Non-intrusive Reduced Order Model Research Articles

Signal Interpolation Augmented Linear Nonintrusive Reduced-Order Model for Aeroelastic Applications

This study aims to develop a linear nonintrusive reduced-order model (NIROM) for facilitating combined fast parametric analysis and reconstruction of unsteady aerodynamic flow variables by using a novel signal interpolation approach, which is combined with a subspace-identification-based method. The NIROM is then applied to airfoil flutter and gust interaction problems where the assumption of linearized dynamics gives reasonably accurate estimates. In the traditional subspace-identification method, the training datasets of force coefficient signals corresponding to each flow parameter (like the Mach number and angle of attack) are required to generate the linear time-invariant system. The generation of the training signal for each flow parameter is the most computationally expensive process in any parametric reduced-order model (ROM). In the present work, the discrete empirical interpolation method (DEIM) combined with the radial basis function (RBF) method is used for the interpolation of training signals at a specific flow parameter to reduce the overall computational cost. The conventional nonintrusive approaches are generally constrained by the number of output channels, and hence not suitable for the reconstruction of unsteady surface force distribution or flowfield variables. A few suitable sensor locations using the DEIM are selected as output channels of the subspace-identification method to realize the full datasets of surface force distribution over the airfoil. Finally, a suitable algorithmic structure is proposed to achieve unsteady distributive flow-variable reconstruction and parametric variation simultaneously. The current signal interpolation-based NIROM is applied to a wing section with two- and four-degree-of-freedom structural motions.

A nonintrusive reduced order modelling approach using Proper Orthogonal Decomposition and locally adaptive sparse grids

Large-scale complex systems require high fidelity models to capture the dynamics of the system accurately. The complexity of these models, however, renders their use to be expensive for applications relying on repeated evaluations, such as control, optimization, and uncertainty quantification. Proper Orthogonal Decomposition (POD) is a powerful Reduced Order Modelling (ROM) technique developed to reduce the computational burden of high fidelity models. In cases where the model is inaccessible, POD can be used in a nonintrusive manner. The accuracy and efficiency of the nonintrusive reduced model are highly dependent on the sampling scheme, especially for high dimensional problems.To that end, we study integrating the locally adaptive sparse grids with the POD method to develop a novel nonintrusive POD-based reduced order model. In our proposed approach, the locally adaptive sparse grid is used to adaptively control the sampling scheme for the POD snapshots, and the hierarchical interpolant is used as a surrogate model for the POD coefficients. An approach to efficiently update the surpluses of the sparse grids with each POD snapshots update is also introduced. The robustness and efficiency of the locally adaptive algorithm are increased by introducing a greediness parameter, and a strategy to validate the reduced model after convergence. The proposed validation algorithm can also enrich the reduced model around regions of detected discrepancies. Three numerical test cases are presented to demonstrate the potential of the proposed POD-Adaptive algorithm. The first is a nuclear reactor point kinetics, the second is a general diffusion problem, and the last is a variation of the analytical Morris function. The results show that the developed algorithm reduced the number of model evaluations compared to the classical sparse grid approach. The built reduced models captured the dynamics of the reference systems with the desired tolerances. The non-intrusiveness and simplicity of the method provide great potential for a wide range of practical large scale applications.

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various DNN architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equation, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a nonintrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered truly nonintrusive since any prior information about the underlying governing equations is not required for generating the reduced order model. Our a posteriori analysis shows that the proposed data-driven approach is remarkably accurate and can be used as a robust predictive tool for nonintrusive model order reduction of complex fluid flows.

