Nonlinear Model Order Reduction Research Articles

Bifurcation analysis in dynamical systems through integration of machine learning and dynamical systems theory

Abstract Characterizing the nonlinear behavior of dynamical systems near the stability boundary is a critical step toward understanding, designing, and controlling systems prone to stability concerns. Traditional methods for bifurcation analysis in both experimental systems and large-dimensional models are often hindered either by the absence of an accurate model or by the analytical complexity involved. This paper presents a novel approach that combines the theoretical frameworks of nonlinear reduced-order modeling and stability analysis in dynamical systems with advanced machine learning techniques to perform bifurcation analysis in dynamical systems. By focusing on a low-dimensional nonlinear invariant manifold, this work proposes a data-driven methodology that simplifies the process of bifurcation analysis in dynamical systems. The core of our approach lies in utilizing carefully designed neural networks to identify nonlinear transformations that map observation space into reduced manifold coordinates in its normal form where bifurcation analysis can be performed. The unique integration of analytical and data-driven approaches in the proposed method enables the learning of these transformations and the performance of bifurcation analysis with a limited number of trajectories. Therefore, this approach improves bifurcation analysis in model-less experimental systems and cost-sensitive high-fidelity simulations. The effectiveness of this approach is demonstrated across several examples.

A nonlinear reduced-order model evaluated with a comprehensive index for dual-excitation flow separation control

Single periodic excitation is widely used in unsteady flow separation control. However, in some two-dimensional (2D) or three-dimensional (3D) flow separation scenarios, dual excitations are necessary or advantageous. Referring to dual-excitation flow separation control, a nonlinear reduced-order model is established in this study to describe the interaction between dual external excitations with the internal flow instability of the separated flow and to provide physical insight into flow control mechanisms. The model is based on the Navier-Stokes equation, flow images of typical 2D or 3D flow separation, and the Stuart–Landau model. Additionally, a comprehensive index is proposed to evaluate flow control performance reflected by the reduced-order model, which accounts for flow losses from time-averaged and fluctuating flow, thus providing a more comprehensive evaluation of the model characteristics. Furthermore, we compare the results of the model to some numerical and experimental results of typical 2D and 3D flow separation control with dual excitations. The agreement of the model results and the numerical and experimental results verify the reduced-order model and its comprehensive index to a certain degree.

Registration-based nonlinear model reduction of parametrized aerodynamics problems with applications to transonic Euler and RANS flows

We develop a registration-based nonlinear model-order reduction (MOR) method for partial differential equations (PDEs) with applications to transonic Euler and Reynolds-averaged Navier–Stokes (RANS) equations in aerodynamics. These PDEs exhibit discontinuous features, namely shocks, whose location depends on problem configuration parameters, and the associated parametric solution manifold exhibits a slowly decaying Kolmogorov N-width. As a result, conventional linear MOR methods, which use linear reduced approximation spaces, do not yield accurate low-dimensional approximations. We present a registration-based nonlinear MOR method to overcome this challenge. Our formulation builds on the following key ingredients: (i) a geometrically transformable parametrized PDE discretization; (ii) localized spline-based parametrized transformations which warp the domain to align discontinuities; (iii) an efficient dilation-based shock sensor and metric to compute optimal transformation parameters; (iv) hyperreduction and online-efficient output-based error estimates; and (v) simultaneous transformation and adaptive finite element training. Compared to existing methods in the literature, our formulation is efficiently scalable to larger problems and is equipped with error estimates and hyperreduction. We demonstrate the effectiveness of the method on two-dimensional inviscid and turbulent flows modeled by the Euler and RANS equations, respectively.

Physics-informed two-tier neural network for non-linear model order reduction

In recent years, machine learning (ML) has had a great impact in the area of non-intrusive, non-linear model order reduction (MOR). However, the offline training phase often still entails high computational costs since it requires numerous, expensive, full-order solutions as the training data. Furthermore, in state-of-the-art methods, neural networks trained by a small amount of training data cannot be expected to generalize sufficiently well, and the training phase often ignores the underlying physical information when it is applied with MOR. Moreover, state-of-the-art MOR techniques that ensure an efficient online stage, such as hyper reduction techniques, are either intrusive or entail high offline computational costs. To resolve these challenges, inspired by recent developments in physics-informed and physics-reinforced neural networks, we propose a non-intrusive, physics-informed, two-tier deep network (TTDN) method. The proposed network, in which the first tier achieves the regression of the unknown quantity of interest and the second tier rebuilds the physical constitutive law between the unknown quantities of interest and derived quantities, is trained using pretraining and semi-supervised learning strategies. To illustrate the efficiency of the proposed approach, we perform numerical experiments on challenging non-linear and non-affine problems, including multi-scale mechanics problems.

