Non-intrusive Reduced Order Model Research Articles

A non-intrusive reduced order model using deep learning for realistic wind data generation for small unmanned aerial systems in urban spaces

Realistic wind data are essential in developing, testing, and ensuring the safety of unmanned aerial systems in operation. Alternatives to Dryden and von Kármán turbulence models are required, aimed explicitly at urban air spaces to generate turbulent wind data. We present a novel method to generate realistic wind data for the safe operation of small unmanned aerial vehicles in urban spaces. We propose a non-intrusive reduced order modeling approach to replicate realistic wind data and predict wind fields. The method uses a well-established large-eddy simulation model, the parallelized large eddy simulation model, to generate high-fidelity data. To create a reduced-order model, we utilize proper orthogonal decomposition to extract modes from the three-dimensional space and use specialized recurrent neural networks and long-term short memory for stepping in time. This paper combines the traditional approach of using computational fluid dynamic simulations to generate wind data with deep learning and reduced-order modeling techniques to devise a methodology for a non-intrusive data-based model for wind field prediction. A simplistic model of an isolated urban subspace with a single building setup in neutral atmospheric conditions is considered a test case for the demonstration of the method.

Multi-fidelity prediction of spatiotemporal fluid flow

Data-driven prediction of spatiotemporal fields in fluid flow problems has received significant interest lately. However, the scarcity of data often plagues the accuracy of the prevalent approaches, particularly in situations where the cost associated with the generation of high-fidelity data is prohibitively expensive. Moreover, the data sources in many spatiotemporal fluid flow problems present a hierarchy of fidelities with respect to their computational cost/accuracy, such that the higher fidelity sources are more accurate and also more expensive than the low-fidelity sources. This article presents a novel multi-fidelity spatiotemporal modeling approach whereby the lower fidelity data source for a dynamical process is gainfully utilized in increasing the accuracy of predicting the higher fidelity fields, thereby reducing the overhead of high-fidelity simulations for data-driven prediction. The methodology is based on non-intrusive reduced order modeling using deep convolutional autoencoders, combined with a latent-space evolution framework based on multi-fidelity Gaussian processes. This framework results in probabilistic spatiotemporal predictions for unknown operating conditions of the dynamical system, which provides the end user with quantified levels of uncertainties associated with the data-driven predictions. The framework is validated on a prototypical fluid flow problem of interest: two-dimensional incompressible viscous flow around a cylinder.

Development of a reduced-order model for large-scale Eulerian–Lagrangian simulations

Multiphase flows with solid particles are commonly encountered in various industries. The CFD–DEM method is extensively used to simulate their dynamical behavior. However, the application of the CFD–DEM method to simulate industrial-scale powder processes unavoidably leads to huge computational costs. With the aim of overcoming this issue, we propose a nonintrusive reduced-order model for Eulerian–Lagrangian simulations (ROM-EL) to efficiently reproduce gas–solid flow in fluidized beds. In the model, a Lanczos based proper orthogonal decomposition (LPOD) is newly employed to efficiently generate a set of POD bases. After the numerical snapshots are projected onto the reduced space spanned by the POD bases, a series of multidimensional functions of POD coefficients are constructed using a surrogate interpolation method. To demonstrate the effectiveness of this model, validation studies are performed based on the simulations of a fluidized bed. The macroscopic properties, such as the particle distribution, bed height, pressure drop, and distribution of bubble size, are shown to agree well in the CFD–DEM model and ROM-EL. Further, our proposed ROM-EL reduces the computational cost by several orders of magnitude compared with the CFD–DEM simulation. Accordingly, the ROM-EL could significantly contribute to the progress of modeling and simulation for industrial granular flows.

