Non-intrusive Reduced Order Model Research Articles

Data assimilation with machine learning for dynamical systems: Modelling indoor ventilation

Data assimilation is a method of combining physical observations with prior knowledge (for instance, a computational simulation) in order to produce an improved estimate of the state of a system; that is, improved over what the physical observations or the computational simulation, alone, could offer. Recently, machine learning techniques have been deployed in order to address the significant computational burden that is associated with the procedures involved in data assimilation.In this paper we propose an approach that uses a non-intrusive reduced-order model (NIROM) as a surrogate for a high-resolution model thereby saving computational effort. The mismatch between observations and the surrogate model is propagated forwards and backwards in time in a manner similar to 4D-variational data assimilation methods. The observations and prior are reconciled in a new way which takes full advantage of the neural network used in the NIROM and also means that there is no need to form the sensitivities explicitly when propagating the mismatch. Instead, the observations are part of the input and output of the network. The key advantages of this data assimilation approach are its simplicity (as we can avoid differentiating the forward model and bypass the use of an optimisation method), and its ability to integrate with control and uncertainty quantification methodologies.Modelling the air quality in a school classroom is the test case for our demonstration. After comparing the proposed data assimilation approach with 4D Variational Data Assimilation, we investigate two scenarios. The first of these is a dual-twin type experiment, for which the proposed approach is shown to perform very well, and the second is a test case which assimilates predictions from the NIROM with observations collected from a classroom in Houndsfield Primary School.

Reduced-order modelling for real-time physics-based variation simulation enhanced with adaptive sampling and optimized interpolation

AbstractManufacturing industries require a right-first-time paradigm to remain competitive. Variation simulation (VS) is a key tool to predict variation of the final shape of flexible assemblies, allowing to reduce defects and waste. VS models involving compliant sheet-metal parts commonly integrate physics-based simulation with statistical approaches (usually Monte Carlo simulation). Although increasingly used as a backbone of synthesis techniques for (stochastic) optimization of assembly systems, the main roadblock of VS methods is the intense computational costs due to time-intensive simulations and high-dimensional design space. Driven by the need of time reduction, this paper presents an innovative real-time physics-based VS model of assembly systems with compliant sheet-metal parts. The proposed methodology involves a non-intrusive reduced-order model (niROM), empowered by a novel adaptive sampling procedure for dataset generation, and a cross-validation-based optimized radial basis function (RBF) formulation for interpolation. Demonstrated through two case studies—(i) a remote laser welding operation to predict mechanical distortions, with two input parameters, and (ii) the assembly of an aircraft vertical stabilizer with five input parameters—the methodology achieves accurate real-time results, with up to a 43% improvement in accuracy compared to traditional sampling techniques. Findings highlight the critical influence of the sampling strategy and the number of input parameters on ROM accuracy. Better results are reached by employing adaptive sampling in combination with optimum RBF, which additionally disengages the user from the choice of the interpolation settings. This study unlocks new avenues in the field of variation simulation and dimensional/quality monitoring by narrowing the gap between any advanced CAE solver and VS models with real-time physics-based simulations.

Adjacency-based, non-intrusive model reduction for vortex-induced vibrations

Vortex-induced Vibrations (VIV) pose computationally expensive problems of high practical interest to several engineering fields. In this work we develop a non-intrusive, reduced-order modeling methodology, applied to two-dimensional (2D), VIV cases subject to a laminar, incompressible flow. Performing a physics-informed approximation of the Arbitrary Lagrangian–Eulerian (ALE) incompressible Navier–Stokes (NS), we derive a discrete-time, quadratic-bilinear model structure for the velocity flowfield on a reference domain. This structure, along with a predefined sparsity pattern motivated by the adjacency-based sparsity of the discretized NS-ALE operators, leads to the formulation of a sparse, full-order model (sFOM) inference problem. Thus, the data-driven inference task requires solving many “local” least squares (LS) problems, isolating the contribution of geometrically “nearest neighbours” for each degree of freedom. Numerical aspects of inference such as data centering and regularization, as well as the direct enforcement of boundary conditions under the sFOM formulation are extensively discussed. The inferred, sFOM operators are then projected to a non-intrusive reduced-order model (ROM) for the velocity flowfield via the Proper Orthogonal Decomposition (POD). The resulting non-intrusive ROM (sFOM-POD) is coupled with the first-principle, 2D solid oscillation equations, resulting to a hybrid physics-informed/first-principle VIV dynamics model, simulated using an implicit time integration scheme. The mapping of the coupled solution from the ALE reference domain to the current configuration is also presented. This methodology is applied to two testcases of an elliptical, non-deformable solid mounted on springs, subject to Re=90,180 flows. Numerical results indicate a successful coupling between the data-driven flowfield and solid dynamics, with prediction errors of less than 3% for both the flowfield and the solid oscillation. A comparative study with respect to the sFOM-POD dimension indicates the robustness and potential of the approach.