A nonlinear manifold model order reduction approach for thermo‐mechanically coupled plasticity

AbstractThis study employs a nonlinear manifold mode order reduction (MOR) approach to address the high computational cost inherent in multi‐physical nonlinear models, which are discretized by the finite element method (FEM), with a particular focus on thermo‐mechanically coupled plasticity under finite strain conditions. Utilizing nodal snapshots obtained from a full‐order simulation, a projection matrix is constructed through singular value decomposition (SVD), facilitating the construction of a reduced system within a lower‐dimensional space. This reduction process entails the multiplication of the discretized residual vector and stiffness matrix by a linear projection matrix, complemented by the incorporation of a nonlinear projection component to ensure accurate reconstruction within the nonlinear part. Subsequently, a comprehensive three‐dimensional benchmark test is conducted to assess the accuracy and efficacy of the reduced‐order model.

Nonlinear Reduced Order Modeling of Heated Structures with Temperature-Dependent Properties

This paper focuses on extending coupled structural–thermal reduced order modeling approaches for the prediction of the nonlinear geometric response of heated structures to efficiently account for temperature-dependent structural and thermal properties. Structurally, a linear dependence of the elasticity tensor and thermal expansion coefficient is assumed consistently with typical material behavior. The resulting structural reduced order model (ROM) equations exhibit polynomials of the temperature generalized coordinates, the coefficients of which are readily identified. A different strategy is proposed for conductance and capacitance properties which typically depend nonlinearly on temperature. It relies on splitting the temperature generalized coordinates into a small set of dominant ones having a nonlinear effect on the ROM conductances and capacitances and a much larger set affecting them linearly. The inclusion of uncertainty in the structural ROM is also addressed extending earlier work limited to temperature independent properties. These strategies are exemplified on a representative hypersonic vehicle panel undergoing a full mission profile and with aero-thermal–structural coupling. The ROM predictions are observed to be very close to the full finite element ones for the mean model. Moreover, an uncertainty analysis demonstrates the strong sensitivity of the response to the structural model in the strongly nonlinear regions of the trajectory.

Nonlinear dynamics of wing-like structures using a momentum subspace-based Koiter-Newton reduction

Cantilevers find a wide range of applications in the design of scientific equipment and large-scale engineering structures such as aircraft wings. Analysis techniques based on linearization approximations are unable to capture the large amplitude oscillation behaviour of such structures and thus, necessitates development of dedicated nonlinear methods. In this work, the recent developments in the Koiter-Newton model reduction method are utilized to obtain nonlinear reduced order models (ROMs) from full finite element structural models in order to simulate large amplitude dynamics of cantilevers. The method describes a nonlinear system of governing equations comprising quadratic and cubic terms which are obtained as higher order derivatives of the in-plane strain energy. To ensure that the large rotations in cantilevers and the resultant foreshortening effect is also accounted for, a ROM updating algorithm is adopted where the ROM parameters are varied with the structural deflections. Linear eigenmodes of the structure are utilized to form the reduction subspace. To validate the methodology, the ROM solution is compared against experimental results and a convergence study is conducted to identify the number of modes needed to replicate the nonlinear response. Finally, a composite wingbox structure is considered for which time domain simulations are conducted and frequency response curves, obtained through a frequency sweep, are presented.

Evaluating the thermal stability of an interior permanent magnet synchronous machine through iterative multi‐physics simulation

AbstractThis paper investigates the thermal operating capability of an interior permanent magnet synchronous machine. An iterative workflow is presented, combining electromagnetic modeling, control, power‐loss modeling, and thermal modeling to identify maximum thermal stable operating points. Special attention is given to the critical temperatures of winding and permanent magnets. A nonlinear reduced order model based on finite element method model was used to simulate the system, including an inverter model with space vector pulse width modulation in combination with field‐oriented control. Furthermore, an advanced iron core loss model and thermal lumped parameter model were employed. The presented approach allows for evaluating losses and their impact on steady‐state temperatures. The obtained results highlight the significant influence of space vector pulse width modulation on iron core losses and the importance of considering both advanced power‐loss models and adequate thermal models when analyzing the machine's thermal state. This research emphasizes the concept of a thermally stable envelope, providing a comprehensive understanding of the thermal boundaries under various operating conditions.