Nonintrusive model order reduction for cross-diffusion systems

In this paper, we investigate tensor based nonintrusive reduced-order models (ROMs) for parametric cross-diffusion equations. The full-order model (FOM) consists of ordinary differential equations (ODEs) in matrix or tensor form resulting from finite difference discretization of the differential operators by taking the advantage of Kronecker structure. The matrix/tensor differential equations are integrated in time with the implicit–explicit (IMEX) Euler method. The reduced bases, relying on a finite sample set of parameter values, are constructed in form of a two-level approach by applying higher-order singular value decomposition (HOSVD) to the space–time snapshots in tensor form, which leads to a large amount of computational and memory savings. The nonintrusive reduced approximation for an arbitrary parameter value is obtained through tensor product of the reduced basis by the parameter dependent core tensor that contains the reduced coefficients. The reduced coefficients for new parameter values are computed with the radial basis functions. The efficiency of the proposed method is illustrated through numerical experiments for two-dimensional Schnakenberg and three-dimensional Brusselator cross-diffusion equations. The spatiotemporal patterns are accurately predicted by the reduced-order models with speedup factors of orders two and three over the full-order models.

Extending the Capabilities of Data-Driven Reduced-Order Models to Make Predictions for Unseen Scenarios: Applied to Flow Around Buildings

We present a data-driven or non-intrusive reduced-order model (NIROM) which is capable of making predictions for a significantly larger domain than the one used to generate the snapshots or training data. This development relies on the combination of a novel way of sampling the training data (which frees the NIROM from its dependency on the original problem domain) and a domain decomposition approach (which partitions unseen geometries in a manner consistent with the sub-sampling approach). The method extends current capabilities of reduced-order models to generalise, i.e., to make predictions for unseen scenarios. The method is applied to a 2D test case which simulates the chaotic time-dependent flow of air past buildings at a moderate Reynolds number using a computational fluid dynamics (CFD) code. The procedure for 3D problems is similar, however, a 2D test case is considered sufficient here, as a proof-of-concept. The reduced-order model consists of a sampling technique to obtain the snapshots; a convolutional autoencoder for dimensionality reduction; an adversarial network for prediction; all set within a domain decomposition framework. The autoencoder is chosen for dimensionality reduction as it has been demonstrated in the literature that these networks can compress information more efficiently than traditional (linear) approaches based on singular value decomposition. In order to keep the predictions realistic, properties of adversarial networks are exploited. To demonstrate its ability to generalise, once trained, the method is applied to a larger domain which has a different arrangement of buildings. Statistical properties of the flows from the reduced-order model are compared with those from the CFD model in order to establish how realistic the predictions are.

Localized non-intrusive reduced-order modelling in the operator inference framework

This paper presents data-driven learning of localized reduced models. Instead of a global reduced basis, the approach employs multiple local approximation subspaces. This localization permits adaptation of a reduced model to local dynamics, thereby keeping the reduced dimension small. This is particularly important for reduced models of nonlinear systems of partial differential equations, where the solution may be characterized by different physical regimes or exhibit high sensitivity to parameter variations. The contribution of this paper is a non-intrusive approach that learns the localized reduced model from snapshot data using operator inference. In the offline phase, the approach partitions the state space into subregions and solves a regression problem to determine localized reduced operators. During the online phase, a local basis is chosen adaptively based on the current system state. The non-intrusive nature of localized operator inference makes the method accessible, portable and applicable to a broad range of scientific problems, including those that use proprietary or legacy high-fidelity codes. We demonstrate the potential for achieving large computational speedups while maintaining good accuracy for a Burgers' equation governing shock propagation in a one-dimensional domain and a phase-field problem governed by the Cahn-Hilliard equation. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Fast Predictions of Aircraft Aerodynamics Using Deep-Learning Techniques

