High-dimensional Nonlinear Systems Research Articles

Data-driven prediction of vortex solitons and multipole solitons in whispering gallery mode microresonator

Deep learning incorporating physics knowledge has become a powerful tool for studying the dynamic behavior of high-dimensional nonlinear systems. In this paper, the two-stage mini-batch resampling of adaptive physics-informed neural network (TMA-PINN) method is proposed to solve the (2 + 1)-dimensional variable-coefficient Lugiato-Lefever equation (vLLE). The vortex soliton in the WGM microresonator with different external excitation is investigated by TMA-PINN. It is found that external excitation can cause the rotation of vortex solitons. In addition, the effect of topological charge and external excitation on the dynamical characteristics of spatial solitons including vortex solitons and multipole solitons are investigated. The results show that the final shape of the rotation of vortex solitons and the number of azimuth lobes of multipole solitons are controlled by topological charges. Compared with classical PINN, TMA-PINN can better handle the gradient balance of various loss terms in (2 + 1)-dimensional vLLE to reconstruct the dynamic behavior of WGM microresonator solitons, having potential applications in other nonlinear systems.

Sufficient variable selection of high dimensional nonparametric nonlinear systems based on Fourier spectrum of density-weighted derivative

Sufficient variable selection of high dimensional nonparametric nonlinear systems based on Fourier spectrum of density-weighted derivative

Model reduction of high-dimensional self-excited nonlinear systems using floquet theory based parameterization method

Model reduction of high-dimensional self-excited nonlinear systems using floquet theory based parameterization method

Conversion mechanisms and transformed waves for the (3 + 1)-dimensional nonlinear equation

In this paper, we focus on investigating the (3 + 1)-dimensional nonlinear equation which is used to describe the propagation of waves in the shallow water. The study begins with the application of the Hirota bilinear method to obtain N-soliton solution. Building on this foundation, the research delves into the construction of first-order breather wave by imposing complex conjugate constraints on the parameters of two solitons. Further analysis of the characteristic lines of breathers leads to the derivation of conversion conditions. Under this specific condition, a series of nonlinear transformed waves are presented, including quasi-kink solitons, W-shaped kink solitons, oscillation W-shaped kink solitons, multipeaks solitons, quasi-periodic waves, and line rogue waves. Each of these transformed waves exhibits unique structural and dynamic properties, enriching the understanding of wave behavior in higher-dimensional nonlinear systems. The study also explores the nonlinear superposition mechanism between solitary wave and periodic wave. This mechanism elucidates the formation process of nonlinear waves, explaining how their locality and oscillatory characteristics emerge from the superposition of different wave components. Moreover, the geometric properties of the two characteristic lines of the waves are analyzed to understand the time-varying nature of the transformed waves. This temporal analysis is crucial for predicting the evolution and interaction of these waves over time. Finally, the research extends to the higher-order breather wave and explores the interactions among various waves. These interactions reveal the complex dynamics that may arise in the (3 + 1)-dimensional nonlinear systems and provide deeper insights into the interactions among different wave structures.

Real-time control of a soft manipulator based on reduced order extended position-based dynamics

Soft robots are high-dimensional nonlinear systems coupled with both geometric and material nonlinearity. Control of such a system is complex and time-consuming. In this study, a real-time trajectory tracking control framework is established based on the reduced order extended position-based dynamics. In contrast to the common nonlinear model order reduction methods that require to collect a large number of data to create the motion subspace, this article's motion subspace is constructed based on the model configuration and material properties. The linear modes of the model and the related modal derivatives provide the reduced order matrix, which streamlines and increases the efficiency of model construction. Then, coupled with the instantaneous optimal control, a real-time reduced order model-based control framework of soft robots can be constructed. Experiments on trajectory tracking of a soft manipulator are conducted to verify the accuracy and efficiency of the proposed controller. The average error of all experiments is within 1 cm; and the single-step calculation time of the controller is about 0.057 s, which is less than the sampling period 0.1 s.