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.

Effect of background noise characteristics on early warning indicators of thermoacoustic instability

In this work, we investigate the effects of background noise characteristics — specifically noise color (or correlation time) and intensity — arising from velocity-coupled and additive noise sources, on the early warning indicators (EWIs) of thermoacoustic instability. We employ a nonlinear reduced-order combustion dynamics model for our investigation. In the absence of noise, the nonlinear model undergoes a subcritical Hopf bifurcation, and our focus lies within the linearly stable region of the system (subthreshold region). The studied class of EWIs encompasses those derived from time series (variance), spectral analysis (coherence factor), Hurst exponent, and nonlinear methods (permutation entropy). We find that when the background noise is purely multiplicative, the trends in EWIs are primarily influenced by noise characteristics rather than the control parameter. Further, the EWIs cannot be estimated at low noise levels for large correlation times. In case of purely additive noise driven system, the coherence factor and variance are reliable EWIs across all range of investigated noise characteristics. The Hurst exponent can serve as effective EWI when the system features large noise correlation times, while permutation entropy is effective only when the system features small noise correlation time, i.e., where white noise assumption is acceptable. When the background noise includes contributions from both multiplicative and additive sources, coherence factor and variance emerges as the most reliable EWIs. These results provide insights for selecting appropriate EWIs to be employed in practical systems, considering potential variations in noise characteristics with simultaneous changes in combustor operating conditions.Novelty and significanceThis study conclusively shows that background noise (multiplicative and additive) characteristics – noise correlation time (color) and intensity – can significantly change the trends in early warning indicators (EWIs) for predicting the impending thermoacoustic oscillations. In gas turbine combustors, where thermoacoustic instability poses a critical challenge, inherent noise dynamics undergo variations with changing operating conditions and combustor designs. The development of effective EWIs to foresee the onset of thermoacoustic instability is crucial for preventing potential damage and ensuring the reliable operation of combustion systems. Previous works on stochastic dynamics of combustors simplify noise with the additive white noise assumption. Thus, our results demonstrated on a nonlinear combustion dynamics model advance the state-of-the-art with respect to the early prediction and control of thermoacoustic instability. The results provide valuable insights for selecting appropriate EWIs for engine monitoring in the absence of information on noise properties and their variation with operating conditions. This practical information is of direct relevance and interest to any gas turbine manufacturer/user.

Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems

Complex mechanical systems often exhibit strongly nonlinear behavior due to the presence of nonlinearities in the energy dissipation mechanisms, material constitutive relationships, or geometric/connectivity mechanics. Numerical modeling of these systems leads to nonlinear full-order models that possess an underlying Lagrangian structure. This work proposes a Lagrangian operator inference method enhanced with structure-preserving machine learning to learn nonlinear reduced-order models (ROMs) of nonlinear mechanical systems. This two-step approach first learns the best-fit linear Lagrangian ROM via Lagrangian operator inference and then presents a structure-preserving machine learning method to learn nonlinearities in the reduced space. The proposed approach can learn a structure-preserving nonlinear ROM purely from data, unlike the existing operator inference approaches that require knowledge about the mathematical form of nonlinear terms. From a machine learning perspective, it accelerates the training of the structure-preserving neural network by providing an informed prior (i.e., the linear Lagrangian ROM structure), and it reduces the computational cost of the network training by operating on the reduced space. The method is first demonstrated on two simulated examples: a conservative nonlinear rod model and a two-dimensional nonlinear membrane with nonlinear internal damping. Finally, the method is demonstrated on an experimental dataset consisting of digital image correlation measurements taken from a lap-joint beam structure from which a predictive model is learned that captures amplitude-dependent frequency and damping characteristics accurately. The numerical results demonstrate that the proposed approach yields generalizable nonlinear ROMs that exhibit bounded energy error, capture the nonlinear characteristics reliably, and provide accurate long-time predictions outside the training data regime.

Analysis and control of large-signal stability with storage in small scale islanded AC microgrids

The analysis of small-signal stability in AC microgrids has yielded numerous findings. However, investigating large-signal stability remains a major difficulty and has received limited attention in research. The article presents the establishment of a non-linear reduced-order model for small-scale AC microgrids with storage. The model considers the changes in converter structure that occur during large disturbances. Next, we will analyse the system’s stability under large-signal conditions and identify the underlying unstable mechanism. The impact of limitation links on system stability is significant, as they result in structural changes. The proposal of “virtual synchronous damping control” (VSD) aims to improve the stability of AC microgrids by addressing the issues caused by an unstable mechanism. The validation of the proposed analysis and control methods are demonstrated by examining relevant case studies in MATLAB/Simulink.

