Multi-fidelity Surrogate Model Research Articles

Characterising harmful data sources when constructing multi-fidelity surrogate models

Surrogate modelling techniques have seen growing attention in recent years when applied to both modelling and optimisation of industrial design problems. These techniques are highly relevant when assessing the performance of a particular design carries a high cost, as the overall cost can be mitigated via the construction of a model to be queried in lieu of the available high-cost source. The construction of these models can sometimes employ other sources of information which are both cheaper and less accurate. The existence of these sources however poses the question of which sources should be used when constructing a model. Recent studies have attempted to characterise harmful data sources to guide practitioners in choosing when to ignore a certain source. These studies have done so in a synthetic setting, characterising sources using a large amount of data that is not available in practice. Some of these studies have also been shown to potentially suffer from bias in the benchmarks used in the analysis. In this study, we approach the characterisation of harmful low-fidelity sources as an algorithm selection problem. We employ recently developed benchmark filtering techniques to conduct a bias-free assessment, providing objectively varied benchmark suites of different sizes for future research. Analysing one of these benchmark suites with the technique known as Instance Space Analysis, we provide an intuitive visualisation of when a low-fidelity source should be used. By performing this analysis using only the limited data available to train a surrogate model, we are able to provide guidelines that can be directly used in an applied industrial setting.

Nacelle optimisation through multi-fidelity neural networks

PurposeAerodynamic shape optimisation is a complex problem usually governed by transonic non-linear aerodynamics, a high dimensional design space and high computational cost. Consequently, the use of a numerical simulation approach can become prohibitive for some applications. This paper aims to propose a computationally efficient multi-fidelity method for the optimisation of two-dimensional axisymmetric aero-engine nacelles.Design/methodology/approachThe nacelle optimisation approach combines a gradient-free algorithm with a multi-fidelity surrogate model. Machine learning based on artificial neural networks (ANN) is used as the modelling technique because of its ability to handle non-linear behaviour. The multi-fidelity method combines Reynolds-averaged Navier Stokes and Euler CFD calculations as high- and low-fidelity, respectively.FindingsRatios of low- and high-fidelity training samples to degrees of freedom of nLF/nDOFs = 50 and nHF/nDOFs = 12.5 provided a surrogate model with a root mean squared error less than 5% and a similar convergence to the optimal design space when compared with the equivalent CFD-in-the-loop optimisation. Similar nacelle geometries and aerodynamic flow topologies were obtained for down-selected designs with a reduction of 92% in the computational cost. This highlights the potential benefits of this multi-fidelity approach for aerodynamic optimisation within a preliminary design stage.Originality/valueThe application of a multi-fidelity technique based on ANN to the aerodynamic shape optimisation problem of isolated nacelles is the key novelty of this work. The multi-fidelity aspect of the method advances current practices based on single-fidelity surrogate models and offers further reductions in computational cost to meet industrial design timescales. Additionally, guidelines in terms of low- and high-fidelity sample sizes relative to the number of design variables have been established.

Deep Learning-Based Multifidelity Surrogate Modeling for High-Dimensional Reliability Prediction

Abstract Multifidelity surrogate modeling offers a cost-effective approach to reducing extensive evaluations of expensive physics-based simulations for reliability prediction. However, considering spatial uncertainties in multifidelity surrogate modeling remains extremely challenging due to the curse of dimensionality. To address this challenge, this paper introduces a deep learning-based multifidelity surrogate modeling approach that fuses multifidelity datasets for high-dimensional reliability analysis of complex structures. It first involves a heterogeneous dimension transformation approach to bridge the gap in terms of input format between the low-fidelity and high-fidelity domains. Then, an explainable deep convolutional dimension-reduction network (ConvDR) is proposed to effectively reduce the dimensionality of the structural reliability problems. To obtain a meaningful low-dimensional space, a new knowledge reasoning-based loss regularization mechanism is integrated with the covariance matrix adaptation evolution strategy (CMA-ES) to encourage an unbiased linear pattern in the latent space for reliability prediction. Then, the high-fidelity data can be utilized for bias modeling using Gaussian process (GP) regression. Finally, Monte Carlo simulation (MCS) is employed for the propagation of high-dimensional spatial uncertainties. Two structural examples are utilized to validate the effectiveness of the proposed method.