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

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

Data driven reduced modeling for fluidized bed with immersed tubes based on PCA and Bi-LSTM neural networks

The fast and accurate reduced-order modeling of fluidized beds is a challenging task in the field of fluid dynamics, owing to their high dimensionality and nonlinear dynamic behavior. In this study, a nonintrusive reduced order modeling method, the reduced order model based on principal component analysis and bidirectional long short-term memory networks (PBLSTM ROM), was developed to capture complex spatio-temporal dynamics of fluidized beds. By combining principal component analysis and Bidirectional long- short-term memory networks, the PBLSTM ROM effectively extracted dynamic evolution information without any prior knowledge of governing equations, enabling reduced-order modeling of unsteady flow fields. The PBLSTM ROM was validated using the solid volume fraction and gas velocity flow fields of a fluidized bed with immersed tubes, showing superior performance over both the PLSTM and PANN ROMs in accurately capturing temporal changes in the fluidization fields, especially in the region near immersed tubes where severe fluctuations appear. Moreover, the PBLSTM ROM improved the simulation speed by five orders of magnitude compared to traditional computational fluid dynamics simulations. These findings suggest that the PBLSTM ROM presents a promising approach for analyzing the complex fluid flows in engineering practice.

Non-Intrusive Reduced-Order Modeling Based on Parametrized Proper Orthogonal Decomposition

A new non-intrusive reduced-order modeling method based on space-time parameter decoupling for parametrized time-dependent problems is proposed. This method requires the preparation of a database comprising high-fidelity solutions. The spatial bases are extracted from the database through first-level proper orthogonal decomposition (POD). The algebraic relationship between the time trajectory/parameter positions and the projection coefficient is described by the linear superposition of the second-level POD bases (temporal bases) and the second-level projection coefficients (parameter-dependent coefficients). This decomposition strategy decouples the space-time parameter effects, providing a stable foundation for fast predictions of parametrized time-dependent problems. The mappings between the parameter locations and the parameter-dependent coefficients are approximated as Gaussian process regression (GPR) models. The accuracy and efficiency of the PPOD-ROM are demonstrated through two numerical examples: flows past a cylinder and turbine flows with a clocking effect.

A novel non-intrusive ROM for randomly excited linear dynamical systems with high stochastic dimension using ANN

Analyzing large stochastic dynamical systems is computationally very expensive. A statistical simulation framework requires invoking the solver multiple times — ranging from thousands to millions. A non-intrusive reduced order model (ROM) serves as a computationally efficient alternative in this framework. Uncertainties in dynamical systems originate from two sources: system parameters and excitation. However, all the existing ROMs have been developed for uncertainty only in the system parameters. To the best of the authors’ knowledge, no ROM exists for dynamical systems under random excitation. A discretization of the input process increases the stochastic dimensionality, typically by a few hundred or thousand. This leads to a high dimensional interpolation or regression in the ROM, which is practically infeasible. This issue is addressed in this work by proposing a novel non-intrusive ROM that bypasses the need for such discretization. Accordingly, a regression is carried out directly on the random excitation using a neural network-based surrogate model. A principal component-based data compression is used in tandem to reduce the stochastic dimensionality of excitations. Detailed numerical studies are conducted to study the accuracy and efficiency of the proposed ROM using two examples: a mistuned bladed disk problem and a soil–structure interaction. The numerical results show that the proposed ROM is accurate and gains a significant speed-up of more than sixty for both examples. Using the proposed ROM, the cost of uncertainty quantification can be reduced significantly within the framework of Monte Carlo simulation.