Adaptive non-intrusive reduced order modeling for compressible flows

An adaptive non-intrusive reduced basis (RB) method based on Gaussian process regression (GPR) is proposed for parametrized compressible flows. Adaptivity is pursued in the offline stage. The reduced basis by proper orthogonal decomposition (POD) is constructed iteratively to achieve a specified tolerance. For GPR, active data selection is used at each iteration, with standard deviation as the error indicator. To improve accuracy for shock-dominated flows, a properly designed simplified problem (SP) is considered as input of the regression models in addition to using parameters directly. Furthermore, a surrogate error model is constructed to serve as an efficient error estimator for the GPR models. Several two- and three-dimensional cases are conducted, including the inviscid nozzle flow, the inviscid NACA0012 airfoil flow and the inviscid M6 wing flow. For all the cases, the trained models are able to make efficient predictions with reasonable accuracy in the online stage. The SP-based approach is observed to result in biased sampling towards transonic regions. The regression models are further applied in sensitivity analysis, from which the solution of the two-dimensional cases are shown to be significantly more sensitive to input parameters than the wing flow. This is consistent to the comparison of convergence histories between the parameter-based and the SP-based models. For cases of high sensitivity, the SP-based approach is superior and can help to significantly reduce the number of required snapshots to achieve a prescribed tolerance.

Error estimation of the parametric non-intrusive reduced order model using machine learning

A novel error estimation method for the parametric non-intrusive reduced order model (P-NIROM) based on machine learning is presented. This method relies on constructing a set of response functions for the errors between the high fidelity full model solutions and P-NIROM using machine learning method, particularly, Gaussian process regression method. This yields closer solutions agreement with the high fidelity full model. The novelty of this work is that it is the first time to use machine learning method to derive error estimate for the P-NIROM. The capability of the new error estimation method is demonstrated using three numerical simulation examples: flow past a cylinder, dam break and 3D fluvial channel. It is shown that the results are closer to those of the high fidelity full model when considering error terms. In addition, the interface between two phases of dam break case is captured well if the error estimator is involved in the P-NIROM.

A domain decomposition method for the non-intrusive reduced order modelling of fluid flow

In this paper we present a new domain decomposition non-intrusive reduced order model (DDNIROM) for the Navier–Stokes equations. The computational domain is partitioned into subdomains and a set of local basis functions is constructed in each subdomain using Proper Orthogonal Decomposition (POD). A radial basis function (RBF) method is then used to generate a set of hypersurfaces for each subdomain. Each local hypersurface represents, not only the fluid dynamics over the subdomain to which it belongs, but also the interactions with the surrounding subdomains. This implicit coupling between the subdomains provides the global coupling necessary to enforce incompressibility and is a means of providing boundary conditions for each subdomain.The performance of this DDNIROM is illustrated numerically by three examples: flow past a cylinder, and air flow over 2D and 3D street canyons. The results show that the DDNIROM exhibits good agreement with the high-fidelity full model while the computational cost is reduced by several orders of magnitude. The domain decomposition (DD) method provides the flexibility to choose different numbers of local basis functions for each subdomain depending on the complexity of the flow therein. The fact that the RBF surface representation takes input only from its current subdomain and the surrounding subdomains, means that, crucially, there is a reduction in the dimensionality of the hypersurface when compared with a more traditional, global NIROM. This comes at the cost of having a larger number of hypersurfaces.

Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem

A non-intrusive reduced-basis (RB) method is proposed for parametrized unsteady flows. A set of reduced basis functions are extracted from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD), and the coefficients of the reduced basis functions are recovered by a feedforward neural network (NN). As a regression model of the RB method for unsteady flows, the neural network approximates the map between the time/parameter value and the projection coefficients of the high-fidelity solution onto the reduced space. The generation of the reduced basis and the training of the NN are accomplished in the offline stage, thus the RB solution of a new time/parameter value can be recovered via direct outputs of the NN in the online stage. Due to its non-intrusive nature, the proposed RB method, referred as the POD-NN, fully decouples the online stage and the high-fidelity scheme, and is thus able to provide fast and reliable solutions of complex unsteady flows. To test this assertion, the POD-NN method is applied to the reduced order modeling (ROM) of the quasi-one dimensional Continuously Variable Resonance Combustor (CVRC) flow. Numerical results demonstrate the efficiency and robustness of the POD-NN method.

