Accurate Surrogate Model Research Articles

An adaptive surrogate model approach for random vibration analysis of the train–bridge system

To improve the efficiency of random vibration analysis of the train–bridge system, a new approach is proposed based on the adaptive sampling (AS) surrogate model in this paper. First, an initial sample set and a candidate sample set are generated by the method based on Generalized F (GF)-discrepancy. Second, a theoretical model is used to calculate the maximum dynamic response of the train–bridge system corresponding to the initial sample set, and a surrogate model based on Gaussian process regression (GPR) is constructed. Third, the learning function is used to identify new samples in the candidate sample set to optimize the current surrogate model until it meets satisfactory prediction accuracy. Finally, the maximum dynamic response of each sample in the representative sample set is predicted by the final surrogate model, and then the statistical properties of the dynamic response of the train–bridge system are analyzed. To verify the effectiveness of the proposed method, the distribution of training sample points and prediction accuracy of the AS surrogate model and the one-stage sampling (OS) surrogate model are compared and analyzed, taking the prediction of the maximum wheel load reduction rate (WLRR) of a train on the bridge as an example. On this basis, the prediction accuracy of the AS and OS surrogate models for the random characteristics of the train on the bridge is evaluated. The results show that the proposed method can significantly improve the prediction accuracy of the surrogate model without increasing the number of training samples.

Uncertainty quantification of two-phase flow in porous media via the Coupled-TgNN surrogate model

The uncertainty quantification (UQ) of subsurface two-phase flow usually requires numerous executions of forward simulations under varying conditions. In this work, a novel coupled theory-guided neural network (TgNN) based surrogate model is built to facilitate computation efficiency under the premise of satisfactory accuracy. The core notion of this proposed method is to bridge two separate blocks on top of an overall network. They underlie the TgNN model in a coupled form, which reflects the coupling nature of pressure and water saturation in the two-phase flow equation. The TgNN model not only relies on labeled data, but also incorporates underlying scientific theory and experiential rules (e.g., governing equations, stochastic parameter fields, boundary and initial conditions, well conditions, and expert knowledge) as additional components into the loss function. The performance of the TgNN-based surrogate model for two-phase flow problems is tested by different numbers of labeled data and collocation points, as well as the existence of data noise. The proposed TgNN-based surrogate model offers an effective way to solve the coupled nonlinear two-phase flow problem, and shows good accuracy and strong robustness when compared with the purely data-driven surrogate model. By combining the accurate TgNN-based surrogate model with the Monte Carlo method, UQ tasks can be performed at a minimum cost to evaluate statistical quantities. Since the heterogeneity of the random fields strongly impacts the results of the surrogate model, corresponding variance and correlation length are added to the input of the neural network to maintain its predictive capacity. In addition, several more complicated scenarios are also considered, including dynamically changing well conditions and dynamically changing variance of random fields. The results show that the TgNN-based surrogate model exhibits satisfactory accuracy, stability, and efficiency in the UQ problem of subsurface two-phase flow.

A general framework for quantifying uncertainty at scale.

In many fields of science, comprehensive and realistic computational models are available nowadays. Often, the respective numerical calculations call for the use of powerful supercomputers, and therefore only a limited number of cases can be investigated explicitly. This prevents straightforward approaches to important tasks like uncertainty quantification and sensitivity analysis. This challenge can be overcome via our recently developed sensitivity-driven dimension-adaptive sparse grid interpolation strategy. The method exploits, via adaptivity, the structure of the underlying model (such as lower intrinsic dimensionality and anisotropic coupling of the uncertain inputs) to enable efficient and accurate uncertainty quantification and sensitivity analysis at scale. Here, we demonstrate the efficiency of this adaptive approach in the context of fusion research, in a realistic, computationally expensive scenario of turbulent transport in a magnetic confinement tokamak device with eight uncertain parameters, reducing the effort by at least two orders of magnitude. In addition, we show that this refinement method intrinsically provides an accurate surrogate model that is nine orders of magnitude cheaper than the high-fidelity model.