Weakly nonlinear analysis of thermoacoustic oscillations in can-annular combustors

Can-annular combustors feature clusters of thermoacoustic eigenvalues, which originate from the weak acoustic coupling between $N$ identical cans at the downstream end. When instabilities occur, one needs to consider the nonlinear interaction between all $N$ modes in the unstable cluster in order to predict the steady-state behaviour. A nonlinear reduced-order model for the analysis of this phenomenon is developed, based on the balance equations for acoustic mass, momentum and energy. Its linearisation yields explicit expressions for the $N$ complex-valued eigenfrequencies that form a cluster. To treat the nonlinear equations semianalytically, a Galerkin projection is performed, resulting in a nonlinear system of $N$ coupled oscillators. Each oscillator represents the dynamics of a global mode that oscillates in the whole can-annular combustor. The analytical expressions of the equations reveal how the geometrical and thermofluid parameters affect the thermoacoustic response of the system. To gain further insights, the method of averaging is applied to obtain equations for the slow-time dynamics of the amplitude and phase of each mode. The averaged system, whose solutions compare very well with those of the full oscillator equations, is shown to be able to predict complex transient dynamics. A variety of dynamical states are identified in the steady-state oscillatory regime, including push–push (in-phase) and spinning oscillations. Notably, the averaged equations are able to predict the existence of synchronised states. These states occur when the frequencies of two (or more) unstable modes with nominally different frequencies lock onto a common frequency as a result of nonlinear interactions.

Non-intrusive, transferable model for coupled turbulent channel-porous media flow based upon neural networks

Turbulent flow over permeable interfaces is omnipresent featuring complex flow topology. In this work, a data-driven, end-to-end machine learning model has been developed to model the turbulent flow in porous media. For the same, we have derived a non-linear reduced order model (ROM) with a deep convolution autoencoder. This model can reduce highly resolved spatial dimensions, which is a prerequisite for direct numerical simulation, by 99%. A downstream recurrent neural network has been trained to capture the temporal trend of reduced modes; thus, it is able to provide future evolution of modes. We further evaluate the trained model's capability on a newer dataset with a different porosity. In such cases, fine-tuning could reduce the efforts (up to two-order of magnitude) to train a model with limited dataset (10%) and knowledge and still show a good agreement on the mean velocity profile. Especially, the fine-tuned model shows a better agreement in the porous domain than the channel and interface areas indicating the topological feature is less challenging for training than the multi-scale nature of the turbulent flows. Leveraging the current model, we find that even quick fine-tuning achieves an impressive order-of-magnitude reduction in training time by approximately O(102) and still results in effective flow predictions. This promising discovery encourages the fast development of a substantial amount of data-driven models tailored for various types of porous media. The diminished training time substantially lowers the computational cost when dealing with changing porous topologies, making it feasible to systematically explore interface engineering with different types of porous media.

Hyper-reduction for parametrized transport dominated problems via adaptive reduced meshes

We propose an efficient residual minimization technique for the nonlinear model-order reduction of parameterized hyperbolic partial differential equations. Our nonlinear approximation space is spanned by snapshots functions over spatial transformations, and we compute our reduced approximation via residual minimization. To speedup the residual minimization, we compute and minimize the residual on a (preferably small) subset of the mesh, the so-called reduced mesh. We show that, similar to the solution, the residual also exhibits transport-type behaviour. To account for this behaviour, we introduce adaptivity in the reduced mesh by “moving” it along the spatial domain depending on the parameter value. Numerical experiments showcase the effectiveness of our method and the inaccuracies resulting from a non-adaptive reduced mesh.

Domain Decomposition Strategy for Combining Nonlinear and Linear Reduced-Order Models

Increasingly ubiquitous reliance on expensive, high-fidelity numerical simulations has led to the emergence of reduced-order modeling as an effective method to predict high-dimensional, spatially discretized field outputs in an efficient manner. This study presents a method to construct parametric, nonintrusive reduced-order models (ROMs) by leveraging domain decomposition (DD) to enable using multiple dimension reduction (DR) techniques within different spatial subdomains of the field to construct a single predictive ROM. To enforce smoothness of solutions across subdomains, points at domain interfaces are reconstructed using the data-driven gappy proper orthogonal decomposition method. The method is assessed on three high-dimensional test cases with shocks, including one with two-dimensional turbulent flow around the RAE2822. Results show that fully nonlinear ROMs predict fields more accurately in the vicinity of discontinuous features compared to their linear counterparts, which are more accurate in regions that are far from these discontinuities. Furthermore, irrespective of the DR method, DD-based ROMs outperformed their global counterparts in all test cases. Finally, models that used a combination of DR methods simultaneously showed superior performance near shockwaves and a significant improvement in total error compared to existing approaches that employ a single DR method in all subdomains.