A domain decomposition non-intrusive reduced order model for turbulent flows

In this paper, a new Domain Decomposition Non-Intrusive Reduced Order Model (DDNIROM) is developed for turbulent flows. The method works by partitioning the computational domain into a number of subdomains in such a way that the summation of weights associated with the finite element nodes within each subdomain is approximately equal, and the communication between subdomains is minimised. With suitably chosen weights, it is expected that there will be approximately equal accuracy associated with each subdomain. This accuracy is maximised by allowing the partitioning to occur through areas of the domain that have relatively little flow activity, which, in this case, is characterised by the pointwise maximum Reynolds stresses.A Gaussian Process Regression (GPR) machine learning method is used to construct a set of local approximation functions (hypersurfaces) for each subdomain. Each local hypersurface represents not only the fluid dynamics over the subdomain it belongs to, but also the interactions of the flow dynamics with the surrounding subdomains. Thus, in this way, the surrounding subdomains may be viewed as providing boundary conditions for the current subdomain.We consider a specific example of turbulent air flow within an urban neighbourhood at a test site in London and demonstrate the effectiveness of the proposed DDNIROM.

Machine learning-based rapid response tools for regional air pollution modelling

A parameterised non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) and machine learning methods has been firstly developed for model reduction of pollutant transport equations. Our motivation is to provide rapid response urban air pollution predictions and controls. The varying parameters in the P-NIROM are pollutant sources. The training data sets are obtained from the high fidelity modelling solutions (called snapshots) for selected parameters (pollutant sources, here) over the parameter space RP. From these training data sets, the machine learning method is used to generate the relationship between the reduced solutions and inputs (pollutant sources) over RP. Furthermore a set of hyper-surface functions associated with each POD basis function is constructed for representing the fluid dynamics over the reduced space. The accuracy of the P-NIROM is highly dependent on the quality of the training set, here obtained from the high fidelity model. Over existing machine learning methods, the P-NIROM algorithm proposed here has the advantages that (1) it is combined with NIROM, thus providing rapid and reasonably accurate solutions; and (2) it is a robust and efficient approach for representation of any parametrised partial differential equations as the model parameters/inputs vary. In this study, we demonstrate the way how to implement the P-NIROM for the pollutant transport equation (but not limited to due to its robustness). Its predictive capability is illustrated in a three-dimensional (3-D) simulation of power plant plumes over a large region in China, where the varying parameters are the emission intensity at three locations. Results indicate that in comparison to the high fidelity model, the CPU cost is reduced by factor up to five orders of magnitude while reasonable accuracy remains.

A reduced order model for turbulent flows in the urban environment using machine learning

To help create a comfortable and healthy indoor and outdoor environment in which to live, there is a need to understand turbulent air flows within the urban environment. To this end, building on a previously reported method [1], we develop a fast-running Non-Intrusive Reduced Order Model (NIROM) for predicting the turbulent air flows found within an urban environment. To resolve larger scale turbulent fluctuations, we employ a Large Eddy Simulation (LES) model and solve the resulting computational model on unstructured meshes. The objective is to construct a rapid-running NIROM from these results that will have ‘similar’ dynamics to the original LES model. Based on Proper Orthogonal Decomposition (POD) and machine learning techniques, this Reduced Order Model (ROM) is six orders of magnitude faster than the high-fidelity LES model and we demonstrate how ‘similar’ it can be to the high-fidelity model by comparing statistical quantities such as the mean flows, Reynolds stresses and probability densities of the velocities. We also include validation of the high-fidelity model against data from wind tunnel experiments.This paper represents a key step towards the use of reduced order modelling for operational purposes with the tantalising possibility of it being used in place of Gaussian plume models, and the potential for greatly improved model fidelity and confidence.

Greedy Nonintrusive Reduced Order Model for Fluid Dynamics

A greedy nonintrusive reduced order method (ROM) is proposed for parameterized time-dependent problems with an emphasis on problems in fluid dynamics. The nonintrusive ROM (NIROM) bases on a two-level proper orthogonal decomposition to extract temporal and spatial reduced basis from a set of candidates, and adopts the radial basis function to approximate undetermined coefficients of extracted reduced basis. Instead of adopting uniform or random sampling strategies, the candidates are determined by an adaptive greedy approach to minimize the overall offline computational cost. Numerical studies are presented for a two-dimensional diffusion problem as well as a lid-driven cavity problem governed by incompressible Navier–Stokes equations. The results demonstrate that the greedy nonintrusive ROM (GNIROM) predicts the flow field accurately and efficiently.