Deep convolutional architectures for extrapolative forecasts in time-dependent flow problems

Physical systems whose dynamics are governed by partial differential equations (PDEs) find numerous applications in science and engineering. The process of obtaining the solution from such PDEs may be computationally expensive for large-scale and parameterized problems. In this work, deep learning techniques developed especially for time-series forecasts, such as LSTM and TCN, or for spatial-feature extraction such as CNN, are employed to model the system dynamics for advection-dominated problems. This paper proposes a Convolutional Autoencoder(CAE) model for compression and a CNN future-step predictor for forecasting. These models take as input a sequence of high-fidelity vector solutions for consecutive time steps obtained from the PDEs and forecast the solutions for the subsequent time steps using auto-regression; thereby reducing the computation time and power needed to obtain such high-fidelity solutions. Non-intrusive reduced-order modeling techniques such as deep auto-encoder networks are utilized to compress the high-fidelity snapshots before feeding them as input to the forecasting models in order to reduce the complexity and the required computations in the online and offline stages. The models are tested on numerical benchmarks (1D Burgers’ equation and Stoker’s dam-break problem) to assess the long-term prediction accuracy, even outside the training domain (i.e. extrapolation). The most accurate model is then used to model a hypothetical dam break in a river with complex 2D bathymetry. The proposed CNN future-step predictor revealed much more accurate forecasting than LSTM and TCN in the considered spatiotemporal problems.

Data-driven reduced-order modelling for blood flow simulations with geometry-informed snapshots

Parametric reduced-order modelling often serves as a surrogate method for hemodynamics simulations to improve the computational efficiency in many-query scenarios or to perform real-time simulations. However, the snapshots of the method require to be collected from the same discretisation, which is a straightforward process for physical parameters, but becomes challenging for geometrical problems, especially for those domains featuring unparameterised and unique shapes, e.g. patient-specific geometries. In this work, a data-driven surrogate model is proposed for the efficient prediction of blood flow simulations on similar but distinct domains. The proposed surrogate model leverages group surface registration to parameterise those shapes and formulates corresponding hemodynamics information into geometry-informed snapshots by the diffeomorphisms constructed between a reference domain and original domains. A non-intrusive reduced-order model for geometrical parameters is subsequently constructed using proper orthogonal decomposition, and a radial basis function interpolator is trained for predicting the reduced coefficients of the reduced-order model based on compressed geometrical parameters of the shape. Two examples of blood flowing through a stenosis and a bifurcation are presented and analysed. The proposed surrogate model demonstrates its accuracy and efficiency in hemodynamics prediction and shows its potential application toward real-time simulation or uncertainty quantification for complex patient-specific scenarios.

Non-intrusive data-driven reduced-order modeling for time-dependent parametrized problems

Reduced-order models are indispensable for multi-query or real-time problems. However, there are still many challenges to constructing efficient ROMs for time-dependent parametrized problems. Using a linear reduced space is inefficient for time-dependent nonlinear problems, especially for transport-dominated problems. The non-linearity usually needs to be addressed by hyper-reduction techniques, such as DEIM, but it is intrusive and relies on the assumption of affine dependence of parameters. This paper proposes and studies a non-intrusive reduced-order modeling approach for time-dependent parametrized problems. It is purely data-driven and naturally split into offline and online stages. During the offline stage, a convolutional autoencoder, consisting of an encoder and a decoder, is trained to perform dimensionality reduction. The encoder compresses the full-order solution snapshots to a nonlinear manifold or a low-dimensional reduced/latent space. The decoder allows the recovery of the full-order solution from the latent space. To deal with the time-dependent problems, a high-order dynamic mode decomposition (HODMD) is utilized to model the trajectories in the latent space for each parameter. During the online stage, the HODMD models are first utilized to obtain the latent variables at a new time, then interpolation techniques are adopted to recover the latent variables at a new parameter value, and the full-order solution is recovered by the decoder. Some numerical tests are conducted to show that the approach can be used to predict the unseen full-order solution at new times and parameter values fast and accurately, including transport-dominated problems.