A comparative study of learning techniques for the compressible aerodynamics over a transonic RAE2822 airfoil

In this study, the modeling of the compressible pressure field on the RAE 2822 airfoil using deep learning (DL) is investigated. The objective is to generate, at low cost, the complete Mach envelope from a given aerodynamic database. A dataset with more than 2.000 RANS simulations at various angles of attack and Mach numbers has been created, and validated, to train fast and accurate surrogate models model of the pressure field. The procedure is inspired by the well-known analytical Prandtl–Glauert transformation, which maps solutions from one Mach number to another, but however fails in the transonic regime because of the strong non-linearities due to shock waves. Thus, instead of a classical direct generation of the pressure fields, DL is used here to perform a geometric transformation from an incompressible pressure field towards a compressible solution. The key question addressed in this study is: how accurate and sample efficient are DL techniques compared with classical surrogate modeling tools? To answer this question, two types of DL are investigated (CNN and GCNN) and compared with Proper Orthogonal Decomposition (POD) coupled to a Gaussian Process Regression (GPR). These methods are trained on various input resolutions and database sizes. Results show that DL models are able to accurately predict the intensity and location of the shock over a wide range of angles of attack and Mach numbers up to the transonic regime. Predictions of the lift coefficient reveal a relative error lower than 2% with respect to the high-fidelity data. Because of their inherent nonlinear nature, CNN and GCNN provide more accurate results even for small training datasets, whereas POD encounters difficulties to reconstruct properly shock waves. Since CNN requires inputs and outputs in a pixel-like format, it suffers from interpolation errors that can only be mitigated with fine resolutions, making the training more difficult. By performing convolutions directly on unstructured data, GCNN eliminates this interpolation error and provides the best accuracy and sample efficiency. This study shows that advanced DL techniques, such as CNN and GCNN, are capable of predicting complex flows, and outperform classical tools on such tasks.

Surrogate model for 3D ground and structural deformations in tunneling by the sequential excavation method

Ground deformation control is one of the key challenges faced during urban tunneling when using the sequential excavation method (SEM). Three-dimensional (3D) numerical analysis is a powerful and essential tool to assess SEM-induced ground and structural deformations. The main challenge in numerical modeling is the uncertainties in data needed to build a model. Various analyses are needed, such as parametric and back-analysis. However, such analyses require thousands to millions of repeated model evaluations making 3D numerical simulations expensive. One technique to make the modeling more manageable is reducing the model size by using a surrogate model that captures the main elements of a full 3D model. This paper examines the capability of four surrogate modeling methods, namely the Polynomial Chaos Expansion (PCE), Kriging, sequential Polynomial-Chaos-Kriging (PCK-SEQ), and optimal Polynomial-Chaos-Kriging (PCK-OPT), to accurately and efficiently capture the ground and structural deformations induced by SEM tunneling. A 3D finite-difference model using FLAC3D was developed to simulate an actual SEM project's excavation and initial support process. A sensitivity analysis was performed to determine the most influential geotechnical input parameters. The Sobol sequence sampling strategy was utilized to conduct a design of experiments. The overall and individual output accuracy of surrogate models, the robustness of training and testing, and the accuracy distribution throughout the input parameter space were evaluated and compared.

Optimal Reduced Basis Method for Aeroacoustics with Impedance Boundary Conditions

Solving the linearized Euler equations in the harmonic frequency domain amounts to solving a large linear system. Doing so repetitively can be costly, especially when some parameters need to be varied (for example, the acoustic impedance of a liner), such as in optimization and impedance eduction problems. To accelerate the calculations, the equations can first be projected onto a basis of reduced dimension, only requiring a limited number of full solutions called snapshots, which are calculated at different impedance values. This paper is concerned with finding an optimal set of impedances allowing the greedy creation of a reduced basis that leads to the most accurate surrogate modeling in the least number of calls to the direct solver. The optimal set of impedance values is first obtained on a two-dimensional duct configuration, and numerical verification tests are performed on a three-dimensional engine nacelle.