Optimization framework for multi-fidelity surrogate model based on adaptive addition strategy—A case study of self-excited oscillation cavity

This study proposes a multi-fidelity efficient global optimization framework for the structural optimization of self-excited oscillation cavity. To construct a high-precision multi-fidelity surrogate model to correlate the structural parameters of a self-excited oscillation cavity with the gas precipitation and energy consumption characteristics by effectively fuzing the information of different fidelity levels, choosing different correlation functions and hyper-parameter estimation methods, and learning the correlation between the data. The optimization framework determines various sampling methods and quantities by calculating the minimum Euclidean distance between sample points and sensitivity index. To enhance computational efficiency, a multi-fidelity sample library is established by utilizing both precise and coarse computational fluid dynamics grids. The expected improvement criterion-based algorithm for global optimization is employed as an additive strategy to incorporate additional data points into the model. This approach considers both local and global search of the model, thereby enhancing sample accuracy while reducing computation time. Moreover, the utilization of the highly generalized Non-dominated Sorting Genetic Algorithm-II (NSGA-II) for identifying the Pareto optimal solution set enhances convergence speed. The proposed optimization framework in this study achieves a remarkable level of model accuracy and provides optimal solutions even with a limited sample size. It can be widely used in engineering optimization problems.

Leveraging deep reinforcement learning for design space exploration with multi-fidelity surrogate model

Design automation is undergoing a new generation of changes caused by artificial intelligence technologies represented by deep learning and reinforcement learning. Notably, the advantages of deep reinforcement learning in addressing solution optimisation and decision-making tasks with cognitive automation functionality have garnered attention in design. In the context of surrogate model-driven engineering design optimisation, this paper addresses current research challenges such as reliance on domain knowledge for local development, shortcomings in the self-learning and adaptive capabilities of optimisation algorithms for global exploration, etc. Centred around the deep reinforcement learning model, Deep Q-learning, and complemented by self-organising maps and neural network technologies, we propose a methodology considering multi-fidelity simulation data for design space exploration. This approach effectively reduces sampling costs and enables the optimisation model to learn the optimal direction for high-precision predictions and achieve rapid, accurate optimisation. Finally, the effectiveness of the proposed method is comprehensively validated through four typical optimisation scenarios and a case study involving the optimisation of a wheeled robot's suspension swing arm structure. This work will be a crucial reference for applying deep reinforcement learning in simulation-driven engineering design optimisation.

Experimental study on the mechanical property of carbon fiber reinforced polymer with the combined application of digital twin and reduced order algorithm

AbstractRecognizing the anisotropy of mechanical property transfer between carbon fiber reinforced polymer (CFRP) layers, a unidirectional reduced‐order model (UROM) is proposed to comprehensively express the mechanical properties of each layer. A crucial focus is on effectively analyzing the real‐time mechanical properties of CFRP specimens, particularly in accurately identifying and predicting the interlaminar mechanical states of anisotropic CFRP laminated specimens. Moreover, the multi fidelity surrogate (MFS) is employed to replace the computationally intensive model, amalgamating high‐fidelity sensor data with low‐fidelity data obtained from the UROM. The UROM MFS is used to reduce the amount of data in the same layer of CFRP laminates, while preserving the characteristics between different layers and preserving the interlaminar information of CFRP laminates. Finally, the stress prediction results of the UROM MFS and the unreduced MFS model were compared and analyzed. The real‐time response speed of the u‐ROM MFS digital twin (DT) model was increased by 658%. The UROM MFS DT model effectively captures the mechanical properties of each layer of CFRP laminates, enables quick calculation of model parameters, and provides accurate predictions.Highlights First, the interlayer mechanical property of CFRP is analyzed with the combined application of experimental data, DT model, and UROM MFS. Second, MFS is employed to merge high‐fidelity sensor data of CFPR laminate with low‐fidelity data obtained from UROM. Third, the digital twin framework for CFPR laminate with high response speed and calculation accuracy is proposed.

Ensemble learning based hierarchical surrogate model for multi-fidelity information fusion

Recently, multi-fidelity information fusion based surrogate modeling methods have made great progress in the engineering design and optimization tasks. Two main issues in this field are: (1) Since the overarching trend of the responses from the high-fidelity (HF) model is captured by the low-fidelity (LF) model, inaccurate LF model may negatively impact the final modeling results. (2) The responsiveness of multi-fidelity surrogate (MFS) models to variations in the relationship between LF and HF models leads to limited prediction performance. To this end, we propose an ensemble learning based MFS modeling method with a hierarchical framework, called EL-MFS. Specifically, to alleviate the impact of issue (1), we present an adaptive ensemble surrogate model, which aims to effectively mitigate the negative impact of inappropriate LF model selection on the HF approximation results. Furthermore, we propose to exploit the feature mapping and hierarchical framework to boost the versatility of the model as a way to mitigate the dependence of MFS performance on the relationship between HF and LF models. To assess the effectiveness of the proposed model, a sequence of numerical problems is tested, and some advanced surrogates are selected as the baseline models. Moreover, to demonstrate the potential of the proposed model in aiding intricate engineering design, one engineering case is investigated as well.

