Model Order Reduction Framework Research Articles

Data‐Driven Learning of Reduced‐Order Dynamics for a Parametrized Shallow Water Equation

AbstractA non‐intrusive data‐driven model order reduction method is introduced that learns low‐dimensional dynamical models for a parametrized non‐traditional shallow water equation (NTSWE) [1]. The reduced‐order model is learnt by setting an appropriate least‐squares optimization problem [2] in a low‐dimensional subspace. The non‐intrusive model order reduction framework is extended to a parametric case using the parameter dependency at the level of the partial differential equation. The efficiency of the proposed non‐intrusive method is illustrated to construct reduced‐order models for NTSWE and compared with an intrusive method, proper orthogonal decomposition with Galerkin projection. Furthermore the predictability of both models outside the range of the training data is discussed.

A GFEM-based reduced-order homogenization model for heterogeneous materials under volumetric and interfacial damage

This manuscript presents a multiscale reduced-order modeling framework for heterogeneous materials that accounts for both cohesive interface failure and continuum damage. The model builds on the eigendeformation-based reduced-order homogenization model (EHM), which relies on the pre-calculation of a set of coefficient tensors that account for the effects of linear and nonlinear material behavior between regions of the domain known as parts. A k-means clustering algorithm is used to optimally construct these parts and a new formulation for the partitioning of interfaces using this method is proposed. The extraction of the volumetric and interfacial influence functions is performed using the Interface-enriched Generalized Finite Element Method (IGFEM), which relies on a finite element discretization that does not conform to the material phase boundaries. A Lagrange multiplier method is used in this preprocessing phase, allowing for the reuse of the matrix factorization for different influence function problems and hence leading to efficiency improvement. A newly proposed traction calculation for the interface partition is also adopted to alleviate the instability caused by traction calculations along interfaces. The accuracy and efficiency of the IGFEM–EHM method is assessed through comparison with reference IGFEM simulations. The method is then used to extract the nonlinear multiscale response of particulate, unidirectional fiber-reinforced, and woven composites.

Learning stable reduced-order models for hybrid twins

AbstractThe concept of “hybrid twin” (HT) has recently received a growing interest thanks to the availability of powerful machine learning techniques. This twin concept combines physics-based models within a model order reduction framework—to obtain real-time feedback rates—and data science. Thus, the main idea of the HT is to develop on-the-fly data-driven models to correct possible deviations between measurements and physics-based model predictions. This paper is focused on the computation of stable, fast, and accurate corrections in the HT framework. Furthermore, regarding the delicate and important problem of stability, a new approach is proposed, introducing several subvariants and guaranteeing a low computational cost as well as the achievement of a stable time-integration.

Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation

Non-intrusive reduced-order models (ROMs) have recently generated considerable interest for constructing computationally efficient counterparts of nonlinear dynamical systems emerging from various domain sciences. They provide a low-dimensional emulation framework for systems that may be intrinsically high-dimensional. This is accomplished by utilizing a construction algorithm that is purely data-driven. It is no surprise, therefore, that the algorithmic advances of machine learning have led to non-intrusive ROMs with greater accuracy and computational gains. However, in bypassing the utilization of an equation-based evolution, it is often seen that the interpretability of the ROM framework suffers. This becomes more problematic when black-box deep learning methods are used which are notorious for lacking robustness outside the physical regime of the observed data. In this article, we propose the use of a novel latent-space interpolation algorithm based on Gaussian process regression. Notably, this reduced-order evolution of the system is parameterized by control parameters to allow for interpolation in space. The use of this procedure also allows for a continuous interpretation of time which allows for temporal interpolation. The latter aspect provides information, with quantified uncertainty, about full-state evolution at a finer resolution than that utilized for training the ROMs. We assess the viability of this algorithm for an advection-dominated system given by the inviscid shallow water equations.