Adaptive machine learning-based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery

In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to finally obtain certified results. Numerical experiments are given to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem.

Thermal cracking of the large molecular alcohols in shale oil by experimental study and kinetic modeling

There is an important influence of alcohols in shale oil on its thermochemical properties. In this work, the fractions and species of the alcohols in shale oil from retorting oil shale were studied experimentally. The alcohols exist mainly in the form of linear carbon chains with different lengths, as well as different sites of hydroxyls. Then, 7-tetradecanol and pentadecanol were selected for further exploring the pyrolysis mechanisms through theoretical calculation and kinetic modeling, due to their abundance and representativeness in alcohols of shale oil. Density functional theory (DFT) method was utilized to reveal the difficulty of the thermal cracking of these two alcohols. The results indicated that the dissociation energies of C-C bonds are lower than those of C-H bonds, the reaction of eliminating water has a low activation energy. The Mulliken charges imply that there is a large electrostatic attraction between the O atom and its attached C atom in 7-tetradecanol, but an electrostatic repulsion between them in pentadecanol, which is thought to have an effect on the elimination of water. Finally, the detailed pyrolysis kinetic mechanisms of these two alcohols were written in a systematic manner by the GRI-Mech3.0 mechanism. A large number of the kinetic parameters were computed by the group-additivity method, which are consistent with the thermochemical data calculated by DFT. In summary, this research provides a theoretical basis for development of the kinetic mechanisms of large molecular alcohols and the accurate surrogate shale oil model.

Design and multi-scenario optimization of a hybrid power system based on a working gas turbine: Energy, Exergy, Exergoeconomic and Environmental evaluation

The rising demand for electricity, along with the need to minimize carbon footprints, has motivated academics to investigate the flexible and efficient integration of energy conversion technologies. A novel hybrid power generation system based on environmentally friendly and cost-effective technologies to recover the waste heat of a working gas turbine is designed and assessed in different scenarios of multi-objective optimization from energy, exergy, exergoeconomic, and environmental (4E) perspectives. In the proposed system, a steam methane reformer and a water gas shift reactor are utilized for hydrogen production, while a polymer electrolyte membrane fuel cell (PEMFC) and steam/organic Rankine cycles are run for generating additional power. Aspen Plus, in conjunction with Fortran, Microsoft Excel, and MATLAB, is used to model and simulate the designed plant. The response surface methodology (RSM) is utilized to determine accurate surrogate models to describe the evaluation criteria, and the Non-dominated Sorting Genetic Algorithm II technique is employed to seek the optimal conditions. Moreover, TOPSIS and LINMAP decision-making approaches are used to find the best final solution among Pareto frontiers. The analysis of variance (ANOVA) and sensitivity analysis are also applied to evaluate the importance of the design variables. In this regard, three single-objective optimizations and four multi-objective optimization scenarios based on the maximization of the ecological coefficient of performance (ECOP) and the minimization of CO2 emissions and total system product cost (Ċp) are carried out. It is demonstrated that the system’s evaluation criteria have the highest and lowest sensitivity to the variation of reformer temperature and ORC pressure, respectively. From the triple-objective optimization procedure, the decision variables including reformer temperature, ORC pressure, Rankine cycle I pressure, and Rankine cycle II pressure are 544 °C, 4.35 bar, 158.12 bar, and 52.82 bar, respectively. At these conditions, the total hybrid system’s energy efficiency, exergy efficiency, exergy destruction, net generated power, and total investment cost rate are 45.96%, 46.83%, 215.72 MW, 203.67 MW, and 9791 $/h, respectively. The findings of this paper conclude that it is necessary to address all objective functions simultaneously in the system’s ultimate optimum design. Furthermore, the objective of this paper becomes even more apparent when there is no choice but to cut greenhouse gas emissions while also addressing the rising global energy demand.