Incremental sampling methods for multi-fidelity surrogate modeling: Application on a furnace operating in MILD combustion conditions

This study introduces a framework for developing a multi-fidelity reduced order model (MF-ROM) of a combustion furnace operating under Moderate and Intense Low-oxygen Dilution (MILD) conditions. It integrates Proper Orthogonal Decomposition for data compression, Procrustes manifold alignment for fidelity transfer, and CoKriging for interpolation. Design parameters such as air injector diameter, fuel composition, and equivalence ratio were used to generate two- and three-dimensional simulations and build the MF-ROM. Additionally, the key question concerning the optimal number of high-fidelity simulations to balance accuracy and training cost is addressed when building the MF-ROM. Through incremental sampling strategies, it is demonstrated that around half of the training cost can be conserved while maintaining comparable error values.

Parametric shape optimization of pin fin arrays using a multi-fidelity surrogate model based Bayesian method

The current work aims to develop a multi-fidelity Bayesian optimization methodology to reduce computation time in CFD based design optimization of an internal cooling channel of a gas turbine blade. To build energy efficient gas turbine engines, it is important to increase the turbine inlet temperature, while preventing material failure of hot gas path components. Power lost due to pumping compressed air in turbine blade internal cooling channels can affect the overall efficiency of a power cycle. While designing such a cooling channel, it is important to ensure high heat transfer within a given pressure drop budget. Design optimization using CFD typically needs a sufficient number of computationally expensive high fidelity simulations to search for an optimum design. In the current work, a multi-fidelity Gaussian process surrogate based Bayesian optimization framework has been used to design a pin fin array channel, typically used for gas turbine blade trailing edge cooling, with the goal of maximizing Nusselt number at a constrained pressure loss. Data from a cheaper, low fidelity computation model has been leveraged, along with an expensive high fidelity one, in order to search for a design configuration that maximizes heat transfer at a given pressure drop constraint. The multi-fidelity approach improves upon single fidelity optimization results, at a low additional computational cost due to usage of simulation data from the cheap lowfidelity CFD model. A multifidelity approach outperformed the previous single fidelity case, with 6% higher objective while satisfying the pressure constraint, with 2% lower computational expense and 8 fewer high fidelity data point computations.

Multi-Task Learning for Design Under Uncertainty With Multi-Fidelity Partially Observed Information

Abstract The assessment of system performance and identification of failure mechanisms in complex engineering systems often requires the use of computation-intensive finite element software or physical experiments, which are both costly and time-consuming. Moreover, when accounting for uncertainties in the manufacturing process, material properties, and loading conditions, the process of reliability-based design optimization (RBDO) for complex engineering systems necessitates the repeated execution of expensive tasks throughout the optimization process. To address this problem, this paper proposes a novel methodology for RBDO. First, a multi-fidelity surrogate modeling strategy is presented, leveraging partially observed information (POI) from diverse sources with varying fidelity and dimensionality to reduce computational cost associated with evaluating expensive high-dimensional complex systems. Second, a multi-task surrogate modeling framework is proposed to address the concurrent evaluation of multiple constraints for each design point. The multi-task framework aids in the development of surrogate models and enhances the effectiveness of reliability analysis and design optimization. The proposed multi-fidelity multi-task machine learning model utilizes a Bayesian framework, which significantly improves the performance of the predictive model and provides uncertainty quantification of the prediction. Additionally, the model provides a highly accurate and efficient framework for reliability-based design optimization through knowledge sharing. The proposed method was applied to two design case studies. By incorporating POI from various sources, the proposed approach improves the accuracy and efficiency of system performance prediction, while simultaneously addressing the cost and complexity associated with the design of complex systems.