The use of GANs and transfer learning in model-order reduction of turbulent wake of an isolated high-rise building

The high cost of computational fluid dynamics (CFD) simulations has limited their use for real-time and long-term simulations. To address this limitation, reduced-order modeling has been developed, and recent advancements in artificial intelligence (AI) algorithms have led to the creation of non-intrusive reduced-order models (NIROMs). The present study proposes a framework for using various AI techniques to develop NIROMs for the turbulent wake of an isolated high-rise building. The framework is demonstrated using datasets from two different flow conditions, isothermal and unstable thermal stratification. A residual adversarial autoencoder (AAE) network with convolutional layers and Wasserstein generative adversarial network (WGAN) is first used to reduce the dimensionality of the unstable dataset. Afterward, a parallel bidirectional long short-term memory (BiLSTM) network computes its evolution through time. The proposed framework provides commendable predictions of the airflow field. The weights of the trained model are utilized to create a NIROM based on transfer learning (TL) for a smaller dataset that represents the isothermal condition. The results showed that when the training dataset is limited and the model cannot be trained on sufficient vortex shedding cycles, the proper use of TL can lead to better outcomes compared to the traditional training method (i.e., training from scratch).

A surrogate non-intrusive reduced order model of quasi-geostrophic turbulence dynamics based on a combination of LSTM and different approaches of DMD

Mathematical modeling is applied to study phenomena and system behavior.In various engineering fields, many physical phenomena are illustrated using a set of differential equations.In many real-world applications, the mathematical models are very complex, and numerical simulations in high-dimensional systems are challenging.Examples of these problems are large-scale physical problems such as geophysical, which have high temporal and spatial variations.In these problems, model order reduction is a useful method for achieving an appropriate approximation because it can significantly decrease computational costs.Deep learning has recently been used to explore information from data and make predictions.There are several methods for dimensionality reduction.In this paper, we combine the dynamic mode decomposition (DMD) and the long short-term memory (LSTM) network.This is because LSTM can predict nonlinear systems and time series data.We use LSTM and DMD to predict nonlinear systems and reduce dimensions, respectively.Four common DMD schemes have been applied for dimensionality reduction.The common geophysical dataset has been used to evaluate the performance of the proposed method.Finally, we compare the variations of the modal coefficients which are obtained from snapshots projection and the reduced-order model.These results show the high accuracy of our proposed method.One of the things that is important is the time complexity of algorithm implementation.The time complexity of the proposed method is 10 times faster when 15 modes are used for modeling than when all features are used.

Deep-learning assisted reduced order model for high-dimensional flow prediction from sparse data

The reconstruction and prediction of full-state flows from sparse data are of great scientific and engineering significance yet remain challenging, especially in applications where data are sparse and/or subjected to noise. To this end, this study proposes a deep-learning assisted non-intrusive reduced order model (named DCDMD) for high-dimensional flow prediction from sparse data. Based on the compressed sensing (CS)-dynamic mode decomposition (DMD), the DCDMD model is distinguished by two novelties. First, a sparse matrix is defined to overcome the strict random distribution condition of sensor locations in CS, thus allowing flexible sensor deployments and requiring very few sensors. Second, a deep-learning-based proxy is invoked to acquire coherent flow modes from the sparse data of high-dimensional flows, thereby addressing the issue of defining sparsity and the stringent incoherence condition in the conventional CSDMD. The two advantageous features, combined with the fact that the model retains flow physics in the online stage, lead to significant enhancements in accuracy and efficiency, as well as superior insensitivity to data noises (i.e., robustness), in both reconstruction and prediction of full-state flows. These are demonstrated by three benchmark examples, i.e., cylinder wake, weekly mean sea surface temperature, and isotropic turbulence in a periodic square area.