The numerical analysis of aerodynamic components based on the Reynolds–averaged Navier–Stokes equations has become critical for the design of transport aircraft but still entails large computational cost. Simulating a multitude of different flow conditions with high-fidelity methods as required for loads analysis or aerodynamic shape optimization is still prohibitive. Within the past few years, the application of machine learning methods has been proposed as a potential way to overcome these shortcomings. This is leading toward a new data-driven paradigm for the modeling of physical problems. The objective of this paper is the development of a deep-learning methodology for the prediction of aircraft surface pressure distributions and the rigorous comparison with existing state-of-the-art nonintrusive reduced-order models. Bayesian optimization techniques are employed to efficiently determine optimal hyperparameters for all deep neural networks. Three data-driven methods which are Gaussian processes, proper orthogonal decomposition combined with an interpolation technique, and deep learning are investigated. The results are compared for a two-dimensional airfoil case and the NASA Common Research Model transport aircraft as a relevant three-dimensional case. Results show that all methods are able to properly predict the surface pressure distribution at subsonic conditions. In transonic flow, when shock waves and separation lead to nonlinearities, deep-learning methods outperform the others by also capturing the shock wave location and strength accurately.

Non-Intrusive Reduced Order Modeling of Convection Dominated Flows Using Artificial Neural Networks with Application to Rayleigh-Taylor Instability

Non-Intrusive Reduced Order Modeling of Convection Dominated Flows Using Artificial Neural Networks with Application to Rayleigh-Taylor Instability

Propulsion optimization of a jellyfish-inspired robot based on a nonintrusive reduced-order model with proper orthogonal decomposition

In this research, the propulsion of the proposed jellyfish-inspired mantle undulated propulsion robot (MUPRo) is optimized. To reliably predict the hydrodynamic forces acting on the robot, the proposed nonintrusive reduced-order model (NIROM) based on proper orthogonal decomposition (POD) additionally considers the POD basis that makes an important contribution to the features on the specified boundary. The proposed model establishes a mapping between the parameter-driven motion of the mantle and the evolution of the fluid characteristics around the swimmer. Moreover, to predict new cases where the input needs to be updated, the input of the proposed model is taken from the kinematics of the robot rather than extracted from full-order high-fidelity models. In this paper, it takes approximately 950 s to perform a simulation using the full-order high-fidelity model. However, the computational cost for one prediction with the proposed POD-NIROM is around 0.54 s, of which about 0.2 s is contributed by preprocessing. Compared with the NIROM based on the classic POD method, the proposed POD-NIROM can effectively update the input and reasonably predict the characteristics on the boundary. The analysis of the hydrodynamic performance of the MUPRo pinpoints that, over a certain period and with a certain undulation amplitude, the hydrodynamic force generated by the swinging-like mantle motion (k < 0.5) is greater, outperforming Aequorea victoria in startup acceleration. It is demonstrated that considering a certain power loss and a certain tail beat amplitude, the wave-like mantle motion (k > 0.5) can produce greater propulsion, which means higher propulsion efficiency.

Reduced basis methods for time-dependent problems

Numerical simulation of parametrized differential equations is of crucial importance in the study of real-world phenomena in applied science and engineering. Computational methods for real-time and many-query simulation of such problems often require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During the last few decades, model order reduction has proved successful in providing low-complexity high-fidelity surrogate models that allow rapid and accurate simulations under parameter variation, thus enabling the numerical simulation of increasingly complex problems. However, many challenges remain to secure the robustness and efficiency needed for the numerical simulation of nonlinear time-dependent problems. The purpose of this article is to survey the state of the art of reduced basis methods for time-dependent problems and draw together recent advances in three main directions. First, we discuss structure-preserving reduced order models designed to retain key physical properties of the continuous problem. Second, we survey localized and adaptive methods based on nonlinear approximations of the solution space. Finally, we consider data-driven techniques based on non-intrusive reduced order models in which an approximation of the map between parameter space and coefficients of the reduced basis is learned. Within each class of methods, we describe different approaches and provide a comparative discussion that lends insights to advantages, disadvantages and potential open questions.