Learning constitutive models from microstructural simulations via a non-intrusive reduced basis method: Extension to geometrical parameterizations

Two-scale simulations are often employed to analyze the effect of the microstructure on a component’s macroscopic properties. Understanding these structure–property relations is essential in the optimal design of materials for specific applications. However, these two-scale simulations are typically computationally expensive and infeasible in multi-query contexts such as optimization and material design. To make such analyses amenable, the microscopic simulations can be replaced by inexpensive-to-evaluate surrogate models. Such surrogate models must be able to handle microstructure parameters in order to be used for material design. A previous work focused on the construction of an accurate surrogate model for microstructures under varying loading and material parameters by combining proper orthogonal decomposition and Gaussian process regression. However, that method works only for a fixed geometry, greatly limiting the design space. This work hence focuses on extending the methodology to treat geometrical parameters. To this end, a method that transforms different geometries onto a parent domain is presented, that then permits existing methodologies to be applied. We propose to solve an auxiliary problem based on linear elasticity to obtain the geometrical transformations. The method has a good reducibility and can therefore be quickly solved for many different geometries. Using these transformations, combined with the nonlinear microscopic problem, we derive a fast-to-evaluate surrogate model with the following key features: (1) the predictions of the effective quantities are independent of the auxiliary problem, (2) the predicted stress fields automatically fulfill the microscopic balance laws and are periodic, (3) the method is non-intrusive, (4) the stress field for all geometries can be recovered, and (5) the sensitivities are available and can be readily used for optimization and material design. The proposed methodology is tested on several composite microstructures, where rotations and large variations in the shape of inclusions are considered. Finally, a two-scale example is shown, where the surrogate model achieves a high accuracy and significant speed up, thus demonstrating its potential in two-scale shape optimization and material design problems.

Quantification of Autoignition Risk in Aeroderivative Gas Turbine Premixers Using Incompletely Stirred Reactor and Surrogate Modeling

Abstract The design and operation of premixers for gas turbines must deal with the possibility of relatively rare events causing dangerous autoignition (AI). Rare AI events may occur in the presence of fluctuations of operational parameters, such as temperature and fuel composition, and must be understood and predicted. This work presents a methodology based on incompletely stirred reactor (ISR) and surrogate modeling to increase efficiency and feasibility in premixer design optimization for rare events. For a representative premixer, a space-filling design is used to sample the variability of three influential operational parameters. An ISR is reconstructed and solved in a postprocessing fashion for each sample, leveraging a well-resolved computational fluid dynamics solution of the non-reacting flow inside the premixer. Via detailed chemistry and reduced computational costs, ISR tracks the evolution of AI precursors and temperature conditioned on a mixture fraction. Accurate surrogate models are then trained for selected AI metrics on all ISR samples. The final quantification of the AI probability is achieved by querying the surrogate models via Monte Carlo sampling of the random parameters. The approach is fast and reliable so that user-controllable, independent variables can be optimized to maximize system performance while observing a constraint on the allowable probability of AI.

Data-driven model for divertor plasma detachment prediction

We present a fast and accurate data-driven surrogate model for divertor plasma detachment prediction leveraging the latent feature space concept in machine learning research. Our approach involves constructing and training two neural networks: an autoencoder that finds a proper latent space representation (LSR) of plasma state by compressing the multi-modal diagnostic measurements and a forward model using multi-layer perception (MLP) that projects a set of plasma control parameters to its corresponding LSR. By combining the forward model and the decoder network from autoencoder, this new data-driven surrogate model is able to predict a consistent set of diagnostic measurements based on a few plasma control parameters. In order to ensure that the crucial detachment physics is correctly captured, highly efficient 1D UEDGE model is used to generate training and validation data in this study. The benchmark between the data-driven surrogate model and UEDGE simulations shows that our surrogate model is capable of providing accurate detachment prediction (usually within a few per cent relative error margin) but with at least four orders of magnitude speed-up, indicating that performance-wise, it has the potential to facilitate integrated tokamak design and plasma control. Comparing with the widely used two-point model and/or two-point model formatting, the new data-driven model features additional detachment front prediction and can be easily extended to incorporate richer physics. This study demonstrates that the complicated divertor and scrape-off-layer plasma state has a low-dimensional representation in latent space. Understanding plasma dynamics in latent space and utilising this knowledge could open a new path for plasma control in magnetic fusion energy research.