Non-intrusive model order reduction for parametric radiation transport simulations

We present a parametric, non-intrusive reduced-order model (ROM) for multigroup radiation transport, addressing the high computational cost and resource requirements associated with transport simulations in scenarios involving multiple queries. To achieve this, we employ Proper Orthogonal Decomposition (POD) to learn a groupwise reduced basis representation of the observed data (obtained from full-order simulations). We utilize Gaussian Process Regression and Multivariate Adaptive Regression Splines to learn the parametric, low-dimensional latent code (the reduced coordinates in the subspace spanned by the observed data). The approach is non-invasive, making it applicable to any software without the need for accessing the source code. We provide numerical results for various parametric problems, including a 3D multigroup transport case. Comparing the ROM-based surrogate to the full-order radiation transport model, solved using discrete ordinates in angle and discontinuous finite elements in space, we achieve a scalar flux solution that is 20,000 times faster with less than 1% relative error in a 3D transport scenario. Moreover, by utilizing the ROM, we demonstrate how a simple uncertainty quantification study can be completed within minutes, which would otherwise require significantly larger computational resources.

Proper Orthogonal Decomposition Mode Coefficient Interpolation: A Non-Intrusive Reduced-Order Model for Parametric Reactor Kinetics

In this paper, a non-intrusive reduced-order model (ROM) for parametric reactor kinetics simulations is presented. Time-dependent ROMs are notoriously data intensive and difficult to implement when nonlinear multiphysics phenomena are considered. These challenges are exacerbated when parametric dependencies are included. The proper orthogonal decomposition mode coefficient interpolation (POD-MCI) ROM presented in this work can be constructed directly from lower-dimensional full-order model (FOM) outputs and is independent of the underlying model. This greatly alleviates the data requirement of many existing ROMs and can be used without modification on arbitrarily complex models or experimental data. The POD-MCI ROM is demonstrated on a number of examples and yields accurate characterizations of the parametric behaviors of both FOM outputs and derived quantities of interest within the selected parameter spaces, at extremely attractive computational speedup factors relative to FOMs.

Reduced-order modeling for stochastic large-scale and time-dependent flow problems using deep spatial and temporal convolutional autoencoders

A non-intrusive reduced-order model based on convolutional autoencoders is proposed as a data-driven tool to build an efficient nonlinear reduced-order model for stochastic spatiotemporal large-scale flow problems. The objective is to perform accurate and rapid uncertainty analyses of the flow outputs of interest for which the input parameters are deemed uncertain. The data are constituted from a set of high-fidelity snapshots, collected using an inhouse high-fidelity flow solver, which correspond to a sample of the uncertain input parameters. The method uses a 1D-convolutional autoencoder to reduce the spatial dimension of the unstructured meshes used by the flow solver. Another convolutional autoencoder is used for the time compression. The encoded latent vectors, generated from the two compression levels, are then mapped to the input parameters using a regression-based multilayer perceptron. The proposed model allows for rapid predictions for unseen parameter values, allowing the output statistical moments to be computed efficiently. The accuracy of the proposed approach is compared to that of the linear reduced-order technique based on an artificial neural network through two benchmark tests (the one-dimensional Burgers and Stoker’s solutions) and a hypothetical dam break flow problem, with an unstructured mesh and over a complex bathymetry river. The numerical results show that the proposed methods present strong predictive capabilities to accurately approximate the statistical moments of the outputs. In particular, the predicted statistical moments are oscillations-free, unlike those obtained with the traditional proper orthogonal decomposition method. The proposed reduction framework is simple to implement and can be applied to other parametric and time-dependent problems governed by partial differential equations, which are commonly encountered in many engineering and science problems.

Deep learning-driven nonlinear reduced-order models for predicting wave-structure interaction

The feasibility of using a Long Short-Term Memory (LSTM) network-driven Non-Intrusive Reduced Order Model (NIROM) to predict the dynamics of a two-dimensional box floating and interacting with surface water waves is assessed in this study. The ground for these wave-structure interactions (WSI) problems, namely box displacements and hydrodynamic forces arising from wave interaction corresponding to a particular surface wave profile, are computed using a single-phase Smoothed Particle Hydrodynamics (SPH). The dimensionality of the system is first reduced using the Discrete Empirical Interpolation Method (DEIM) and the LSTM is applied to the reduced system resulting in a DEIM-LSTM network for developing a surrogate for prediction. This is further enhanced by incorporating the physics information into the loss function resulting in a physics-informed LSTM (LSTM-PINN) for predicting the rigid body dynamics of box motion. The performance of these networks in predicting the dynamics of the floating box is compared as a proof-of-concept demonstration.

