Construction Of Reduced-order Model Research Articles

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Reduced-Order Model Approaches for Predicting Airfoil Performance

This study delves into the construction of reduced-order models (ROMs) of a flow field over a NACA 0012 airfoil at a moderate Reynolds number and an angle of attack of 8∘. Numerical simulations were computed through the finite-volume solver OpenFOAM. The analysis considers two different reduction techniques: the standard Galerkin projection method, which involves projecting the governing equations onto proper orthogonal decomposition modes (POD−ROMs), and the cluster-based network model (CNM), a fully data-driven nonlinear approach. An analysis of the topology of the dominant POD modes was conducted, uncovering a traveling wave pattern in the wake dynamics. We compared the performances of both ROM techniques regarding their prediction of flow field behavior and integral quantities. The ROM framework facilitates the practical actuation of control strategies with significantly reduced computational demands compared to the full-order approach.

Estimation of the Born data in inverse scattering of layered media

The first term in the Born series, as we call the Born data, is linear with the scatterers. Here we present a scheme to map the total field data to the Born data in layered media using only the single-input single-output (SISO) setup. This nonlinear mapping is based on the reduced order model (ROM) approach, which constructs ROMs of the original wave operator. Normally, the construction of ROMs requires multi-input multi-output data. By introducing fictitious sensors, we estimate the Born data with SISO data in layered media. We give a simple way of using the time-domain Green’s function to estimate the received data for other fictitious sensors without calculating the complicated Sommerfeld integral. The resulting Born data contains only the single-scattering component, which can be helpful for many imaging applications. A numerical example is given incorporating the direct imaging back-propagation method. It validates the linearity of the Born data by providing an artifact-free image without the optimization process.

Data-driven modelling of brain activity using neural networks, diffusion maps, and the Koopman operator.

We propose a machine-learning approach to construct reduced-order models (ROMs) to predict the long-term out-of-sample dynamics of brain activity (and in general, high-dimensional time series), focusing mainly on task-dependent high-dimensional fMRI time series. Our approach is a three stage one. First, we exploit manifold learning and, in particular, diffusion maps (DMs) to discover a set of variables that parametrize the latent space on which the emergent high-dimensional fMRI time series evolve. Then, we construct ROMs on the embedded manifold via two techniques: Feedforward Neural Networks (FNNs) and the Koopman operator. Finally, for predicting the out-of-sample long-term dynamics of brain activity in the ambient fMRI space, we solve the pre-image problem, i.e., the construction of a map from the low-dimensional manifold to the original high-dimensional (ambient) space by coupling DMs with Geometric Harmonics (GH) when using FNNs and the Koopman modes per se. For our illustrations, we have assessed the performance of the two proposed schemes using two benchmark fMRI time series: (i) a simplistic five-dimensional model of stochastic discrete-time equations used just for a "transparent" illustration of the approach, thus knowing a priori what one expects to get, and (ii) a real fMRI dataset with recordings during a visuomotor task. We show that the proposed Koopman operator approach provides, for any practical purposes, equivalent results to the FNN-GH approach, thus bypassing the need to train a non-linear map and to use GH to extrapolate predictions in the ambient space; one can use instead the low-frequency truncation of the DMs function space of L2-integrable functions to predict the entire list of coordinate functions in the ambient space and to solve the pre-image problem.

Novel attention-based convolutional autoencoder and ConvLSTM for reduced-order modeling in fluid mechanics with time derivative architecture

To construct reduced-order models, we propose a convolutional autoencoder and a convolutional LSTM (CAE-ConvLSTM). Using convolutional LSTMs, our model preserves spatial information at the level of representative features, which are extracted using convolutional autoencoding. A block attention algorithm was proposed for improving the convolutional layer to capture long-range dependency in features. According to the results, the concatenated version of this attention mechanism can improve the performance of the model. A first- and second-order time derivative architecture is employed to modify CAE-ConvLSTM. Future forecasting in this approach is not only based on current conditions but also on dynamic representations of input that enable better features to be captured. The results indicate that the model’s accuracy has improved significantly. On the basis of the obtained results, more than 3000 and 30 times better accuracy were achieved on cylinder and dam break test cases, as well as the ability to predict the next 20 steps.

Adaptive Model Reduction of High-Order Solutions of Compressible Flows via Optimal Transport

The solution of conservation laws with parametrised shock waves presents challenges for both high-order numerical methods and model reduction techniques. We introduce an r-adaptivity scheme based on optimal transport and apply it to develop reduced order models for compressible flows. The optimal transport theory allows us to compute high-order r-adaptive meshes from a starting reference mesh by solving the Monge–Ampère equation. A high-order discretization of the conservation laws enables high-order solutions to be computed on the resulting r-adaptive meshes. Furthermore, the Monge–Ampère solutions contain mappings that are used to reduce the spatial locality of the resulting solutions and make them more amenable to model reduction. We use a non-intrusive model reduction method to construct reduced order models of both the mesh and the solution. The procedure is demonstrated on three supersonic and hypersonic test cases, with the hybridisable discontinuous Galerkin method being used as the full order model.