Parameterised non-intrusive reduced order methods for ensemble Kalman filter data assimilation

This paper presents a novel Ensemble Kalman Filter (EnKF) data assimilation method based on a parameterised non-intrusive reduced order model (P-NIROM) which is independent of the original computational code. EnKF techniques involve the expensive calculations of ensembles. In this work, the recently developed P-NIROM Xiao et al. [40] is incorporated into EnKF to speed up the ensemble simulations. A reduced order flow dynamical model is generated from the solution snapshots, which are obtained from a number of the high fidelity full simulations over the specific parametric space RP. The varying parameter is the background error covariance σ ∈ RP. Using the Smolyak sparse grid method, a set of parameters in the Gaussian probability density function is selected as the training points. The proposed method uses a two-level interpolation method for constructing the P-NIROM using a Radial Basis Function (RBF) interpolation method. The first level interpolation approach is used for generating the solution snapshots and POD basis functions for any given background error covariance while the second level interpolation approach for forming a set of hyper-surfaces representing the reduced system.The EnKF in combination with P-NIROM (P-NIROM-EnKF) has been implemented within an unstructured mesh finite element ocean model and applied to a three dimensional wind driven circulation gyre case. The numerical results show that the accuracy of ensembles and updated solutions using the P-NIROM-EnKF is maintained while the computational cost is significantly reduced by several orders of magnitude in comparison to the full-EnKF.

A parametric and non-intrusive reduced order model of car crash simulation

Industrials have an intensive use of numerical simulations in order to avoid physical testing and to speed up the design stages of their products. The numerical testing is indeed quicker to set-up, less expensive, and supplies a lot of information about the system under study. Moreover, it can be much closer to the physical tests as the computation power increases. Despite the rise of this power, time consuming simulations remain challenging to be used in design process, especially in an optimization study. Crash simulations belong to this category. These rapid dynamic computations are used by RENAULT during the sizing of the vehicle structure in order to ensure that it meets specifications set up to reach safety criteria in case of accidents. They are completed using finite element software such as VPS (Virtual Performance Solver) developed by ESI group that will be used in this study. For car manufacturers, the goal of the optimization study is to minimize the mass of the vehicle (and thus its consumption) by modifying the thicknesses of some parts (from 20 to 100 variables). Industrials such as RENAULT currently perform optimization studies based on numerical design of experiments. The number of computations required by this technique is from 3 to 10 times the number of variables. This is too much in order to be intensively used in a design process.In order to decrease the time-to-market and to explore alternative technical solutions, we explore the potential of using a parametrized reduced order model in the optimization studies. The parametrized reduced order model gives an estimation of the high-fidelity result for a new set of parameters without using the solver, by analysing the existing results of previous computations with various sets of parameters. The developed reduced order model is called ReCUR. It is partly based on a CUR approach embedded in a regression analysis. The regression statistical model uses the data of a few calculations made with the solver. Other tools such as clustering and linear programming are used to get the regression analysis more efficient.It is hoped to drastically reduce the number of required simulations of a standard optimization study. In this paper, the construction of the reduced order model will be presented. Then, the relevancy of using the reduced order model into a design process will be exhibited through the treatment of two industrial test-cases. Some improvements of the method as well as several potential uses will then be outlined. The applications will highlight the promising power of the method to shorten design process using optimization and long-run simulations.