Multi-output multi-physics-informed neural network for learning dimension-reduced probability density evolution equation with unknown spatio-temporal-dependent coefficients

The Dimension-Reduced Probability Density Evolution Equation (DR-PDEE) offers a promising approach for evaluating probability density evolution in stochastic dynamical systems. Physics-Informed Neural Networks (PINNs) are well-suited for solving DR-PDEE due to their ability to encode physical laws into the learning process. However, challenges arise from the spatio-temporal-dependence of unknown intrinsic drift and diffusion coefficients, which drive DR-PDEE, along with their derivatives. To address these challenges, a novel framework called Multi-Output Multi-Physics-Informed Neural Network (MO-MPINN) is proposed to predict the evolution of time-varying coefficients and response probability density simultaneously. MO-MPINN features multiple output neurons, eliminating the necessity for distinct identification of unknown spatio-temporal-dependent coefficients separately. It uses parallel subnetworks to reduce training complexity and embeds multiple physical laws in the loss function to ensure an accurate representation of the underlying principles. Leveraging automatic differentiation, MO-MPINN efficiently computes derivatives of coefficients without resorting to numerical differentiation. The framework is applicable to high-dimensional stochastic nonlinear systems with double randomness in structural parameters and excitations. Several structures are presented to validate the performance of the MO-MPINN. This study introduces a new paradigm for solving partial differential equations involving differentiation of spatio-temporal-dependent coefficients.

Projected data assimilation using sliding window proper orthogonal decomposition

Prediction of the state evolution of complex high-dimensional nonlinear systems is challenging due to the nonlinear sensitivity of the evolution to small inaccuracies in the model. Data Assimilation (DA) techniques improve state estimates by combining model simulations with real-time data. Few DA techniques can simultaneously handle nonlinear evolution, non-Gaussian uncertainty, and the high dimension of the state. We recently proposed addressing these challenges using a POD technique that projects the physical and data models into a reduced-dimensional subspace. Proper Orthogonal Decomposition (POD) is a tool to extract spatiotemporal patterns (modes) that dominate the observed data. We combined the POD-based projection operator, computed in an offline fashion, with a DA scheme that models non-Gaussian uncertainty in lower dimensional subspace. If the model parameters change significantly during time evolution, the offline computation of the projection operators ceases to be useful. We address this challenge using a Sliding Window POD (SW-POD), which recomputes the projection operator based on a sliding subset of snapshots from the entire evolution. The physical model projection is updated dynamically in terms of modes and number of modes, and the data model projection is also chosen to promote a sparse approximation. We test the efficacy of this technique on a modified Lorenz'96 model (L96) with a time-varying forcing and compare it with the time-invariant offline projected algorithm. In particular, dynamically determined physical and data model projections decrease the Root Mean Squared Error (RMSE) and the resampling rate.

Dynamic Mode Decomposition With Gaussian Process Regression for Control of High-Dimensional Nonlinear Systems

Abstract In this work, we consider the problem of learning a reduced-order model of a high-dimensional stochastic nonlinear system with control inputs from noisy data. In particular, we develop a hybrid parametric/nonparametric model that learns the “average” linear dynamics in the data using dynamic mode decomposition with control (DMDc) and the nonlinearities and model uncertainties using Gaussian process (GP) regression and compare it with total least-squares dynamic mode decomposition (tlsDMD), extended here to systems with control inputs (tlsDMDc). The proposed approach is also compared with existing methods, such as DMDc-only and GP-only models, in two tasks: controlling the stochastic nonlinear Stuart–Landau equation and predicting the flowfield induced by a jet-like body force field in a turbulent boundary layer using data from large-scale numerical simulations.

Application of Numerical Bifurcation Tracking Strategy to Blade-Tip/Casing Interactions in Aircraft Engines

Abstract Building on the regularized-Lanczos harmonic balance method (RL-HBM), a previously developed frequency method, this paper presents a numerical bifurcation tracking strategy dedicated to high-dimensional nonlinear mechanical systems. In order to demonstrate its applicability to industrial applications, it is here used to obtain original results in the context of blade-tip/casing interactions in aircraft engines. The emphasis is put specifically on the tracking of predicted limit point (LP) bifurcations as key parameters—such as the amplitude of the aerodynamic forcing applied on the blade, the friction coefficient or the operating clearances—vary. Overall, presented results underline that the employed frequency method is well-suited to tackle the numerical challenges inherent to such computations on high-dimensional systems. For the mechanical system of interest, the industrial fan blade National Aeronautics and Space Administration (NASA) rotor 67, it is shown that the application of the presented strategy yields an efficient way to identify isolated branches of solutions, which may be of critical importance from a design standpoint.