Non-intrusive reduced order model of urban airflow with dynamic boundary conditions

To address the problem that urban wind field simulations are limited by high computational requirements, a supervised machine learning framework for the non-intrusive model order reduction of urban airflow is proposed to provide fast predictions with dynamic boundary conditions. An integrated atmospheric modeling system is established by coupling the Parallelized LES Model (PALM) with the Urban Canopy Model (UCM) which embedded in the Weather Research and Forecasting (WRF) model to simulate the high-resolution urban wind field. Within this machine learning framework, encoders and decoders are trained to compress and reproduce the full-order airflow data. The eXtreme Gradient Boost regression models (XGBoost) are trained to map the boundary conditions to the latent vectors in the encoders and decoders. Thus, high resolution urban airflow can be predicted rapidly by the XGBoost-decoder models with the input of boundary conditions. By comparing four types of encoding-decoding models (i.e., the proper orthogonal decomposition models, autoencoders with linear/nonlinear fully connected neural networks and autoencoders with Convolutional Neural Network (AE-CNN)), we find that the AE-CNN model with advantages of portable model size and convenient updating procedure is capable for high resolution simulations of urban airflow with dynamic boundary conditions at a low computational cost.

A framework for parametric reduction in large-scale nonlinear dynamical systems

This manuscript presents a general parametric model order reduction (pMOR) framework for nonlinear dynamical systems. At first, a family of local nonlinear reduced order models (ROMs) is generated for different parametric space vectors using the nonlinear moment matching (NLMM) scheme along-with the discrete empirical interpolation method (DEIM). Then, the nonlinear reduced order model for any new parameter is obtained by interpolating the neighbouring reduced order models after projecting them onto a universal subspace. The advantage of such a scheme is that the parametric dependency is maintained in the reduced nonlinear models. Finally, we substantiate our observations by a suite of numerical tests.

Reduced order framework for convection dominant and pure diffusive problems based on combination of deep long short‐term memory and proper orthogonal decomposition/dynamic mode decomposition methods

AbstractIn many real‐world applications, mathematical models are highly complex, and numerical simulations in high‐dimensional systems are challenging. Model order reduction is a useful method to obtain a reasonable approximation by significantly reducing the computational cost of such problems. Deep learning technology is a recent improvement in artificial neural networks that can find more hidden information from the data. Deep learning has the advantage of processing data in its raw form and trains the nonlinear system with different levels of representation and predicts the data. In this article, a reduced order model framework based on a combination of deep learning [long short‐term memory (LSTM)] and proper orthogonal decomposition/dynamic mode decomposition (POD/DMD) modes is presented. Due to the robustness and stability of the LSTM recurrent neural network in predicting chaotic dynamical systems, we consider LSTM architecture to develop our data‐driven reduced order modeling (ROM). We investigate the proposed method performance by solving two well‐known canonical cases: a steady shear flow exhibiting the Kelvin‐Helmholtz instability, and two‐dimensional and unsteady mass diffusion equation. The focus of this article is to use LSTM deep recursive neural network to learn the time dynamics and POD/DMD to generate the order reduction model. The results show that the proposed method is very accurate in predicting time dynamics and input reconstruction.

Fast interactive CFD evaluation of hemodynamics assisted by RBF mesh morphing and reduced order models: the case of aTAA modelling

The medical digital twin is emerging as a viable opportunity to provide patient-specific information useful for treatment, prevention and surgical planning. A bottleneck toward its effective use when computational fluid dynamics (CFD) techniques and tools are adopted for the high fidelity prediction of blood flow, is the significant computing cost required. Reduced order models (ROM) looks to be a promising solution for facing the aforementioned limit. In fact, once ROM data processing is accomplished, the consumption stage can be performed outside the computer-aided engineering software adopted for simulation and, in addition, it could be also implemented on interactive software visualization interfaces that are commonly employed in the medical context. In this paper we demonstrate the soundness of such a concept by numerically investigating the effect of the bulge shape for the ascending thoracic aorta aneurysm case. Radial basis functions (RBF) based mesh morphing enables the implementation of a parametric shape, which is used to build up the ROM framework and data. The final result is an inspection tool capable to visualize, interactively and almost in real-time, the effect of shape parameters on the entire flow field. The approach is first verified considering a morphing action representing the progression from an average healthy patient to an average aneurismatic one (Capellini et al. in Proceedings VII Meeting Italian Chapter of the European Society of Biomechanics (ESB-ITA 2017), 2017; Capellini et al. in J. Biomech. Eng. 140(11):111007-1–111007-10, 2018). Then, a set of shape parameters, suitable to consistently represent a widespread number of possible bulge configurations, are defined and accordingly generated. The concept is showcased taking into account the steady flow field at systolic peak conditions, using ANSYS®Fluent®and its ROM environment for CFD and ROM calculations respectively, and the RBF MorphTM software for shape parametrization.