Deep adaptive arbitrary polynomial chaos expansion: A mini-data-driven semi-supervised method for uncertainty quantification

All kinds of uncertainties influence the reliability of the engineering system. Thus, uncertainty quantification is significant to the system reliability analysis. Polynomial chaos expansion (PCE) is an effective method for uncertainty quantification while it requires sufficient labeled data to quantify uncertainty accurately. To overcome this problem, this paper proposes the adaptive arbitrary polynomial chaos (aPC) and proves two properties of the adaptive expansion coefficients. Sequentially, a semi-supervised deep adaptive arbitrary polynomial chaos expansion (Deep aPCE) method is proposed based on the adaptive aPC and the deep neural network (DNN). The Deep aPCE method uses two properties of the adaptive aPC to assist in training the DNN by a small amount of labeled data and abundant unlabeled data, significantly reducing the training data cost. On the other hand, the Deep aPCE method adopts the DNN to fine-tune the adaptive expansion coefficients dynamically, improving the accuracy of uncertainty quantification. Besides, the Deep aPCE method can directly construct accurate surrogate models of the high dimensional stochastic systems. Five numerical examples are used to verify the effectiveness of the Deep aPCE method. Finally, the Deep aPCE method is applied to the reliability analysis of an axisymmetric conical aircraft.

Stochastic low-order modelling of hydrogen autoignition in a turbulent non-premixed flow

Autoignition risk in initially non-premixed flowing systems, such as premixing ducts, must be assessed to help the development of low-NOx systems and hydrogen combustors. Such situations may involve randomly fluctuating inlet conditions that are challenging to model in conventional mixture-fraction-based approaches. A Computational Fluid Dynamics (CFD)-based surrogate modelling strategy is presented here for fast and accurate predictions of the stochastic autoignition behaviour of a hydrogen flow in a hot air turbulent co-flow. The variability of three input parameters, i.e., inlet fuel and air temperatures and average wall temperature, is first sampled via a space-filling design. For each sampled set of conditions, the CFD modelling of the flame is performed via the Incompletely Stirred Reactor Network (ISRN) approach, which solves the reacting flow governing equations in post-processing on top of a Large Eddy Simulation (LES) of the inert hydrogen plume. An accurate surrogate model, namely a Gaussian Process, is then trained on the ISRN simulations of the burner, and the final quantification of the variability of autoignition locations is achieved by querying the surrogate model via Monte Carlo sampling of the random input quantities. The results are in agreement with the observed statistics of the autoignition locations. The methodology adopted in this work can be used effectively to quantify the impact of fluctuations and assist the design of practical combustion systems.

Interval optimization design of a submersible surface ship form considering the uncertainty of surrogate model

The integrated design of the ship form is a key issue in the design of submersible surface ship (SSS). In the simulation-based design (SBD) framework, lack of sample data leads to uncertainty in the surrogate model. In this paper, first, a hull cross-section shape modification technology, the self-fusion method, was used to change the ship form, and a fourth-order response surface model was introduced to establish surrogate model of hydrodynamic performance. Second, total resistance in the surface condition and one at a diving depth of 0.48 m were selected as the optimization targets. Third, an automated computing platform was developed to integrate the modules to improve computing efficiency and accuracy. Finally, to explore the adaptability of the ship form scheme, the interval optimization analysis method was introduced into uncertainty optimization. Among them, the interval variable was the surrogate model accuracy, and the targets were the minimum median and the minimum radius of the weighted resistance of cross-domain cases. A two-layer nested optimization system was chosen. Results show that resistance of the optimal SSS was reduced. The interval optimization results were compared with the deterministic optimization results, and the feasibility of hull shape interval optimization was verified.