Model order reduction strategies for weakly dispersive waves

We focus on the numerical modeling of water waves by means of depth averaged models. We consider in particular PDE systems which consist in a nonlinear hyperbolic model plus a linear dispersive perturbation involving an elliptic operator. We propose two strategies to construct reduced order models for these problems, with the main focus being the control of the overhead related to the inversion of the elliptic operators, as well as the robustness with respect to variations of the flow parameters. In a first approach, only a linear reduction strategies is applied only to the elliptic component, while the computations of the nonlinear fluxes are still performed explicitly. This hybrid approach, referred to as pdROM, is compared to a hyper-reduction strategy based on the empirical interpolation method to reduce also the nonlinear fluxes. We evaluate the two approaches on a variety of benchmarks involving a generalized variant of the BBM–KdV model with a variable bottom, and a one-dimensional enhanced weakly dispersive shallow water system. The results show the potential of both approaches in terms of cost reduction, with a clear advantage for the pdROM in terms of robustness, and for the EIMROM in terms of cost reduction.

DEIM-embedded hybrid snapshot simulation for reduced order model generation

PurposeThe purpose of this paper is to establish an intelligent framework to generate the data representatives in snapshot simulation in order to construct the online reduced-order model based on the generated data information. It could greatly reduce the computational time in snapshot simulation and accelerate the computational efficiency in the real-time computation of reduced-order modeling.Design/methodology/approachThe snapshot simulation, which generates the data to construct reduced-order models (ROMs), usually is computationally demanding. In order to accelerate the snapshot generation, this paper presents a discrete element interpolaiton method (DEIM)-embedded hybrid simulation approach, in which the entire snapshot simulation is partitioned into multiple intervals of equal length. One of the three models: the full order model (FOM), local ROM, or local ROM-DEIM which represents a hierarchy of model approximations, fidelities and computational costs, will be adopted in each interval.FindingsThe outcome of the proposed snapshot simulation is an efficient ROM-DEIM applicable to various online simulations. Compared with the traditional FOM and the hybrid method without DEIM, the proposed method is able to accelerate the snapshot simulation by 54.4%–63.91% and 10.5%–27.85%, respectively. In the online simulation, ROM-DEIM only takes 4.81%–8.56% of the computational time of FOM, while preserving excellent accuracy (with relative error <1%).Originality/value1. A DEIM-embedded hybrid snapshot simulation methodology is proposed to accelerate snapshot data generation and reduced-order model (ROM)-DEIM development. 2. The simulation alternates among FOM, ROM and ROM-DEIM to adaptively generate snapshot data of salient subspace representation while minimizing computational load. 3. The DEIM-embedded hybrid snapshot approach demonstrates excellent accuracy (<1% error) and computational efficiency in both online snapshot simulation and online ROM-DEIM verification simulation.

Local Lagrangian reduced-order modeling for the Rayleigh-Taylor instability by solution manifold decomposition

The Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various disciplines of science and engineering, including astrophysics, atmospheric sciences and climate, geophysics, and fusion energy. Analytical methods cannot be applied to explain the long-time behavior of the Rayleigh-Taylor instability, and therefore, numerical simulation of the full problem is required. However, in order to capture the growth of amplitude of perturbations accurately, both the spatial and temporal discretizations need to be extremely fine for traditional numerical methods, and long-time simulation may become prohibitively expensive. In this paper, we propose efficient reduced order model techniques to accelerate the simulation of the Rayleigh-Taylor instability in compressible gas dynamics. We introduce a general framework for decomposing the solution manifold to construct the temporal domain partition and temporally-local reduced order model construction with varying Atwood number. We propose two practical approaches in this framework, namely decomposition by physical time and by penetration distance. Numerical results are presented to examine the performance of the proposed approaches.

A Simple Matlab Code for Material Design Optimization Using Reduced Order Models.

The main part of the computational cost required for solving the problem of optimal material design with extreme properties using a topology optimization formulation is devoted to solving the equilibrium system of equations derived through the implementation of the finite element method (FEM). To reduce this computational cost, among other methodologies, various model order reduction (MOR) approaches can be utilized. In this work, a simple Matlab code for solving the topology optimization for the design of materials combined with three different model order reduction approaches is presented. The three MOR approaches presented in the code implementation are the proper orthogonal decomposition (POD), the on-the-fly reduced order model construction and the approximate reanalysis (AR) following the combined approximations approach. The complete code, containing all participating functions (including the changes made to the original ones), is provided.