An AI-based non-intrusive reduced-order model for extended domains applied to multiphase flow in pipes

The modeling of multiphase flow in a pipe presents a significant challenge for high-resolution computational fluid dynamics (CFD) models due to the high aspect ratio (length over diameter) of the domain. In subsea applications, the pipe length can be several hundreds of meters vs a pipe diameter of just a few inches. Approximating CFD models in a low-dimensional space, reduced-order models have been shown to produce accurate results with a speed-up of orders of magnitude. In this paper, we present a new AI-based non-intrusive reduced-order model within a domain decomposition framework (AI-DDNIROM), which is capable of making predictions for domains significantly larger than the domain used in training. This is achieved by (i) using a domain decomposition approach; (ii) using dimensionality reduction to obtain a low-dimensional space in which to approximate the CFD model; (iii) training a neural network to make predictions for a single subdomain; and (iv) using an iteration-by-subdomain technique to converge the solution over the whole domain. To find the low-dimensional space, we compare Proper Orthogonal Decomposition with several types of autoencoder networks, known for their ability to compress information accurately and compactly. The comparison is assessed with two advection-dominated problems: flow past a cylinder and slug flow in a pipe. To make predictions in time, we exploit an adversarial network, which aims to learn the distribution of the training data, in addition to learning the mapping between particular inputs and outputs. This type of network has shown the potential to produce visually realistic outputs. The whole framework is applied to multiphase slug flow in a horizontal pipe for which an AI-DDNIROM is trained on high-fidelity CFD simulations of a pipe of length 10 m with an aspect ratio of 13:1 and tested by simulating the flow for a pipe of length 98 m with an aspect ratio of almost 130:1. Inspection of the predicted liquid volume fractions shows a good match with the high fidelity model as shown in the results. Statistics of the flows obtained from the CFD simulations are compared to those of the AI-DDNIROM predictions to demonstrate the accuracy of our approach.

Reduced Order Modeling Using Advection-Aware Autoencoders

Physical systems governed by advection-dominated partial differential equations (PDEs) are found in applications ranging from engineering design to weather forecasting. They are known to pose severe challenges to both projection-based and non-intrusive reduced order modeling, especially when linear subspace approximations are used. In this work, we develop an advection-aware (AA) autoencoder network that can address some of these limitations by learning efficient, physics-informed, nonlinear embeddings of the high-fidelity system snapshots. A fully non-intrusive reduced order model is developed by mapping the high-fidelity snapshots to a latent space defined by an AA autoencoder, followed by learning the latent space dynamics using a long-short-term memory (LSTM) network. This framework is also extended to parametric problems by explicitly incorporating parameter information into both the high-fidelity snapshots and the encoded latent space. Numerical results obtained with parametric linear and nonlinear advection problems indicate that the proposed framework can reproduce the dominant flow features even for unseen parameter values.

Fast modelling of gas reservoir performance with proper orthogonal decomposition based autoencoder and radial basis function non-intrusive reduced order models

Two new, non-intrusive reduced order frameworks for the faster modelling of gas reservoirs with time-varying production are presented and compared. The first method is an extension of a method using proper orthogonal decomposition (POD) in conjunction with radial basis functions (RBFs) that has previously been applied to predicting the performance of oil reservoirs undergoing a constant rate waterflood. The second method uses an autoencoder rather than RBFs to estimate the flow dynamics (pressure distributions) in hyperspace for unseen cases. Both frameworks are ‘trained’ using sample outputs from off-line, commercial reservoir simulations of a realistic heterogeneous gas reservoir with time-varying production controls typical of gas field operation. These controls include time-varying rate and switching between bottom hole pressure and rate control as well as cases where wells get shut-in. Both POD-based models produce reasonable forecasts of the reservoir performance for new unseen/prediction cases and are between 0.22 and 300 times faster than conventional simulation, including the time spent performing training simulations with conventional simulation solutions. The POD-RBF models are more accurate and consistent with reference commercial simulation outputs than the POD-AE models. In addition, the POD-AE models required more trial and error to set up as the number of hidden layers needed, depends on the particular scenario being modelled. There is no ab initio way of predicting the best number of layers for a given type of scenario. This makes them less suitable for practical application by reservoir engineers. Overall the POD-RBF framework is the most robust and accurate of the two methods. • Non Intrusive Reduced Order Modelling of real gas field from commercial simulator solutions. • Evaluation of machine learning and numerical techniques. • Modelling time-varying production scenarios with mixed boundary conditions and shut-in wells. • Estimation of spatial distribution of pressure in gas field.

