High-dimensional Nonlinear Systems Research Articles

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.

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.

Nonlinear vibration of porous truncated conical shell under unified boundary condition and mechanical load

The aim of this article is to research nonlinear vibration responses of a metal porous truncated conical shell undergoing mechanical harmonic loads in thickness direction. The artificial spring technique is employed to unify boundary condition which can arrive at arbitrary classical boundary condition in real-word by changing the stiffness of springs. Utilizing Hamilton's principle, theoretical formulations of the conical shell are developed in view of FSDT and von-Kármán geometric relation. Afterwards, by employing generalized differential quadrature method (GDQM), the governing equation of nonlinear vibration in the form of partial differential equation is truncated and discretized into an ordinary differential equation which is a high-dimensional nonlinear dynamic system described in physical space rather than the mode space. On this basis, the analysis of effect of exciting frequency on the amplitude is carried out for three different distributing types of porous truncated conical shells by numerical integration. The frequency-amplitude responses of the system are depicted utilizing the sweeping frequency. The bifurcation diagrams, time series and phase diagrams are utilized to analyze the nonlinear dynamics of the porous truncated conical shell under unified boundary condition. It is found that the multiple solutions high-incidence area is near the 1st primary resonance, and vibration amplitude is dominated by the 1st order primary resonance and the 3rd one. The hardening-spring nature decreases for Type3, Type2 and Type1 successively.

A decoupled approach for determination of the joint probability density function of a high-dimensional nonlinear stochastic dynamical system via the probability density evolution method

The joint probability density function (PDF) of multiple response processes of a system is a crucial topic in the fields of science and engineering. It is adopted to describe the dependent uncertainty propagation in complex stochastic dynamical systems, including the displacement and velocity of one degree of freedom (DOF), and the nonlinear probabilistic dependencies between multiple variables. However, the partial differential equations (PDEs) governing the time evolution of joint PDFs for multiple stochastic processes are often high-dimensional and coupled, making them analytically or numerically intractable for systems with high dimensionality. The probability density evolution method (PDEM) has provided a new perspective from the standpoint of random events description of the principle of preservation of probability. For an n-dimensional system, if merely the joint PDF of m processes is needed, where m is not larger than n, then it is sufficient to solve the generalized density evolution equation (GDEE), i.e., the Li–Chen equation, in m dimensions. This approach is particularly convenient for cases where n is large while m is small. However, for cases where m is not less than two, it is still challenging to avoid solving high-dimensional PDEs. This paper proposes a new method for the determination of joint PDF of multiple responses based on the principle of preservation of probability. To capture the joint PDF of m processes, the m-dimensional PDE can be transformed into m one-dimensional PDEs, significantly reducing the computational complexity of numerical implementation. The proposed method gives an enhanced version to the PDEM, enabling it to not only determine the individual probabilistic response, but also efficiently determine the dependent probabilistic distribution of multiple responses of high-dimensional stochastic dynamical systems without solving the coupling PDE. It reduces the computational cost from exponential growth of m powers to linear growth. Some numerical examples are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Nonlinear wave excitations in the (2+1)-D asymmetric Nizhnik-Novikov-Veselov system

The (2+1)-dimensional nonlinear asymmetric Nizhnik-Novikov-Veselov system is one of the extended versions of the Korteweg-de Vries equation, and as such of considerable significance in nonlinear wave dynamics. In this work, it is transformed into a bilinear form by using the Hirota transformation and then treated by the variable separation method. In particular, by separating variables in a specific manner, a class of solutions is obtained, which are constructed with the help of three auxiliary functions, Py, Ry, and Qxt. By selecting the three functions conveniently, we obtain novel nonlinear excitations for the asymmetric Nizhnik-Novikov-Veselov system. To make our results more explicit and interesting, various types of nonlinear profiles were displayed, colloquially named as the parallelogram, walnut, dark breather, half-moon breather, and the top-compressed quadrilateral breather structures. It is found that these local wave packets can be well controlled by appropriately selecting the three auxiliary functions, and that the variable separation approach can be extended to other high-dimensional nonlinear systems.

An algorithm for solving the equilibrium points of high-dimensional nonlinear vehicle dynamic system based on a combination of genetic algorithm, sequential quadratic programming method, and continuation method

The paper proposes an algorithm that combines Genetic Algorithm, Sequential Quadratic Programming, and Continuation Method to solve the equilibrium points of a multi-degree-of-freedom vehicle nonlinear system. The algorithm’s effectiveness is demonstrated by applying it to search for equilibrium points of a 5-degree-of-freedom (5DOF) nonlinear vehicle model, which considers both longitudinal and lateral motion. The dynamic equilibrium points of front-wheel-drive, rear-wheel-drive, and all-wheel-drive vehicles with different front-wheel steering angle inputs are calculated. Taking the front-wheel-drive system as an example, the system equilibrium points are further analyzed using phase space analysis and eigenvalue analysis for verification. In addition, the impact of the driving effect on the equilibrium points bifurcation is investigated. The results show that compared with the Genetic Algorithm alone and the combination method of Genetic Algorithm and Sequential Quadratic Programming, the proposed algorithm can effectively and accurately solve the equilibrium points of the 5DOF vehicle model. The study also reveals that the driving effect significantly influences the vehicle equilibrium point bifurcation.

Dimension of Activity in Random Neural Networks.

Neural networks are high-dimensional nonlinear dynamical systems that process information through the coordinated activity of many connected units. Understanding how biological and machine-learning networks function and learn requires knowledge of the structure of this coordinated activity, information contained, for example, in cross covariances between units. Self-consistent dynamical mean field theory (DMFT) has elucidated several features of random neural networks-in particular, that they can generate chaotic activity-however, a calculation of cross covariances using this approach has not been provided. Here, we calculate cross covariances self-consistently via a two-site cavity DMFT. We use this theory to probe spatiotemporal features of activity coordination in a classic random-network model with independent and identically distributed (i.i.d.) couplings, showing an extensive but fractionally low effective dimension of activity and a long population-level timescale. Our formulas apply to a wide range of single-unit dynamics and generalize to non-i.i.d. couplings. As an example of the latter, we analyze the case of partially symmetric couplings.

Flow time history representation and reconstruction based on machine learning

Based on deep learning technology, a new spatiotemporal flow data representation and reconstruction scheme is proposed by using flow time history (FTH) data instead of flow snapshots. First, the high-dimensional nonlinear flow system is reduced to a low-dimensional representation latent code using the FTH autoencoder model. Second, the mapping from physical space to latent code space is built using mathematical and machine-learning schemes. Finally, FTH at unavailable positions in physical space is generated by the FTH generator. The proposed scheme is validated by three case studies: (i) representing and recovering the FTH data of periodic laminar flow around a circular cylinder at Re = 200 and generating high-resolution laminar flow data; (ii) reconstructing complex FTH of flow past cylinder at Re = 3900 which including laminar and turbulent flow region and generating three-dimensional high-resolution turbulent flow data, respectively; (iii) representing and generating multi-variable turbulent flow data simultaneously using the multi-channel model. The results show that the proposed scheme is an effective low-dimensional representation for complex flow time variant features, which is suitable for both laminar and turbulent FTH data to generate spatiotemporal high-resolution FTH data in three-dimensional space.