Study on source identification and source-sink relationship of LNAPLs pollution in groundwater by the adaptive cyclic improved iterative process and Monte Carlo stochastic simulation

• Biodegradation is considered in the transport of LNAPLs by adding a biodegradation model. • The VDGS method is used to optimize the hyperparameters of ResNet-18 surrogate model. • The adaptive cyclic improved iterative process is constructed to improve the solution effect of groundwater LNAPLs pollution source identification problem. • The groundwater pollution source-sink relationship is studied by using the Monte Carlo method. It is important to study the source-sink relationship of pollution. However, research on the groundwater pollution sources and pollution sinks has often been conducted separately in the past, which makes it difficult to study the source-sink relationship of pollutants. In this study, the source and sink of pollution in a site contaminated by a light non-aqueous phase liquid (LNAPL) is considered to study the relationship between them. The simulation model needs to be called repeatedly in the research process. To save time and reduce the calculation load, the ResNet-18 network structure is first utilized to create a substitute of simulation model. However, the performance of ResNet-18 surrogate model largely depends on the choice of hyperparameters. The variable-density grid search method is thus utilized to optimize the hyperparameters. In addition, the quality of training samples is an important factor to determine the accuracy of surrogate models. And the accuracy of the surrogate model will affect the effect of groundwater pollution identification. Therefore, the ResNet-18 surrogate model and the ensemble smoother algorithm are combined to construct an adaptive cyclic improved iterative process. Through the adaptive cyclic improved iterative process, the training samples, surrogate model and identification results of variables to be solved can be improved and updated together with the iterative process, to improve the solution effect of groundwater LNAPLs pollution source identification problem. Biodegradation is an important sink of LNAPL pollutants. Therefore, the biodegradation on the LNAPL pollutants in groundwater is considered by adding a biodegradation model to the original model of multi-phase flow. Finally, based on the numerical simulation models of multiphase flow and biodegradation, the Monte Carlo stochastic simulation are used to study the source-sink relationship of LNAPL pollutants. A hypothetical case based on real conditions was used to illustrate the validity of the methods. The results implied that the proposed methods can not only improve the effect of LNAPL pollution identification, but also study the source-sink relationship of LNAPL pollutant.

Non-parametric surrogate model method based on machine learning with application on low-pressure steam turbine exhaust system

Current surrogate model methods that are widely used in optimization and design processes rely on manual parameterization to describe the geometry of objects. The loss of geometric information in this process limits the prediction accuracy of surrogate model. To tackle this problem, the new method directly picks important geometric features from surface meshes of fluid domain using Graph Neural Networks (GNNs) and predicts contours of fluid variables based on extracted information with Convolutional Neural Networks (CNNs). The prediction error of CNNs propagates backwards to train GNNs to select sensitive features from surface meshes. This framework reduces uncertainties introduced by manual parameterization and the loss of geometric information because the input of this new method is from the meshes used in the numerical simulations. With CNN and larger amount of extracted geometric information, this method can also predict higher dimensions distributions of flow variables rather than only several performance metrics. The nature of non-parametric representation of geometry also allows users to access designs defined by other parameterization methods to create a larger database. Additionally, thanks to the generic nature of the new method, it can be used for any other design or optimization processes governed by partial differential equations involving complicated geometries. To demonstrate this new method, a non-parametric surrogate model is built for a low-pressure steam turbine exhaust system (LPES). The new surrogate model uses 10 surfaces meshes of the LPES as input and it is used to predicts the energy flux contours at the exit of the last stage of the turbine. Altogether 582 designs have been generated, which contains two types of geometries defined by different methods. Among them, 550 cases are used for training, and 32 cases for testing. The power output of the last two stages of the turbine predicted by the surrogate model has average 0.86% difference compared with those of numerical simulations over a wide range of power ratings. The structural similarity index measure (SSIM) is used to measure the differences between the simulated and predicted contours at the exit of the last rotor, where the average SSIM of 640 contours is 0.9594 (1.0 being identical).