Probabilistic response determination of high-dimensional nonlinear dynamical systems enforced by parametric multiple Poisson white noises

Probabilistic response determination of high-dimensional nonlinear dynamical systems enforced by parametric multiple Poisson white noises

A novel method for response probability density of nonlinear stochastic dynamic systems

AbstractThis paper presents a novel method for analyzing high-dimensional nonlinear stochastic dynamic systems. In particular, we attempt to obtain the solution of the Fokker–Planck–Kolmogorov (FPK) equation governing the response probability density of the system without using the FPK equation directly. The method consists of several important components including the radial basis function neural networks (RBFNN), Feynman–Kac formula and the short-time Gaussian property of the response process. In the area of solving partial differential equations (PDEs) using neural networks, known as physics-informed neural network (PINN), the proposed method presents an effective alternative for obtaining solutions of PDEs without directly dealing with the equation, thus avoids evaluating the derivatives of the equation. This approach has a potential to make the neural network-based solution more efficient and accurate. Several highly challenging examples of nonlinear stochastic systems are presented in the paper to illustrate the effectiveness of the proposed method in comparison to the equation-based RBFNN approach.

A general and efficient harmonic balance method for nonlinear dynamic simulation

This paper presents a general and efficient harmonic balance (GE-HB) method to expeditiously evaluate the periodic responses of nonlinear systems. By employing the fast Fourier transform (FFT) technique, it rapidly computes harmonic expansion coefficients of the nonlinear terms and their corresponding constant tensor matrices. Notably, the nonlinear part of the Jacobian matrix is just the product of the coefficients and the constant tensor matrices, loop computations are no longer requisite for calculating the Jacobian matrix. This, facilitates accelerated determination of the nonlinear portion of the Jacobian matrix, thus obtaining the periodic responses of the system in an efficient manner. Furthermore, the constant tensor matrices are exclusively contingent upon the harmonic bases of the system's periodic response, devoid of reliance on the equation's specific formulation, facilitates obtaining the Jacobian matrix in a general manner. Simulation results on a 20-degree-of-freedom nonlinear oscillator chain system demonstrate that the efficiency of the GE-HB method is about tenfold that of the harmonic balance-alternating frequency/time domain (HB-AFT) method and twenty-fivefold that of the fourth-order Runge-Kutta (RK4) method. Further, the simulation of a 284-degree-of-freedom nonlinear rotor system exhibits the GE-HB method's computational efficiency surpassing HB-AFT method by over 500 times and Newmark method by over 3000 times. Moreover, in these two examples, the GE-HB method captures all solution branches, including unstable periodic solutions, providing comprehensive insights into periodic responses of the systems. The GE-HB method proposed here has the potential to explore dynamic characteristics for high-dimensional complex nonlinear systems in engineering.

Domain estimation and coupled controller design for high-dimensional nonlinear multi-agent systems

This paper investigates the consensus problem of a class of high-dimensional nonlinear multi-agent systems. The agents are homogeneous and have coupled interactions with their nearby agents. Firstly, we study the consensus of this class of nonlinear multi-agent systems and present some consensus conditions based on the linearization technique. By transforming the consensus problem into a stability problem, we show that there exists a domain such that if all pairs of connected agents stay in the domain, the system will achieve consensus under the consensus conditions. Then, we propose a method to estimate the consensus domain since the accurate description of the domain is still challenging. Moreover, a method of coupled controller design is presented to make the multi-agent systems achieve consensus. At last, two numerical examples and a practical example are used to show the effectiveness of the results proposed in this paper.