Hybrid uncertainty propagation based on multi-fidelity surrogate model

There always exist multiple uncertainties including random uncertainty, interval uncertainty, and fuzzy uncertainty in engineering structures. In the presence of hybrid uncertainties, the hybrid uncertainty propagation analysis can be a challenging problem, which suffers from the computational burden of double-loop procedure when numerical simulation techniques are employed. In this work, a novel method for efficient hybrid uncertainty propagation analysis with the three types of uncertainties is proposed. Generally, multi-fidelity surrogate models, such as Co-Kriging, can greatly improve the computational efficiency by leveraging information from a low-fidelity model to build a high-fidelity approximate model. However, the traditional multi-fidelity surrogate model methods always calculate the hybrid uncertainty propagation result by combining with several numerical simulation techniques. This process can introduce post-processing errors unless unlimited number of samples are used, which is impossible in engineering application. In order to address this issue, the analytical solutions of the output mean and output variance are derived based on the Co-Kriging, and the resulting mean and variance are both random variables. Moreover, a new adaptive framework is established to strengthen the estimation accuracy of the hybrid uncertainty propagation result, by combining the augmented expected improvement function and the derived mean random variable. Several applications are introduced to demonstrate the effectiveness of the proposed method for solving hybrid uncertainty propagation problems.

A Bayesian Multi-fidelity Neural Network to Predict Nonlinear Frequency Backbone Curves

Abstract The use of structural mechanics models during the design process often leads to the development of models of varying fidelity. Often low-fidelity models are efficient to simulate but lack accuracy, while the high-fidelity counterparts are accurate with less efficiency. This paper presents a multi-fidelity surrogate modeling approach that combines the accuracy of a high-fidelity finite element model with the efficiency of a low-fidelity model to train an even faster surrogate model that parameterizes the design space of interest. The objective of these models is to predict the nonlinear frequency backbone curves of the Tribomechadynamics Research Challenge benchmark structure which exhibits simultaneous nonlinearities from frictional contact and geometric nonlinearity. The surrogate model consists of an ensemble of neural networks that learn the mapping between low and high-fidelity data through nonlinear transformations. Bayesian neural networks are used to assess the surrogate model's uncertainty. Once trained, the multi-fidelity neural network is used to perform sensitivity analysis to assess the influence of the design parameters on the predicted backbone curves. Additionally, Bayesian calibration is performed to update the input parameter distributions to correlate the model parameters to the collection of experimentally measured backbone curves.

A novel multi-fidelity surrogate modeling method for non-hierarchical data fusion

A novel multi-fidelity surrogate modeling method for non-hierarchical data fusion

A multi-fidelity surrogate modeling method in the presence of non-hierarchical low-fidelity data

Multi-fidelity (MF) surrogate models are increasingly utilized in engineering design due to their capability to achieve desired accuracy at a reduced cost. However, many existing MF modeling methods presuppose a hierarchical relationship between different fidelity models, which may not be applicable when multiple low-fidelity (LF) models exhibit varying fidelity levels across the design space due to different simplification methods. To tackle this issue, a multi-fidelity surrogate modeling method based on local correlation-weighted fusion (LCWF-MFS) is developed. In this proposed approach, each LF model is assigned variable weights depending on its local correlation with the high-fidelity (HF) model, thereby maximizing the utilization of information from these LF data sources. Furthermore, an innovative parameter tuning technique rooted in penalty error optimization is introduced, aiming to establish a more reliable scale factor between HF and LF models. Several numerical examples as well as a maximum deformation prediction problem pertaining to the micro aerial vehicle (MAV) fuselage during landing or impact are used to illustrate the effectiveness of the proposed method. Results demonstrate the superior accuracy and robustness of the proposed LCWF-MFS method in comparison to alternative methods.

Multi-fidelity reduced-order surrogate modelling

High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated. Multi-fidelity surrogate modelling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are scarce. However, low-fidelity models, while often displaying the qualitative solution behaviour, fail to accurately capture fine spatio-temporal and dynamic features of high-fidelity models. To address this shortcoming, we present a data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states—time-parameter-dependent expansion coefficients of the POD basis—using a multi-fidelity long short-term memory network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality of this method is demonstrated by a collection of PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features.

A novel multi-fidelity surrogate modeling framework integrated with sequential sampling criterion for non-hierarchical data

A novel multi-fidelity surrogate modeling framework integrated with sequential sampling criterion for non-hierarchical data

A BAYESIAN NEURAL NETWORK APPROACH TO MULTI-FIDELITY SURROGATE MODELING

This paper deals with surrogate modeling of a computer code output in a hierarchical multi-fidelity context, i.e., when the output can be evaluated at different levels of accuracy and computational cost. Using observations of the output at low- and high-fidelity levels, we propose a method that combines Gaussian process (GP) regression and the Bayesian neural network (BNN), called the GPBNN method. The low-fidelity output is treated as a single-fidelity code using classical GP regression. The high-fidelity output is approximated by a BNN that incorporates, in addition to the highfidelity observations, well-chosen realizations of the low-fidelity output emulator. The predictive uncertainty of the final surrogate model is then quantified by a complete characterization of the uncertainties of the different models and their interaction. The GPBNN is compared to most of the multi-fidelity regression methods allowing one to quantify the prediction uncertainty.