Multi-fidelity bayesian optimization using model-order reduction for viscoplastic structures

One of the main issues when dealing with the numerical optimization of mechanical structures is the balance between computation time and model accuracy. The work presented herein aims at accelerating global optimization by using the framework of Bayesian optimization on a quantity of interest together with multiple levels of fidelity. These multi-fidelity data are generated from a model-order reduction framework: the LATIN Proper Generalized Decomposition. Within this framework, a reduced-order basis is generated on-the-fly and re-exploited to reduce the computational cost of observations. This strategy is illustrated on two elasto-viscoplastic test-cases for which significant speedups can be observed.

A POD-DEIM reduced model for compressible gas reservoir flow based on the Peng-Robinson equation of state

The efficient simulation of gas flow in porous media is highly required in petroleum engineering, CO2 sequestration, etc. However, it still remains a great challenge attributing to the gas compressibility compared with incompressible fluid flows. The commonly-used equation of states (EOS) of the gas are cubic equation and need to be updated in each iteration, and it also leads to the nonlinearity in flow equations, thus the computational cost mainly stems from the solution of gas EOS. In this paper, a hybrid reduced order model (ROM) coupling the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) is presented to accelerate the calculation of compressible single-phase gas reservoir flow, in which the Peng-Robinson EOS (P-R EOS) is considered to describe the gas states. To this end, in the hybrid ROM framework POD is applied to solve the flow equation and DEIM is used to solve the P-R EOS, respectively. The selection of POD modes and DEIM interpolation points, which plays a crucial role in the hybrid ROM, is discussed and carried out carefully. Performances of the proposed POD-DEIM-ROM are evaluated and demonstrated by two numerical cases. Simulation results illustrate that the proposed hybrid ROM displays a satisfactory computational speed-up (two orders of magnitudes faster) without sacrificing numerical accuracy significantly compared with the standard finite difference method. In addition, DEIM shows excellent acceleration and it is a perfect choice for solving the cubic gas EOS.

Stable compact modeling of piezoelectric energy harvester devices

Purpose Based on the framework of Krylov subspace-based model order reduction (MOR), compact models of the piezoelectric energy harvester devices can be generated. However, the stability of reduced piezoelectric model often cannot be preserved. In previous research studies, “MOR after Schur,” “Schur after MOR” and “multiphysics structure preserving MOR” methods have proven successful in obtaining stable reduced piezoelectric energy harvester models. Though the stability preservation of “MOR after Schur” and “Schur after MOR” methods has already been mathematically proven, the “multiphysics structure preserving MOR” method was not. This paper aims to provide the missing mathematical proof of “multiphysics structure preserving MOR.” Design/methodology/approach Piezoelectric energy harvesters can be represented by system of differential-algebraic equations obtained by the finite element method. According to the block structure of its system matrices, “MOR after Schur” and “Schur after MOR” both perform Schur complement transformations either before or after the MOR process. For the “multiphysics structure preserving MOR” method, the original block structure of the system matrices is preserved during MOR. Findings This contribution shows that, in comparison to “MOR after Schur” and “Schur after MOR” methods, “multiphysics structure preserving MOR” method performs the Schur complement transformation implicitly, and therefore, stabilizes the reduced piezoelectric model. Originality/value The stability preservation of the reduced piezoelectric energy harvester model obtained through “multiphysics structure preserving MOR” method is proven mathematically and further validated by numerical experiments on two different piezoelectric energy harvester devices.

Model order reduction assisted by deep neural networks (ROM-net)

In this paper, we propose a general framework for projection-based model order reduction assisted by deep neural networks. The proposed methodology, called ROM-net, consists in using deep learning techniques to adapt the reduced-order model to a stochastic input tensor whose nonparametrized variabilities strongly influence the quantities of interest for a given physics problem. In particular, we introduce the concept of dictionary-based ROM-nets, where deep neural networks recommend a suitable local reduced-order model from a dictionary. The dictionary of local reduced-order models is constructed from a clustering of simplified simulations enabling the identification of the subspaces in which the solutions evolve for different input tensors. The training examples are represented by points on a Grassmann manifold, on which distances are computed for clustering. This methodology is applied to an anisothermal elastoplastic problem in structural mechanics, where the damage field depends on a random temperature field. When using deep neural networks, the selection of the best reduced-order model for a given thermal loading is 60 times faster than when following the clustering procedure used in the training phase.