Spatiotemporal dual-periodic soliton pulsation in a multimode fiber laser

Spatiotemporal mode-locked (STML) fiber lasers have become a new platform for investigating nonlinear phenomena. In this work, spatiotemporal dual-periodic soliton pulsation (SDSP) is firstly observed in an STML fiber laser. It is found that in the SDSP, the long-period pulsations (LPPs) of different transverse modes are synchronous, while the short-period pulsations (SPPs) exhibit asynchronous modulations. The numerical simulation confirms the experimental results and further reveals that the proportion of transverse mode components can manipulate the periods of the LPP and SPP but does not affect the synchronous and asynchronous pulsations of different transverse modes. The obtained results bring the study of spatiotemporal dissipative soliton pulsation into the multi-period modulation stage, which helps to understand the complex spatiotemporal dynamics in STML fiber lasers and discover new dynamics in high-dimensional nonlinear systems.

Separable Gaussian neural networks for high-dimensional nonlinear stochastic systems

This paper extends the recently developed method of separable Gaussian neural networks (SGNN) to obtain solutions of the Fokker–Planck–Kolmogorov (FPK) equation in high-dimensional state space. Several challenges when extending SGNN to high-dimensional state space are addressed including proper definition of domain for placing Gaussian neurons and region for data sampling, and numerical integration issue of evaluating marginal probability density functions. Three benchmark nonlinear dynamic systems with increasing complexity and dimension are examined with the SGNN method. In particular, the steady-state probability density of the response is obtained with the SGNN method and compared with the results of extensive Monte Carlo simulations. It should be pointed out that some solutions of high-dimensional FPK equations for nonlinear dynamic systems would be very difficult to obtain without SGNN.

A hybrid proper orthogonal decomposition and next generation reservoir computing approach for high-dimensional chaotic prediction: Application to flow-induced vibration of tube bundles.

Chaotic time series prediction is a central science problem in diverse areas, ranging from engineering, economy to nature. Classical chaotic prediction techniques are limited to short-term prediction of low- or moderate-dimensional systems. Chaotic prediction of high-dimensional engineering problems is notoriously challenging. Here, we report a hybrid approach by combining proper orthogonal decomposition (POD) with the recently developed next generation reservoir computing (NGRC) for the chaotic forecasting of high-dimensional systems. The hybrid approach integrates the synergistic features of the POD for model reduction and the high efficiency of NGRC for temporal data analysis, resulting in a new paradigm on data-driven chaotic prediction. We perform the first chaotic prediction of the nonlinear flow-induced vibration (FIV) of loosely supported tube bundles in crossflow. Reducing the FIV of a continuous beam into a 3-degree-of-freedom system using POD modes and training the three time coefficients via a NGRC network with three layers, the hybrid approach can predict time series of a weakly chaotic system with root mean square prediction error less than 1% to 19.3 Lyapunov time, while a three Lyapunov time prediction is still achieved for a highly chaotic system. A comparative study demonstrates that the POD-NGRC outperforms the other existing methods in terms of either predictability or efficiency. The efforts open a new avenue for the chaotic prediction of high-dimensional nonlinear dynamic systems.

LyZNet with Control: Physics-Informed Neural Network Control of Nonlinear Systems with Formal Guarantees

Optimal control for high-dimensional nonlinear systems remains a fundamental challenge. One bottleneck is that classical approaches for solving the Hamilton-Jacobi-Bellman (HJB) equation suffer from the curse of dimensionality. Recently, physics-informed neural networks have demonstrated potential in overcoming the curse of dimensionality in solving certain classes of PDEs, including special cases of HJB equations. However, one perceived limitation of neural networks is their lack of formal guarantees in the solutions they provide. To address this issue, we have built LyZNet, a Python tool that combines physics-informed learning with formal verification. The previous version of the tool demonstrated the capability for stability analysis and region of attraction estimates. In this paper, we present the tool for solving optimal control problems. We expand the functionalities of the tool to support the formulation and solving of optimal control problems for control-affine systems via physics-informed neural network policy iteration (PINN-PI). We outline the methodology that enables the learning and verification of PINN for optimal stabilization tasks. We demonstrate with a classical control example that the learned optimal controller indeed has significantly improved performance and verifiable regions of attraction.

