performance has been observed.

Nonlinear Dynamics of Complex Systems: Modeling and Analysis

Nonlinear dynamics plays a pivotal role in understanding and modeling complex systems. This article provides an overview of key concepts, methodologies, and recent advances in the modeling and analysis of nonlinear dynamics in complex systems.

Design of a novel robust recurrent neural network for the identification of complex nonlinear dynamical systems

Design of a novel robust recurrent neural network for the identification of complex nonlinear dynamical systems

A novel grid multi-structure chaotic attractor and its application in medical image encryption

Grid multi-scroll/wing chaotic systems are complex non-linear dynamic systems, which are widely used in secure communication. The grid multi-scroll/wing chaotic systems are usually realized by using the function control method, which has a complex realization method, many control parameters, and a simple unit attractor structure. In this paper, based on the Hopfield neural network, a memristive Hopfield neural network model is proposed by using the memristor synapse control method. The model can generate novel grid multi-structure chaotic attractors, which have the characteristics of a simple implementation method, few control parameters, and complex unit attractor structure. Firstly, the generation mechanism of the grid multi-structure chaotic attractors is analyzed by the equilibrium points and stability. Secondly, its basic dynamical characteristics including the Lyapunov exponent spectrum, fractal dimension, time series, power spectrum, bifurcation diagram, and Poincaré section are analyzed. Thirdly, an analog circuit of the neural network model is designed and realized by Multisim. Finally, combined with the chaos encryption principle, an image encryption scheme is designed based on the generated grid multi-structure attractors. Experimental results show that compared with the existing schemes, the proposed scheme has larger information entropy, higher key sensitivity, and a good application prospect.

Efficient forecasting of chaotic systems with block-diagonal and binary reservoir computing.

The prediction of complex nonlinear dynamical systems with the help of machine learning has become increasingly popular in different areas of science. In particular, reservoir computers, also known as echo-state networks, turned out to be a very powerful approach, especially for the reproduction of nonlinear systems. The reservoir, the key component of this method, is usually constructed as a sparse, random network that serves as a memory for the system. In this work, we introduce block-diagonal reservoirs, which implies that a reservoir can be composed of multiple smaller reservoirs, each with its own dynamics. Furthermore, we take out the randomness of the reservoir by using matrices of ones for the individual blocks. This breaks with the widespread interpretation of the reservoir as a single network. In the example of the Lorenz and Halvorsen systems, we analyze the performance of block-diagonal reservoirs and their sensitivity to hyperparameters. We find that the performance is comparable to sparse random networks and discuss the implications with regard to scalability, explainability, and hardware realizations of reservoir computers.

Multidiscipline Design Optimization for Large-Scale Complex Nonlinear Dynamic System Based on Weak Coupling Interfaces

For high-tech manufacturing industries, developing large-scale complex nonlinear dynamic systems must be taken as one of the basic works, formulating problems to be solved, steering system design in a more preferable direction, and making correct strategic decisions. By using effective tools of big data mining, Dynamic Design Methodology was proposed to establish a technical platform for Multidiscipline Design Optimization such as High-Speed Rolling Stock, including three key technologies: analysis graph of full-vehicle stability properties and variation patterns, improved transaction strategy of flexible body to MBS interface, seamless collaboration with weldline fatigue damage assessments through correct Modal Stress Recovery. By applying the above methodology, a self-adaptive improved solution was formulated with optimal parameter configuration, which is one of the more favorable options for higher-speed bogies. While within a velocity (140–200) km/h at λe < 0.10, car body instability’s influence on ride comfort can be easily improved by using a semi-active vibration reduction technique between inter-vehicles through outer windshields. Comprehensive evaluations show that only under rational conditions of wheel-rail matching, i.e., 0.10 ≥ λeN > λemin and λemin = (0.03–0.06), can this low-cost solution achieve the three goals of low track conicity, optimal route planning, and investment benefit maximization. So, rail vehicle experts are necessary to collaborate and innovate intensively with passenger transportation and steel rail ones. Specifically, by adopting rail grinding treatment, occurrence probability is controlled at 85% and 5% for track conicity of (0.03–0.10) and (0.25–0.35). By optimizing routing planning, operating across dedicated lines of different speed grades can achieve self-cleaning of central hollow tread wear over time. According to the inherent rigid-flex coupling relationship, geometric nonlinearities of worn wheel-rail contact should be avoided as much as possible for HSR practices. Only under weak coupling interfaces in the floor frame can the structural integrity of an aluminum alloy car body be ensured.