A Rigorous Model Order Reduction Framework for Waste Heat Recovery Systems Based on Proper Orthogonal Decomposition and Galerkin Projection

A proper orthogonal decomposition (POD) and Galerkin projection-based model order reduction framework is developed for the evaporator heat exchanger in heavy-duty diesel engine organic Rankine cycle waste heat recovery system. The dynamics in the heat exchanger are first modeled by a finite-volume model, composed of highly nonlinear, coupled, partial differential equations, and then used to generate snapshots from which basis functions are defined. Reduced order models (ROMs) are then derived using the Galerkin projection approach. The accuracy and the execution time of different POD ROMs are evaluated against the high-fidelity finite-volume model. The results show that the POD ROM dimension can be selected based on the specific requirements of accuracy and computation time demanded by the intended model utilization. The proposed ROM framework can be utilized to generate ROMs with various dimensions for different purposes, such as estimator design, model-based control, and benchmark generation.

Data-driven recovery of hidden physics in reduced order modeling of fluid flows

In this article, we introduce a modular hybrid analysis and modeling (HAM) approach to account for hidden physics in reduced order modeling (ROM) of parameterized systems relevant to fluid dynamics. The hybrid ROM framework is based on using first principles to model the known physics in conjunction with utilizing the data-driven machine learning tools to model the remaining residual that is hidden in data. This framework employs proper orthogonal decomposition as a compression tool to construct orthonormal bases and a Galerkin projection (GP) as a model to build the dynamical core of the system. Our proposed methodology, hence, compensates structural or epistemic uncertainties in models and utilizes the observed data snapshots to compute true modal coefficients spanned by these bases. The GP model is then corrected at every time step with a data-driven rectification using a long short-term memory (LSTM) neural network architecture to incorporate hidden physics. A Grassmann manifold approach is also adopted for interpolating basis functions to unseen parametric conditions. The control parameter governing the system’s behavior is, thus, implicitly considered through true modal coefficients as input features to the LSTM network. The effectiveness of the HAM approach is then discussed through illustrative examples that are generated synthetically to take hidden physics into account. Our approach, thus, provides insights addressing a fundamental limitation of the physics-based models when the governing equations are incomplete to represent underlying physical processes.

Interpolatory Rational Model Order Reduction of Parametric Problems Lacking Uniform Inf-Sup Stability

We present a technique for the approximation of a class of Hilbert space-valued maps which arise within the framework of Model Order Reduction for parametric partial differential equations, whose solution map has a meromorphic structure. Our MOR stategy consists in constructing an explicit rational approximation based on few snapshots of the solution, in an interpolatory fashion. Under some restrictions on the structure of the original problem, we describe a priori convergence results for our technique, hereafter called minimal rational interpolation, which show its ability to identify the main features (e.g. resonance locations) of the target solution map. We also investigate some procedures to obtain a posteriori error indicators, which may be employed to adapt the degree and the sampling points of the minimal rational interpolant. Finally, some numerical experiments are carried out to confirm the theoretical results and the effectiveness of our technique.

Nonintrusive reduced order modeling framework for quasigeostrophic turbulence.

In this study, we present a nonintrusive reduced order modeling (ROM) framework for large-scale quasistationary systems. The framework proposed herein exploits the time series prediction capability of long short-term memory (LSTM) recurrent neural network architecture such that (1) in the training phase, the LSTM model is trained on the modal coefficients extracted from the high-resolution data snapshots using proper orthogonal decomposition (POD) transform, and (2) in the testing phase, the trained model predicts the modal coefficients for the total time recursively based on the initial time history. Hence, no prior information about the underlying governing equations is required to generate the ROM. To illustrate the predictive performance of the proposed framework, the mean flow fields and time series response of the field values are reconstructed from the predicted modal coefficients by using an inverse POD transform. As a representative benchmark test case, we consider a two-dimensional quasigeostrophic ocean circulation model which, in general, displays an enormous range of fluctuating spatial and temporal scales. We first demonstrate that the conventional Galerkin projection-based reduced order modeling of such systems requires a high number of POD modes to obtain a stable flow physics. In addition, ROM-Galerkin projection (ROM-GP) does not seem to capture the intermittent bursts appearing in the dynamics of the first few most energetic modes. However, the proposed nonintrusive ROM framework based on LSTM (ROM-LSTM) yields a stable solution even for a small number of POD modes. We also observe that the ROM-LSTM model is able to capture quasiperiodic intermittent bursts accurately, and yields a stable and accurate mean flow dynamics using the time history of a few previous time states, denoted as the lookback time window in this paper. We show several features of ROM-LSTM framework such as significantly higher accuracy than ROM-GP, and faster performance using larger time step size. Throughout the paper, we demonstrate our findings in terms of time series evolution of the field values and mean flow patterns, which suggest that the proposed fully nonintrusive ROM framework is robust and capable of predicting chaotic nonlinear fluid flows in an extremely efficient way compared to the conventional projection-based ROM framework.