Extended Hierarchical Kriging Method for Aerodynamic Model Generation Incorporating Multiple Low-Fidelity Datasets

Multi-fidelity surrogate modeling (MFSM) methods are gaining recognition for their effectiveness in addressing simulation-based design challenges. Prior approaches have typically relied on recursive techniques, combining a limited number of high-fidelity (HF) samples with multiple low-fidelity (LF) datasets structured in hierarchical levels to generate a precise HF approximation model. However, challenges arise when dealing with non-level LF datasets, where the fidelity levels of LF models are indistinguishable across the design space. In such cases, conventional methods employing recursive frameworks may lead to inefficient LF dataset utilization and substantial computational costs. To address these challenges, this work proposes the extended hierarchical Kriging (EHK) method, designed to simultaneously incorporate multiple non-level LF datasets for improved HF model construction, regardless of minor differences in fidelity levels. This method leverages a unique Bayesian-based MFSM framework, simultaneously combining non-level LF models using scaling factors to construct a global trend model. During model processing, unknown scaling factors are implicitly estimated through hyperparameter optimization, resulting in minimal computational costs during model processing, regardless of the number of LF datasets integrated, while maintaining the necessary accuracy in the resulting HF model. The advantages of the proposed EHK method are validated against state-of-the-art MFSM methods through various analytical examples and an engineering case study involving the construction of an aerodynamic database for the KP-2 eVTOL aircraft under various flying conditions. The results demonstrated the superiority of the proposed method in terms of computational cost and accuracy when generating aerodynamic models from the given multi-fidelity datasets.

Research on rapid calculation method of wind turbine blade strain for digital twin

Digital twin (DT) technology provides unlimited possibilities for remote intelligent operation and maintenance of wind turbine blade (WTB) in operation. However, the urgent demand for high accuracy and real-time performance limit the engineering applications of this technology. This article presents a novel rapid calculation method of WTB strain, which aims to break the technical difficulties of real-time calculation in digital twin and realize synchronous mapping of WTB health status. Firstly, the load acting on the WTB is simplified according to the equivalent conversion theory. Then, a simplified finite element model, namely mechanism calculation model, of WTB is built based on mechanics. The multi-fidelity surrogate model is innovatively adopted to calibrate the results of the mechanism calculation via a small amount of data measured via sensors on the WTB. Based on the calibrated results, a unique strain field estimation model is built to provide more comprehensive information about WTB. Finally, the proposed method is verified via an experiment about WTB strain measurement and numerical simulation. The results demonstrate a considerable advancement in real-time performance with approximately 1444× compared to the traditional finite element method. The shortest elapsed time is 0.91s. Moreover, the minimum relative error is 0.11 % and the average relative error is less than 0.76 %.

Parallel multi-objective Bayesian optimization approaches based on multi-fidelity surrogate modeling

Aerospace product design optimizations, such as micro-aerial vehicle fuselage design, often involve multiple objectives. Multi-objective Bayesian optimization (MOBO) is an efficient approach in solving problems concerning multiple conflicting objectives. Conventional MOBO approaches are often limited by sequential optimizations, which can only add one sample in each iteration. The parallel computing strategy is a desirable way to accelerate the optimization process by adding a batch of samples in one iteration. Unfortunately, existing parallel MOBO approaches are constrained to handling single-fidelity data, thereby missing out on the potential benefits of utilizing auxiliary low-fidelity data. To further improve the optimization efficiency of the parallel MOBO approach, this paper proposes two parallel MOBO approaches based on multi-fidelity surrogate modeling. Specially, the cheap auxiliary low-fidelity data can be utilized to improve the performance of the parallel MOBO approach by multi-fidelity surrogate modeling. The updating points and fidelity levels are determined by a modified hypervolume excepted improvement function, and two parallel computing strategies are developed for multi-point sampling. Additionally, a constraint handling strategy is developed for problems with constraints by adopting the probability of feasibility functions. The proposed approaches are demonstrated through numerical benchmark examples and two real-world applications involving the multi-objective optimizations of a micro-aerial vehicle fuselage and a metamaterial vibration isolator. Results in the real-world applications show that the proposed approaches significantly improve the optimization efficiency with faster convergence speed and exhibit superior overall performance compared with the state-of-the-art MOBO methods.