Numerical Assessment of a Nonintrusive Surrogate Model Based on Recurrent Neural Networks and Proper Orthogonal Decomposition: Rayleigh–Bénard Convection

Recent developments in diagnostic and computing technologies offer to leverage numerous forms of nonintrusive modelling approaches from data where machine learning can be used to build computationally cheap and accurate surrogate models. To this end, we present a nonlinear proper orthogonal decomposition (POD) framework, denoted as NLPOD, to forge a nonintrusive reduced-order model for the Boussinesq equations. In our NLPOD approach, we first employ the POD procedure to obtain a set of global modes to build a linear-fit latent space and utilise an autoencoder network to compress the projection of this latent space through a nonlinear unsupervised mapping of POD coefficients. Then, long short-term memory (LSTM) neural network architecture is utilised to discover temporal patterns in this low-rank manifold. While performing a detailed sensitivity analysis for hyperparameters of the LSTM model, the trade-off between accuracy and efficiency is systematically analysed for solving a canonical Rayleigh–Bénard convection system.

Surrogate-Assisted Autoencoder-Embedded Evolutionary Optimization Algorithm to Solve High-Dimensional Expensive Problems

Surrogate-assisted evolutionary algorithms (EAs) have been intensively used to solve computationally expensive problems with some success. However, traditional EAs are not suitable to deal with high-dimensional expensive problems (HEPs) with high-dimensional search space even if their fitness evaluations are assisted by surrogate models. The recently proposed autoencoder-embedded evolutionary optimization (AEO) framework is highly appropriate to deal with high-dimensional problems. This work aims to incorporate surrogate models into it to further boost its performance, thus resulting in surrogate-assisted AEO (SAEO). It proposes a novel model management strategy that can guarantee reasonable amounts of re-evaluations; hence, the accuracy of surrogate models can be enhanced via being updated with new evaluated samples. Moreover, to ensure enough data samples before constructing surrogates, a problem-dimensionality-dependent activation condition is developed for incorporating surrogates into the SAEO framework. SAEO is tested on seven commonly used benchmark functions and compared with state-of-the-art algorithms for HEPs. The experimental results show that SAEO can further enhance the performance of AEO on most cases and SAEO performs significantly better than other algorithms. Therefore, SAEO has great potential to deal with HEPs.

An Ensemble Surrogate-Based Framework for Expensive Multiobjective Evolutionary Optimization

Surrogate-assisted evolutionary algorithms (SAEAs) have become very popular for tackling computationally expensive multiobjective optimization problems (EMOPs), as the surrogate models in SAEAs can approximate EMOPs well, thereby reducing the time cost of the optimization process. However, with the increased number of decision variables in EMOPs, the prediction accuracy of surrogate models will deteriorate, which inevitably worsens the performance of SAEAs. To deal with this issue, this article suggests an ensemble surrogate-based framework for tackling EMOPs. In this framework, a global surrogate model is trained under the entire search space to explore the global area, while a number of surrogate submodels are trained under different search subspaces to exploit the subarea, so as to enhance the prediction accuracy and reliability. Moreover, a new infill sampling criterion is designed based on a set of reference vectors to select promising samples for training the models. To validate the generality and effectiveness of our framework, three state-of-the-art evolutionary algorithms [nondominated sorting genetic algorithm III (NSGA-III), multiobjective evolutionary algorithm based on decomposition with differential evolution (MOEA/D-DE) and reference vector-guided evolutionary algorithm (RVEA)] are embedded, which significantly improve their performance for solving most of the test EMOPs adopted in this article. When compared to some competitive SAEAs for solving EMOPs with up to 30 decision variables, the experimental results also validate the advantages of our approach in most cases.