A frequency domain approach for reduced- order transonic aerodynamic modelling

AbstractThis paper describes a new efficient method for the construction of an approximately balanced aerodynamic Reduced Order Model (ROM) via the frequency domain using Computational Fluid Dynamics data. Time domain ROM construction requires CFD data, which is obtained from the DLR TAU RANS or Euler Linearised Frequency Domain (LFD) solver. The ROMs produced with this approach, using a small number of frequency simulations, are shown to exhibit a strong ability to reconstruct the system response for inviscid flow about the NLR7301 aerofoil and the FFAST wing; and viscous flow about the NASA Common Research Model. The latter demonstrates that the reduced order model approach can reconstruct the full order frequency response of a viscous aircraft configuration with excellent accuracy using a strip wise approach. The time domain models are built using the frequency domain, but also give promising results when applied to reconstruct non-periodic motions. Results are compared to time domain simulations, showing good agreement even with small ROM sizes, but with a substantial reduction in calculation time. The main advantage of the current model order reduction approach is that the method does not require the formation and storage of large matrices, such as in POD approaches.

Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response

In Part I of this paper, we have used spectral submanifold (SSM) theory to construct reduced-order models for harmonically excited mechanical systems with internal resonances. In that setting, extracting forced response curves formed by periodic orbits of the full system was reduced to locating the solution branches of equilibria of the corresponding reduced-order model. Here, we use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system. Specifically, we identify Hopf bifurcations of equilibria and limit cycles in reduced models on SSMs to predict the existence of two-dimensional and three-dimensional quasi-periodic attractors and repellers in periodically forced mechanical systems of arbitrary dimension. We illustrate the accuracy and efficiency of these computations on finite-element models of beams and plates.

Weakly-invasive LATIN-PGD for solving time-dependent non-linear parametrized problems in solid mechanics

A weakly-invasive version of the LATIN-PGD method is introduced to compute reduced-order models in solid mechanics for time-dependent non-linear parametrized problems under quasi-static assumption. Its main interest is to be easily implemented in any general-purpose industrial finite element software while taking advantage of all facilities offered, including the ability to handle any kind of non-linearities. This way of proceeding provides unified tools – for the construction of reduced-order models in non-linear context – all integrated in one certified product in consistency with the purpose of having end-to-end processes. Possibilities of our approach are illustrated thanks to the first implementation conducted within Simcenter Samcef™ software developed by Siemens Digital Industries Software without ever redeveloping any non-linear part of the software. Finally, interesting performance gains are highlighted on a non-linear time-dependent test-case involving some parameters.

Parameterized Periodic Steady-State Analysis via Reduced-Order Modeling

This article presents a new approach to construct reduced-order models for large nonlinear circuits excited by periodical or quasi-periodical sources. The key feature in the constructed reduced model is that it enables, through solving a much smaller system, tracing the variations in the periodical steady-state response of the original circuit with regards to variations in selected design parameters. The proposed approach is based on the notion of using discrete empirical interpolation to project the nonlinear operator of the circuit equation onto a smaller subspace. In addition, the original system equations are projected onto the subspace spanned by the first few derivatives of the solution with respect to the selected design parameters. Computational savings are made possible through solving a reduced system of equations instead of the full system. Numerical examples are presented to validate the accuracy and efficiency of the proposed approach.

Machine Learning Tools Set for Natural Gas Fuel Cell System Design

This study is focusing on leveraging the system design tools set for next-generation solid oxide fuel cell (SOFC) based natural gas fuel cell (NGFC) system. Conventionally, system design and optimization of NGFC systems rely heavily on traditional reduced order model (ROM) techniques and designers’ experience level. For overcoming the technical barriers of system design, multiple multi-physics models and machine learning (ML) tools have been utilized to automate the conceptual design process and enhance the reliability of solutions for the NGFC system. The proposed tools set includes a physics-informed ML tool for automated ROM construction that leverages advances in deep neural networks to significantly reduce ROM prediction error for the NGFC power island compared to traditional approaches. The constructed physics-informed ML ROM can be used in system design and optimization tools set Institute for the Design of Advanced Energy Systems (IDAES) Process Systems Engineering (PSE) framework. The tools set also provides user-friendly graphic user interface built within Jupyter Notebooks, and the whole tools set is open-source public available.