Non-Intrusive Reduced Order Modelling for the fast simulation of gas reservoirs

Non-Intrusive Reduced Order Modelling (NIROM) is evaluated for the prediction of gas reservoir performance, including the time evolution of the pressure distribution within the reservoir. The NIROM projects simulation solutions into hyperspace using proper orthogonal decomposition (POD), then radial basis function (RBF) interpolation is used to describe the dynamics of the fluid flow in hyperspace. POD-RBF NIROMs have been used previously for the estimation of saturation distributions over time in oil reservoirs during pressure maintenance (e.g. by gas flooding) but have not been tested for gas reservoir depletion problems. In this work, the accuracy and efficiency of the NIROM were evaluated by comparing its outputs to those obtained from conventional simulations for two test cases: a homogeneous gas reservoir model with gas rate or bottom hole pressure production controls, and a realistic heterogeneous reservoir model with a similar geological description to the Norne field. We show that the NIROM can provide accurate estimates of pressure and saturation distributions as well as production data and it runs between 4 and 40 times faster. This is despite having been implemented using an interpreted language. We show that a 99.9% energy criterion can be used for determining the number of POD basis functions for pressure and saturation distributions, but more basis functions are needed for estimating production data. In all cases, the linear radial basis function gives the best estimation.

Greedy Non-Intrusive Reduced-Order Model’s application in dynamic blowing and suction flow control to suppress the flow separation

The Greedy Non-Intrusive Reduced-Order Model (GNIROM) is applied in dynamic blowing and suction’s flow control to suppress the GAW1 airfoil’s flow separation. The effects of the uniform blowing and uniform suction on the flow fields and aerodynamic coefficients have been studied firstly. By comparing a uniform suction control with a dynamic blowing and suction control, it has been found that the dynamic blowing and suction can provide a higher lift coefficient and a lower energy loss. Finally, for the sake of achieving an optimal dynamic blowing and suction control, an optimization of the dynamic blowing and suction’s parameters is taken based on the GNIROM, consisting of a Non-Intrusive ROM (NIROM) built by the two-level POD and Radial Basis Function (RBF), and the relative greedy method. A cost function has been proposed by considering suppressing the flow separation, improving the lift-drag characteristics and reducing the energy loss together. The GNIROM has been built for unsteady vorticity fields with the parameters of the dynamic blowing and suction varying. Based on the GNIROM’s approximation, the optimal control parameters’ values have been determined quickly. Compared to the initial dynamic blowing and suction control, the optimal one has reduced the drag coefficient by 43.4 percents and the energy loss by 42 percents.

Local non‐intrusive reduced order modeling based on soft clustering and classification algorithm

AbstractThe use of non‐intrusive reduced order modeling (NIROM) to approximate high‐fidelity computer models has been steadily increased over the past decade. Recently, local NIROM has been proposed to improve the model accuracy in highly nonlinear problems in which distinct characteristic regimes coexist. The core concept of local NIROM is the decomposition of the parameter domains into a subregime to create multiple models. However, the existing local NIROM not only partitions the individual models in a mutually exclusive manner, but also uses a single model for prediction. This results in the extrapolation of surrogate models and the generation of artificial discontinuities. To mitigate these problems, a local NIROM that allows flexible overlapping of individual NIROMs is developed. This method softly partitions and combines individual NIROMs using machine learning techniques, such as fuzzy c‐means and multinomial logistic regression. Furthermore, a variance‐based adaptive sampling technique that can consider both local exploitation and global exploration is applied to improve model accuracy. The proposed method is validated against the transonic flow and in‐flight icing problem, and demonstrates superior performance relative to its local counterpart by up to 16.5% and 33.9% in terms of normalized root‐mean‐square error and exclusive OR error, respectively.