A graph convolutional autoencoder approach to model order reduction for parametrized PDEs

The present work proposes a framework for nonlinear model order reduction based on a Graph Convolutional Autoencoder (GCA-ROM). In the reduced order modeling (ROM) context, one is interested in obtaining real-time and many-query evaluations of parametric Partial Differential Equations (PDEs). Linear techniques such as Proper Orthogonal Decomposition (POD) and Greedy algorithms have been analyzed thoroughly, but they are more suitable when dealing with linear and affine models showing a fast decay of the Kolmogorov n-width. On one hand, the autoencoder architecture represents a nonlinear generalization of the POD compression procedure, allowing one to encode the main information in a latent set of variables while extracting their main features. On the other hand, Graph Neural Networks (GNNs) constitute a natural framework for studying PDE solutions defined on unstructured meshes. Here, we develop a non-intrusive and data-driven nonlinear reduction approach, exploiting GNNs to encode the reduced manifold and enable fast evaluations of parametrized PDEs. We show the capabilities of the methodology for several models: linear/nonlinear and scalar/vector problems with fast/slow decay in the physically and geometrically parametrized setting. The main properties of our approach consist of (i) high generalizability in the low-data regime even for complex behaviors, (ii) physical compliance with general unstructured grids, and (iii) exploitation of pooling and un-pooling operations to learn from scattered data.

Data-driven nonlinear parametric model order reduction framework using deep hierarchical variational autoencoder

A data-driven parametric model order reduction (MOR) method using a deep artificial neural network is proposed. The present network, a least-squares hierarchical variational autoencoder (LSH-VAE), is capable of performing nonlinear MOR for the parametric interpolation of a nonlinear dynamic system with a significant number of degrees of freedom. LSH-VAE differs from existing networks in two major respects: a deep hierarchical structure and a hybrid weighted, probabilistic loss function. The enhancements result in significantly improved accuracy and stability compared with conventional nonlinear MOR methods, autoencoders, and variational autoencoders. In LSH-VAE, the parametric MOR framework is based on the spherically linear interpolation of the latent manifold. The present framework is validated and evaluated on three nonlinear and multiphysics dynamic systems. First, the present framework is evaluated on the fluid–structure interaction benchmark problem to assess its efficiency and accuracy. Then, a highly nonlinear aeroelastic phenomenon, limit cycle oscillation, is analyzed. Finally, the framework is applied to three-dimensional fluid flow to demonstrate its capability to efficiently analyze a significantly large number of degrees of freedom. The superior performance of LSH-VAE is emphasized by comparing its results against those of widely used nonlinear MOR methods, a convolutional autoencoder, and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-VAE. The proposed framework exhibits significantly enhanced accuracy compared with that of conventional methods, while the computational efficiency continues to remain significantly higher.

Learning-based nonlinear model order reduction for image reconstruction

Learning-based nonlinear model order reduction for image reconstruction

Analysis of medium-frequency oscillation based on bifurcation theory and joint modelling of SVG and direct-drive wind turbine

With the expansion of wind power clusters in power system, medium and high frequency oscillations in wind farm area are preliminarily emerging. At present, linear analysis such as impedance analysis are mainly used to analyse the grid-connected oscillation of renewable energy generation. However, the explanation of the oscillation mechanism is not intuitive and does not take into account the nonlinearity of converter. Considering the influence of external loop control of SVG and direct-drive wind turbine on the medium-frequency characteristics of the parallel system, a reduced-order nonlinear model under voltage time scale is established. The conjoint analysis of Jacobi matrix, bifurcation diagram and phase space trajectory are applied to study the influence rule of machine-side output power and short-circuit ratio on the oscillation behaviour of the parallel system. In addition, the oscillation evolution process is analysed from the dynamic standpoint. It is found that the system operates on the limit-loop state when the medium-frequency oscillation occurs, which reveals the mapping relationship between the oscillation amplitude and the limit-loop trajectory. The experimental results ultimately validate the theoretical analysis conducted on the StarSim-HIL (hardware-in-the-loop) experiment platform.