Extended dynamic mode decomposition for inhomogeneous problems

Dynamic mode decomposition (DMD) is a powerful data-driven technique for construction of reduced-order models of complex dynamical systems. Multiple numerical tests have demonstrated the accuracy and efficiency of DMD, but mostly for systems described by homogeneous partial differential equations (PDEs) with homogeneous boundary conditions. We propose an extended dynamic mode decomposition (xDMD) approach to cope with the potential unknown sources/sinks in PDEs. Motivated by similar ideas in deep neural networks, we equip our xDMD with two new features. First, it has a bias term, which accounts for inhomogeneity of PDEs and/or boundary conditions. Second, instead of learning a flow map, xDMD learns the residual increment by subtracting the identity operator. Our theoretical error analysis demonstrates the improved accuracy of xDMD relative to standard DMD. Several numerical examples are presented to illustrate this result.

Machine Learning Tools Set for Natural Gas Fuel Cell System Design

This study is focusing on leveraging the system design tools set for next-generation solid oxide fuel cell (SOFC) based natural gas fuel cell (NGFC) system. Conventionally, system design and optimization of NGFC systems rely heavily on traditional reduced order model (ROM) techniques and designers’ experience level. For overcoming the technical barriers of system design, multiple multi-physics models and machine learning (ML) tools have been utilized to automate the conceptual design process and enhance the reliability of solutions for the NGFC system. The proposed tools set includes a physics-informed ML tool for automated ROM construction that leverages advances in deep neural networks to significantly reduce ROM prediction error for the NGFC power island compared to traditional approaches. The constructed physics-informed ML ROM can be used in system design and optimization tools set Institute for the Design of Advanced Energy Systems (IDAES) Process Systems Engineering (PSE) framework. The tools set also provides user-friendly graphic user interface built within Jupyter Notebooks, and the whole tools set is open-source public available.

Data-driven discovery of Koopman eigenfunctions for control

Data-driven transformations that reformulate nonlinear systems in a linear framework have the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics, and its eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly. Previous studies have used finite-dimensional approximations of the Koopman operator for model-predictive control approaches. In this work, we illustrate a fundamental closure issue of this approach and argue that it is beneficial to first validate eigenfunctions and then construct reduced-order models in these validated eigenfunctions. These coordinates form a Koopman-invariant subspace by design and, thus, have improved predictive power. We show then how the control can be formulated directly in these intrinsic coordinates and discuss potential benefits and caveats of this perspective. The resulting control architecture is termed Koopman Reduced Order Nonlinear Identification and Control (KRONIC). It is further demonstrated that these eigenfunctions can be approximated with data-driven regression and power series expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. Validating discovered eigenfunctions is crucial and we show that lightly damped eigenfunctions may be faithfully extracted from EDMD or an implicit formulation. These lightly damped eigenfunctions are particularly relevant for control, as they correspond to nearly conserved quantities that are associated with persistent dynamics, such as the Hamiltonian. KRONIC is then demonstrated on a number of relevant examples, including (a) a nonlinear system with a known linear embedding, (b) a variety of Hamiltonian systems, and (c) a high-dimensional double-gyre model for ocean mixing.

DYNAMIC MODE DECOMPOSITION FOR CONSTRUCTION OF REDUCED-ORDER MODELS OF HYPERBOLIC PROBLEMS WITH SHOCKS

Construction of reduced-order models (ROMs) for hyperbolic conservation laws is notoriously challenging mainly due to the translational property and nonlinearity of the governing equations. While the Lagrangian framework for ROM construction resolves the translational issue, it is valid only before a shock forms. Once that occurs, characteristic lines cross each other and projection from a high-fidelity model space onto a ROM space distorts a moving grid, resulting in numerical instabilities. We address this grid distortion issue by developing a physics-aware dynamic mode decomposition (DMD) method based on hodograph transformation. The latter provides a map between the original nonlinear system and its linear counterpart, which coincides with the Koopman operator. This strategy is consistent with the spirit of physics-aware DMDs in that it retains information about shock dynamics. Several numerical examples are presented to validate the proposed physics-aware DMD approach to constructing accurate ROMs.

Balanced Reduced-Order Models for Iterative Nonlinear Control of Large-Scale Systems

We propose a new framework to design controllers for high-dimensional nonlinear systems. The control is designed through the iterative linear quadratic regulator (ILQR), an algorithm that computes control by iteratively applying the linear quadratic regulator on the local linearization of the system at each time step. Since ILQR is computationally expensive, we propose to first construct reduced-order models (ROMs) of the high-dimensional nonlinear system. We derive nonlinear ROMs via projection, where the basis is computed via balanced truncation (BT) and LQG balanced truncation (LQG-BT). Numerical experiments are performed on a semi-discretized nonlinear Burgers equation. We find that the ILQR algorithm produces good control on ROMs constructed either by BT or LQG-BT, with BT-ROM based controllers outperforming LQG-BT slightly for very low-dimensional systems.