Causal deep learning models for studying the Earth system

Abstract. Earth is a complex non-linear dynamical system. Despite decades of research and considerable scientific and methodological progress, many processes and relations between Earth system variables remain poorly understood. Current approaches for studying relations in the Earth system rely either on numerical simulations or statistical approaches. However, there are several inherent limitations to existing approaches, including high computational costs, uncertainties in numerical models, strong assumptions about linearity or locality, and the fallacy of correlation and causality. Here, we propose a novel methodology combining deep learning (DL) and principles of causality research in an attempt to overcome these limitations. On the one hand, we employ the recent idea of training and analyzing DL models to gain new scientific insights into relations between input and target variables. On the other hand, we use the fact that a statistical model learns the causal effect of an input variable on a target variable if suitable additional input variables are included. As an illustrative example, we apply the methodology to study soil-moisture–precipitation coupling in ERA5 climate reanalysis data across Europe. We demonstrate that, harnessing the great power and flexibility of DL models, the proposed methodology may yield new scientific insights into complex non-linear and non-local coupling mechanisms in the Earth system.

Brief review of the development of theories of robustness, roughness and bifurcations of dynamic systems

The development issues of theories of robustness, roughness and bifurcations of dynamic systems are considered. In the modern theory of dynamic systems and automatic control systems, researches of the properties of roughness and robustness of systems are becoming more and more important. The work considers methods of research and ensuring robust stability of interval dynamic systems of both algebraic and frequency directions of robust stability. The main results of the original algebraic method of robust stability for continuous and discrete time are given. In the frequency direction of robust stability, the issues of a frequency-robust method to the analysis and synthesis of robust multidimensional control systems based on the use of the frequency condition number of the transfer matrix of the “input-output” ratio are considered. The main provisions of the theory and method of topological roughness of dynamic systems based on the concept of roughness according to Andronov-Pontryagin are presented with the introduction of a measure of roughness of systems in the form of a condition number of matrices of reduction to a diagonal (quasi-diagonal) basis at special points of phase space. Criteria for dynamic systems bifurcations are formulated. Applications of the topological roughness method to synergetic systems and chaos have been used to investigate many systems, such as Lorenz and Rössler attractors, Belousov-Jabotinsky, Chua systems, “predator-prey” and “predator-prey-food”, Hopf bifurcation, Schumpeter and Caldor economic systems, Henon mapping, and others. For research of weakly formalized and non-formalized systems, the use of the approach of analogies of theoretical-multiple topology and the abstract method to such systems is proposed. Further research suggests the development of roughness and bifurcation theories for complex nonlinear dynamical systems.

Research on the state of health estimation of lithium-ion batteries for energy storage based on XGB-AKF method

With the advantages of high energy density, long cycle life and high stability, lithium-ion batteries have been used in a large number of fields such as electric vehicles and grid scale energy storage. To ensure the safe and reliable operation of battery systems, it is important to make an accurate and rapid estimation of the state of health (SOH) of Li-ion cells. A Li-ion cell is a complex nonlinear dynamic system. The SOH of a Li-ion can not be measured directly in actual working conditions; it can only be estimated indirectly by external characteristic parameters that reflects the extent of cell aging. It is difficult to ensure the reliability of method based on a single aging feature or model. Therefore, this paper proposes a multi-feature SOH estimation method that combines data-driven XGBoost and a Kalman filter. Firstly, a principal component analysis algorithm to reconstruct multiple battery aging features based on data is used, and an XGBoost online estimation model incorporating multiple features based on the reconstructed feature data is constructed. Finally, the joint optimal estimation of SOH of Li-ion cells by introducing a time-domain Kalman filter based on the real-time correction of the XGBoost model is achieved in this method. The results show that the method improves the accuracy and robustness of the estimation model and achieves a high-precision joint estimation of SOH for Li-ion cells.

Nonlinear Dynamics of Complex Systems: Modeling and Analysis

Nonlinear Dynamics of Complex Systems: Modeling and Analysis

Data-Driven Probabilistic Methodology for Aircraft Conflict Detection Under Wind Uncertainty