Dynamic order allocation in a duopoly hybrid workforce of competition: A machine learning approach

We develop a continuous-time stochastic differential game model that aims to capture market demand and stochastic cross-network effects, and we seek to find equilibrium order allocation strategies between the firm and the platform. By solving the Hamilton–Jacobi–Bellman (HJB) partial differential equation system, we obtain the feedback equilibrium. For a simple scenario, we derive the analytical solution which indicates that the equilibrium expenditures depend only on the marginal market thickness, and that market thickness is consistently advantageous for the value function. For the complex scenario, we propose a machine learning approach based on the Deep Galerkin Method to solve high-dimensional nonlinear HJB systems, and we demonstrate its good convergence properties. Based on reliable parameter values, our simulation results show that: (1) For higher market thickness, value functions exhibit greater sensitivity to changes in cross-network effects. (2) For a given cross-network effect, the equilibrium acquisition and retention expenditures display significant sensitivity to market thickness and time, respectively. To show the interaction between the two platforms, we present the following two results. One is that for the platform adopting a high pricing strategy, its acquisition expenditures exceed those of its competitor, while retention expenditures are the opposite. The other is that as the cross-network effect of the platform increases, its maximum profit initially rises and then declines, and the time of occurrence of the maximum profit monotonically decreases. In contrast, the competitor’s maximum profit initially declines and then rises, and the time of occurrence of the maximum profit monotonically increases.

On the detection of nonlinear normal mode-related isolated branches of periodic solutions for high-dimensional nonlinear mechanical systems with frictionless contact interfaces

This contribution introduces a methodology for the detection of isolated branches of periodic solutions to the nonlinear mechanical equation of motion for systems featuring frictionless contact interfaces. This methodology relies on a harmonic balance method-based solving procedure combined with the application of the Melnikov energy principle. It is able to predict the location of isolated branches of solutions in the vicinity of families of autonomous periodic solutions, namely nonlinear normal modes. The methodology is first applied on a two-degree-of-freedom phenomenological system in order to illustrate its relevance and accuracy. In particular, for this academic application, the proposed methodology yields an understanding of the discontinuous evolution of the first nonlinear resonance frequency as the forcing amplitude increases. Isolated branches of solutions featuring high amplitudes of vibration are detected far beyond nonlinear resonance frequencies obtained with a typical application of the harmonic balance method coupled with an arc-length continuation algorithm. Demonstration of the applicability of the proposed methodology to high-dimensional industrial finite element models is made with the nonlinear vibration analysis of a transsonic compressor blade, NASA rotor 37, subjected to an harmonic loading and blade-tip/casing structural contacts. The proposed methodology yields numerous isolated branches of solutions, which relevance is assessed by means of time integration simulations. In the end, the presented results underline that typical continuation algorithms may yield a significant underestimation of nonlinear resonance frequencies.

Investigation on the 1:2:3 internal resonance of high dimensional nonlinear dynamic system for FGGP reinforced rotating pre-twisted blade under the aerodynamic forces

The study of internal resonances is one of the main research areas in nonlinear dynamics. The energy transfer behaviors between multiple vibrational modes caused by internal resonances lead to complex nonlinear vibration behaviors. In this paper, the rotating blade is modeled as a functionally graded graphene platelet (FGGP) reinforced rotating pre-twisted plate. The nonlinear vibration of a high dimensional nonlinear dynamic system for FGGP reinforced rotating pre-twisted blade under the aerodynamic forces with 1:2:3 internal resonance is investigated. The axial excitations on the blade are primarily caused by axial aerodynamic force in the tip clearance and contact forces at the top of the blade. The subsonic airflow is treated as a transverse excitation and is calculated using the vortex lattice method. Rayleigh-Ritz method is used to obtain the mode shapes of the rotating composite pre-twisted blade. The three-degree-of-freedom ordinary differential equations of the rotating FGGP reinforced pre-twisted plate are derived by using Lagrange equation. Based on the multiscale method, the averaged equations of the rotating FGGP reinforced pre-twisted plate in the case of primary resonance and 1:2:3 internal resonance under axial and transverse excitations are acquired. We present a discussion on the amplitude-frequency response, force-amplitude response, bifurcations, and chaotic motions of a rotating FGGP reinforced pre-twisted cantilever plate under both axial and transverse excitations. A crescent-shaped independent resonance region is found in the amplitude-frequency response curves. The system exhibits complex nonlinear vibration behavior as axial or transverse excitation amplitude changes.