Non-intrusive reduced order modeling of nonlinear problems using neural networks

We develop a non-intrusive reduced basis (RB) method for parametrized steady-state partial differential equations (PDEs). The method extracts a reduced basis from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD) and employs artificial neural networks (ANNs), particularly multi-layer perceptrons (MLPs), to accurately approximate the coefficients of the reduced model. The search for the optimal number of neurons and the minimum amount of training samples to avoid overfitting is carried out in the offline phase through an automatic routine, relying upon a joint use of the Latin hypercube sampling (LHS) and the Levenberg–Marquardt (LM) training algorithm. This guarantees a complete offline-online decoupling, leading to an efficient RB method – referred to as POD-NN – suitable also for general nonlinear problems with a non-affine parametric dependence. Numerical studies are presented for the nonlinear Poisson equation and for driven cavity viscous flows, modeled through the steady incompressible Navier–Stokes equations. Both physical and geometrical parametrizations are considered. Several results confirm the accuracy of the POD-NN method and show the substantial speed-up enabled at the online stage as compared to a traditional RB strategy.

Non-intrusive model reduction for a 3D unstructured mesh control volume finite element reservoir model and its application to fluvial channels

A non-intrusive model reduction computational method using hypersurfaces representation has been developed for reservoir simulation and further applied to 3D fluvial channel problems in this work. This is achieved by a combination of a radial basis function (RBF) interpolation and proper orthogonal decomposition (POD) method. The advantage of the method is that it is generic and non-intrusive, that is, it does not require modifications to the original complex source code, for example, a 3D unstructured mesh control volume finite element (CVFEM) reservoir model used here. The capability of this non-intrusive reduced order model (NIROM) based on hypersurfaces representation has been numerically illustrated in a horizontally layered porous media case, and then further applied to a 3D complex fluvial channel case. By comparing the results of the NIROM against the solutions obtained from the high fidelity full model, it is shown that this NIROM results in a large reduction in the CPU computation cost while much of the details are captured. [Received: May 19, 2017; Accepted: December 1, 2017]

Non-intrusive model reduction for a 3D unstructured mesh control volume finite element reservoir model and its application to fluvial channels

A non-intrusive model reduction computational method using hypersurfaces representation has been developed for reservoir simulation and further applied to 3D fluvial channel problems in this work. This is achieved by a combination of a radial basis function (RBF) interpolation and proper orthogonal decomposition (POD) method. The advantage of the method is that it is generic and non-intrusive, that is, it does not require modifications to the original complex source code, for example, a 3D unstructured mesh control volume finite element (CVFEM) reservoir model used here. The capability of this non-intrusive reduced order model (NIROM) based on hypersurfaces representation has been numerically illustrated in a horizontally layered porous media case, and then further applied to a 3D complex fluvial channel case. By comparing the results of the NIROM against the solutions obtained from the high fidelity full model, it is shown that this NIROM results in a large reduction in the CPU computation cost while much of the details are captured. [Received: May 19, 2017; Accepted: December 1, 2017]

Towards non-intrusive reduced order 3D free surface flow modelling

In this article, we describe a novel non-intrusive reduction model for three-dimensional (3D) free surface flows. However, in this work we limit the vertical resolution to be a single element. So, although it does resolve some non-hydrostatic effects, it does not examine the application of reduced modelling to full 3D free surface flows, but it is an important step towards 3D modelling. A newly developed non-intrusive reduced order model (NIROM) (Xiao et al., 2015a) has been used in this work. Rather than taking the standard POD approach using the Galerkin projection, a Smolyak sparse grid interpolation method is employed to generate the NIROM. A set of interpolation functions is constructed to calculate the POD coefficients, where the POD coefficients at previous time steps are the inputs of the interpolation function. Therefore, this model is non-intrusive and does not require modifications to the code of the full system and is easy to implement.By using this new NIROM, we have developed a robust and efficient reduced order model for free surface flows within a 3D unstructured mesh finite element ocean model. What distinguishes the reduced order model developed here from other existing reduced order ocean models is (1) the inclusion of 3D dynamics with a free surface (the 3D computational domain and meshes are changed with the movement of the free surface); (2) the incorporation of wetting-drying; and (3) the first implementation of non-intrusive reduced order method in ocean modelling. Most importantly, the change of the computational domain with the free surface movement is taken into account in reduced order modelling. The accuracy and predictive capability of the new non-intrusive free surface flow ROM have been evaluated in Balzano and Okushiri tsunami test cases. This is the first step towards 3D reduced order modelling in realistic ocean cases. Results obtained show that the accuracy of free surface problems relative to the high fidelity model is maintained in ROM whilst the CPU time is reduced by several orders of magnitude.