Assuming the availability of a reliable aircraft trajectory planner, this paper presents a probabilistic methodology to detect conflicts between aircraft in the cruise phase of flight in the presence of wind velocity forecasting uncertainty. This uncertainty is quantified by ensemble weather forecasts, the members of which are regarded as realizations of correlated random processes and used to derive the eastward and northward components of the wind velocity. First, the Karhunen-Loève expansion is used to obtain a series expansion of the components of the wind velocity in terms of a set of uncorrelated random variables and deterministic coefficients. Then, the uncertainty generated by these uncorrelated random variables in the outputs of the aircraft trajectory planner is quantified using the arbitrary polynomial chaos technique. Finally, the probability density function of the great circle distance between each pair of aircraft is derived from the polynomial expansions using a Gaussian kernel density estimator and used to estimate the probability of conflict. The arbitrary polynomial chaos technique allows the effects of uncertainty in complex nonlinear dynamical systems, such as those underlying aircraft trajectory planners, to be quantified with high computational efficiency, only requiring the existence of a finite number of statistical moments of the random variables of the Karhunen-Loève expansion while avoiding any assumptions on their probability distributions. To demonstrate the effectiveness of the proposed conflict detection method, numerical experiments are conducted via an optimal control-based aircraft trajectory planner for a given wind velocity forecast represented by an ensemble prediction system.

Picture Processing Optimization Technology Based on Mask R-CNN Algorithm

Picture processing is applied in all kind of fields, such as space science research, medical imaging, photography art. Because the human vision system is a complex nonlinear dynamic system, the traditional image enhancement methods often can not meet the requirements of real-time in practical applications. In the face of a large number of images, it is of great practical significance to obtain practical information from these data. With the rapid development of computer CnTech, image compression coding CnTech has also made great progress. And through the Mask R-CNN algorithm for effective segmentation and image recognition, can help people face massive image information, to their own needs as the goal, to achieve efficient retrieval, analysis, induction. The traditional methods of image classification by manually selecting features or manually extracting template matching have some defects. In this paper, a Mask R-CNN image optimization processing CnTech is proposed, which can accurately identify images and has higher accuracy for image feature extraction. It is meaningful to achieve automatic and accurate segmentation of microscopic images. Through training on COCO data set, it is verified through experiments that there are problems in image segmentation and recognition. In Mask R-CNN model, resource consumption is reduced and the efficiency of image segmentation and recognition is improved. Compared with the current cutting-edge algorithms, the image object detection on the strength of Mask R-CNN model has significant advantages.

Machine learning-assisted extreme events forecasting in Kerr ring resonators

Predicting complex nonlinear dynamical systems has been even more urgent because of the emergence of extreme events such as earthquakes, volcanic eruptions, extreme weather events (lightning, hurricanes/cyclones, blizzards, tornadoes), and giant oceanic rogue waves, to mention a few. The recent milestones in the machine learning framework o↵er a new prospect in this area. For a high dimensional chaotic system, increasing the system’s size causes an augmentation of the complexity and, finally, the size of the artificial neural network. Here, we propose a new supervised machine learning strategy to locally forecast bursts occurring in the turbulent regime of a fiber ring cavity.

Analysis of Numerical Integration in RNN-Based Residuals for Fault Diagnosis of Dynamic Systems

Data-driven modeling and machine learning are widely used to model the behavior of dynamic systems. One application is the residual evaluation of technical systems where model predictions are compared with measurement data to create residuals for fault diagnosis applications. While recurrent neural network models have been shown capable of modeling complex non-linear dynamic systems, they are limited to fixed steps discrete-time simulation. Modeling using neural ordinary differential equations, however, make it possible to evaluate the state variables at specific times, compute gradients when training the model and use standard numerical solvers to explicitly model the underlying dynamic of the time-series data. Here, the effect of solver selection on the performance of neural ordinary differential equation residuals during training and evaluation is investigated. The paper includes a case study of a heavy-duty truck's after-treatment system to highlight the potential of these techniques for improving fault diagnosis performance.

Linearizing nonlinear dynamics using deep learning

The majority of systems of practical interest are characterized by nonlinear dynamics. This renders the control and optimization of such systems a complex task owing to their nonlinear behaviour. Standard methods of dealing with nonlinear systems such as linearizing around a fixed point may not be an effective strategy for many systems, thus requiring an alternative approach. For this reason, we propose a novel deep learning framework to discover a transformation of a nonlinear dynamical system to an equivalent higher dimensional linear system using data generated from identification experiments. We demonstrate that the resulting learned linear representation accurately captures the dynamics of the original system for a wide range of conditions defined by the training dataset used. As a result of this, we show that the learned linear model can subsequently be used for the successful control of the original system. We demonstrate this by applying the proposed framework to three examples; a benchmark example from the literature, and two practical, complex nonlinear dynamical systems from the chemical engineering domain: the Continuous Stirred Tank Reactor (CSTR) system and finally the four-tank system.