A non-intrusive reduced-order model for wind farm wake analysis based on SPOD-DNN

Wind farm wake modeling is of great significance for wind turbine layout optimization design and yaw control strategy. In this work, we combine deep neural network (DNN) with spectral proper orthogonal decomposition (SPOD) to discover dynamic characteristics of wake under different inflow conditions. Then an assessment of the proposed SPOD-DNN surrogate modeling method of parameterized fluid is performed by comparing the predicted results. Meanwhile, we demonstrate the robustness of the SPOD-DNN through a comparison with POD-DNN, where SPOD produces fewer modes than POD but can achieve the same cumulative contribution rate and wake prediction accuracy. In the end, the method is developed to predict the wake of single wind turbine in untrained inflow condition and Wake of six wind turbines with different yaw angles. The results reveals that the model has good generalization performance and can robustly reconstruct the wake of multiple wind turbines in different directions.

Machine learning and reduced order modelling for the simulation of braided stent deployment.

Endoluminal reconstruction using flow diverters represents a novel paradigm for the minimally invasive treatment of intracranial aneurysms. The configuration assumed by these very dense braided stents once deployed within the parent vessel is not easily predictable and medical volumetric images alone may be insufficient to plan the treatment satisfactorily. Therefore, here we propose a fast and accurate machine learning and reduced order modelling framework, based on finite element simulations, to assist practitioners in the planning and interventional stages. It consists of a first classification step to determine a priori whether a simulation will be successful (good conformity between stent and vessel) or not from a clinical perspective, followed by a regression step that provides an approximated solution of the deployed stent configuration. The latter is achieved using a non-intrusive reduced order modelling scheme that combines the proper orthogonal decomposition algorithm and Gaussian process regression. The workflow was validated on an idealized intracranial artery with a saccular aneurysm and the effect of six geometrical and surgical parameters on the outcome of stent deployment was studied. We trained six machine learning models on a dataset of varying size and obtained classifiers with up to 95% accuracy in predicting the deployment outcome. The support vector machine model outperformed the others when considering a small dataset of 50 training cases, with an accuracy of 93% and a specificity of 97%. On the other hand, real-time predictions of the stent deployed configuration were achieved with an average validation error between predicted and high-fidelity results never greater than the spatial resolution of 3D rotational angiography, the imaging technique with the best spatial resolution (0.15mm). Such accurate predictions can be reached even with a small database of 47 simulations: by increasing the training simulations to 147, the average prediction error is reduced to 0.07mm. These results are promising as they demonstrate the ability of these techniques to achieve simulations within a few milliseconds while retaining the mechanical realism and predictability of the stent deployed configuration.

Non-intrusive reduced-order modeling of radiation transport in the atmosphere

In this work, we investigate a non-intrusive parametric reduced-order model (ROM) for atmospheric scale neutral particle transport via an application to neutron transport. The spatial domain of interest consists of an 80-km tall cylinder with a 20-km radius. The system parameters are the elevation and emission spectrum of the radiation source, as well as the relative humidity, which alters the cross sections of the atmospheric air. The full-order model (FOM) employs a first-collision source approach to mitigate ray effects. The separate computation of the uncollided and collided flux components is exploited in the reduced-order model (ROM). The ROM utilizes various non-linear transformations of the FOM solution to address the many orders of magnitude present in the FOM data, which can vary by up to 26 orders of magnitude. Subspace discovery is performed on this transformed dataset and the parametric dependence in this latent space is learned via Gaussian process regression. Speed-up factors of 5,100× to 9,700× are observed, while maintaining less than 10% pointwise error throughout most of the spatial domain. Based on the choice of transformation, small regions of high error can be present as a result of the variable source position.