A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces

Reduced order modeling (ROM) provides an efficient framework to compute solutions of parametric problems. Basically, it exploits a set of precomputed high-fidelity solutions—computed for properly chosen parameters, using a full-order model—in order to find the low dimensional space that contains the solution manifold. Using this space, an approximation of the numerical solution for new parameters can be computed in real-time response scenario, thanks to the reduced dimensionality of the problem. In a ROM framework, the most expensive part from the computational viewpoint is the calculation of the numerical solutions using the full-order model. Of course, the number of collected solutions is strictly related to the accuracy of the reduced order model. In this work, we aim at increasing the precision of the model also for few input solutions by coupling the proper orthogonal decomposition with interpolation (PODI)—a data-driven reduced order method—with the active subspace (AS) property, an emerging tool for reduction in parameter space. The enhanced ROM results in a reduced number of input solutions to reach the desired accuracy. In this contribution, we present the numerical results obtained by applying this method to a structural problem and in a fluid dynamics one.

A Proper Generalized Decomposition (PGD) approach to crack propagation in brittle materials: with application to random field material properties

Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in material properties are frequently modeled through random fields, which typically induces the need to solve finite element problems for a large number of realizations. In this context, we make use of reduced order modeling to solve these problems at an affordable computational cost. This paper proposes a reduced order modeling framework to predict crack propagation in brittle materials with random heterogeneities. The framework is based on a combination of the Proper Generalized Decomposition (PGD) method with Griffith’s global energy criterion. The PGD framework provides an explicit parametric solution for the physical response of the system. We illustrate that a non-intrusive sampling-based technique can be applied as a post-processing operation on the explicit solution provided by PGD. We first validate the framework using a global energy approach on a deterministic two-dimensional linear elastic fracture mechanics benchmark. Subsequently, we apply the reduced order modeling approach to a stochastic fracture propagation problem.

Computational Aeroelasticity Using Modal-Based Structural Nonlinear Analysis

This paper presents the development of an aeroelastic analysis formulation for the dynamic response of a flexible wing. The geometrically nonlinear structure is represented by a reduced-order model that accounts for the large-deformation effects on the inertia and stiffness terms. It is based on a system identification of a built-up finite element model and does not rely on a geometric nonlinear beam-based model as most of the state-of-the-art nonlinear aeroelastic frameworks do. The complexity of the wing structural layout is captured in the built-up finite element model as in traditional industry applications. To preserve a small number of states to represent the structural dynamics of the system, a modal approach is used in a nonlinear sense. Moreover, this approach is easily linked to computational fluid dynamics. For this study, the structural reduced-order model framework is integrated within the aeroelastic framework of the code CFL3D. Using this integrated solution, the aeroelastic response of a high-aspect-ratio wing is investigated.

An adaptive domain-based POD/ECM hyper-reduced modeling framework without offline training

This work presents a reduced-order modeling framework that precludes the need for offline training and adaptively adjusts its lower-order solution space as the analysis progresses. The analysis starts with a fully-solved step and elements are clustered based on their strain response. Elements with the highest strains are solved with a local/global approach in which degrees of freedom from elements undergoing the highest amount of nonlinearity are fully-solved and the rest is approximated by a Proper Orthogonal Decomposition (POD) reduced model with full integration. Elements belonging to the remaining clusters are subjected to a hyper-reduction step using the Empirical Cubature Method (ECM). Online error estimators are used to trigger a retraining process once the reduced solution space becomes inadequate. The performance of the framework is assessed through a series of numerical examples featuring a material model with pressure-dependent plasticity.