Randomized dynamic mode decomposition for nonintrusive reduced order modelling

SummaryThis paper focuses on a new framework for obtaining a nonintrusive (i.e., not requiring projecting of the governing equations onto the reduced basis modes) reduced order model for two‐dimensional fluid problems. To overcome the shortcomings of intrusive model order reduction usually derived by combining the Proper Orthogonal Decomposition and the Galerkin projection methods, we developed a novel technique on the basis of randomized dynamic mode decomposition (DMD) as a fast and accurate option in model order reduction. Our approach utilizes an adaptive randomized DMD to obtain a reduced basis in the offline stage, and then the temporal values of the reduced order model are obtained in the online stage through an interpolation using radial basis functions. The rank of the reduced DMD model is given as the unique solution of a constrained optimization problem. The Saint‐Venant (shallow water) equations in a channel on the rotating earth are employed to provide the numerical data. We emphasize the excellent behavior of the nonintrusive reduced order model by performing a qualitative analysis. In addition, we gain a significantly reduction of CPU time in computation of the reduced order models compared with the classical DMD method. Copyright © 2016 John Wiley & Sons, Ltd.

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

A novel parameterized non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) has been developed. This P-NIROM is a generic and efficient approach for model reduction of parameterized partial differential equations (P-PDEs). Over existing parameterized reduced order models (P-ROM) (most of them are based on the reduced basis method), it is non-intrusive and independent on partial differential equations and computational codes. During the training process, the Smolyak sparse grid method is used to select a set of parameters over a specific parameterized space (Ωp∈RP). For each selected parameter, the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model. More generally, the snapshots and basis function sets for any parameters over Ωp can be obtained using an interpolation method. The P-NIROM can then be constructed by using our recently developed technique (Xiao et al., 2015 [41,42]) where either the Smolyak or radial basis function (RBF) methods are used to generate a set of hyper-surfaces representing the underlying dynamical system over the reduced space.The new P-NIROM technique has been applied to parameterized Navier–Stokes equations and implemented with an unstructured mesh finite element model. The capability of this P-NIROM has been illustrated numerically by two test cases: flow past a cylinder and lock exchange case. The prediction capabilities of the P-NIROM have been evaluated by varying the viscosity, initial and boundary conditions. The results show that this P-NIROM has captured the quasi-totality of the details of the flow with CPU speedup of three orders of magnitude. An error analysis for the P-NIROM has been carried out.

A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting

This work presents the first application of a non-intrusive reduced order method to model solid interacting with compressible fluid flows to simulate crack initiation and propagation. In the high fidelity model, the coupling process is achieved by introducing a source term into the momentum equation, which represents the effects of forces of the solid on the fluid. A combined single and smeared crack model with the Mohr–Coulomb failure criterion is used to simulate crack initiation and propagation. The non-intrusive reduced order method is then applied to compressible fluid and fractured solid coupled modelling where the computational cost involved in the full high fidelity simulation is high. The non-intrusive reduced order model (NIROM) developed here is constructed through proper orthogonal decomposition (POD) and a radial basis function (RBF) multi-dimensional interpolation method.The performance of the NIROM for solid interacting with compressible fluid flows, in the presence of fracture models, is illustrated by two complex test cases: an immersed wall in a fluid and a blasting test case. The numerical simulation results show that the NIROM is capable of capturing the details of compressible fluids and fractured solids while the CPU time is reduced by several orders of magnitude. In addition, the issue of whether or not to subtract the mean from the snapshots before applying POD is discussed in this paper. It is shown that solutions of the NIROM, without mean subtracted before constructing the POD basis, captured more details than the NIROM with mean subtracted from snapshots.