A Nonintrusive Reduced-Order Model for Uncertainty Quantification in Numerical Solution of One-Dimensional Free-Surface Water Flows Over Stochastic Beds

Free-surface water flows over stochastic beds are complex due to the uncertainties in topography profiles being highly heterogeneous and imprecisely measured. In this study, the propagation and influence of several uncertainty parameters are quantified in a class of numerical methods for one-dimensional free-surface flows. The governing equations consist of both single-layer and two-layer shallow water equations on either flat or nonflat topography. For this purpose, the free-surface profiles are computed for different realizations of the random variables when the bed is excited with sources whose statistics are well defined. Many research studies have been dedicated to the development of numerical methods to achieve some order of accuracy in free-surface flows. However, little concern was given to examine the performance of these numerical methods in the presence of uncertainty. This work addresses this specific area in computational hydraulics with regards to the uncertainty generated from bathymetric forces. As numerical solvers for the one-dimensional shallow water equations, we implement four finite volume methods. To reduce the required number of samples for uncertainty quantification, we combine the proper orthogonal decomposition method with the polynomial chaos expansions for efficient uncertainty quantifications of complex hydraulic problems with large number of random variables. Numerical results are shown for several test examples including dam-break problems for single-layer and two-layer shallow water flows. The problem of flow exchange through the Strait of Gibraltar is also solved in this study. The obtained results demonstrate that in some hydraulic applications, a highly accurate numerical method yields an increase in its uncertainty and makes it very demanding to use in an operational manner with measured data from the field. On the other hand, when the complexity of physics increases, these highly accurate numerical methods display less uncertainty compared to the low accurate methods.

Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: Comparison with linear subspace techniques

Natural convection in porous media is a highly nonlinear multiphysical problem relevant to many engineering applications (e.g., the process of CO2 sequestration). Here, we extend and present a non-intrusive reduced order model of natural convection in porous media employing deep convolutional autoencoders for the compression and reconstruction and either radial basis function (RBF) interpolation or artificial neural networks (ANNs) for mapping parameters of partial differential equations (PDEs) on the corresponding nonlinear manifolds. To benchmark our approach, we also describe linear compression and reconstruction processes relying on proper orthogonal decomposition (POD) and ANNs. We present comprehensive comparisons among different models through three benchmark problems. The reduced order models, linear and nonlinear approaches, are much faster than the finite element model, obtaining a maximum speed-up of 7×106 because our framework is not bound by the Courant–Friedrichs–Lewy condition; hence, it could deliver quantities of interest at any given time contrary to the finite element model. Our model’s accuracy still lies within a relative error of 7% in the worst-case scenario. We illustrate that, in specific settings, the nonlinear approach outperforms its linear counterpart and vice versa. We hypothesize that a visual comparison between principal component analysis (PCA) and t-Distributed Stochastic Neighbor Embedding (t-SNE) could indicate which method will perform better prior to employing any specific compression strategy.

Non-Intrusive Reduced-Order Modeling of Radiation Transport in the Atmosphere

Non-Intrusive Reduced-Order Modeling of Radiation Transport in the Atmosphere

A Novel Approach Combining Bootstrapped Non-Intrusive Reducedorder Models and Unscented Transform for the Robust and Efficientcfd Analysis of Accidental Gas Releases in Congested Plants

A Novel Approach Combining Bootstrapped Non-Intrusive Reducedorder Models and Unscented Transform for the Robust and Efficientcfd Analysis of Accidental Gas Releases in Congested Plants