Multi-fidelity Surrogate Model Research Articles

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

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

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.

A multi-fidelity surrogate model based on design variable correlations

Multi-fidelity surrogate (MFS) models have garnered significant attention in the field of engineering optimization due to their ability to attain the desired accuracy at a reduced cost. However, most previous MFS models employed a single scaling factor for the global design space, which posed challenges in adaptively adjusting the scaling factor within local design spaces. To address this issue, this paper proposes a novel MFS model based on design variable correlations (MFS-DVC). MFS-DVC introduces local characteristics to the scaling factors by leveraging correlations between design variables, enabling adaptive adjustments of the scaling factors at different positions within the design space. Moreover, MFS-DVC offers a more comprehensive exploration of relationships and information among high-fidelity (HF) data points by utilizing the correlations between design variables. This enhancement contributes to improving the accuracy and robustness of the model. The performance of MFS-DVC was evaluated by comparing it with three benchmark MFS models and two single-fidelity surrogate models on 22 test functions and one engineering problem involving a hydraulic press. Additionally, the cost ratio and combination of HF and low-fidelity samples were studied to assess their impact on the performance of the MFS-DVC model. The results demonstrate that MFS-DVC consistently delivers competitive performance in terms of prediction accuracy and robustness.

Enhanced multi-fidelity modeling for digital twin and uncertainty quantification

The increasing significance of digital twin technology across engineering and industrial domains, such as aerospace, infrastructure, and automotive, is undeniable. However, the lack of detailed application-specific information poses challenges to its seamless implementation in practical systems. Data-driven models play a crucial role in digital twins, enabling real-time updates and predictions by leveraging data and computational models. Nonetheless, the fidelity of available data and the scarcity of accurate sensor data often hinder the efficient learning of surrogate models, which serve as the connection between physical systems and digital twin models. To address this challenge, we propose a novel framework that begins by developing a robust multi-fidelity surrogate model, subsequently applied for tracking digital twin systems. Our framework integrates polynomial correlated function expansion (PCFE) with the Gaussian process (GP) to create an effective surrogate model called H-PCFE. Going a step further, we introduce deep-HPCFE, a cascading arrangement of models with different fidelities, utilizing nonlinear auto-regression schemes. These auto-regressive schemes effectively address the issue of erroneous predictions from low-fidelity models by incorporating space-dependent cross-correlations among the models. To validate the efficacy of the multi-fidelity framework, we first assess its performance in uncertainty quantification using benchmark numerical examples. Subsequently, we demonstrate its applicability in the context of digital twin systems.

A Single-Fidelity Surrogate Modeling Method Based on Nonlinearity Integrated Multi-Fidelity Surrogate

Abstract Surrogate model provides a promising way to reasonably approximate complex underlying relationships between system parameters. However, the expensive modeling cost, especially in large problem sizes, hinders its applications in practical problems. To overcome this issue, with the advantages of the multi-fidelity surrogate (MFS) model, this paper proposes a single-fidelity surrogate model with a hierarchical structure, named nonlinearity integrated correlation mapping surrogate (NI-CMS) model. The NI-CMS model first establishes the low-fidelity model to capture the underlying landscape of the true function, and then, based on the idea of MFS model, the established low-fidelity model is corrected by minimizing the mean square error to ensure prediction accuracy. Especially, a novel MFS model (named NI-MFS), is constructed to enhance the stability of the proposed NI-CMS model. More specifically, a nonlinear scaling term, which assumes the linear combination of the projected low-fidelity predictions in a high-dimensional space can reach the high-fidelity level, is introduced to assist the traditional scaling term. The performances of the proposed model are evaluated through a series of numerical test functions. In addition, a surrogate-based digital twin of an XY compliant parallel manipulator is used to validate the practical performance of the proposed model. The results show that compared with the existing models, the NI-CMS model provides a higher performance under the condition of a small sample set, illustrating the promising potential of this surrogate modeling technique.

Variable-Stiffness Composite Optimization Using Dynamic and Exponential Multi-Fidelity Surrogate Models

Variable-stiffness composite laminates with spatially varied orientation angles always require refined finite element models to accurately model the spatial variation characteristics, thus resulting in high computation costs. Further, practical restrictions in fiber steering should be imposed to generate manufacturable designs, making the design problem more challenging. To address these challenges, this paper presents a new framework assisted by multi-fidelity surrogate models for variable-stiffness composite optimization with manufacturing constraints. An initial sampling strategy is originally developed for the case of involving the fiber steering constraints, improving the accuracy of the surrogate model in the concerned space. Based on Gaussian process regressions, a new type of multi-fidelity model corrected with an exponential function is proposed by fusing many cheap low-fidelity models and a few expensive high-fidelity models. Using genetic algorithm as the optimizer, new data points are generated from the optimization process and then employed to dynamically update the constructed multi-fidelity model. The proposed optimization strategy is applied to case studies of buckling optimization for both a composite plate and a composite cylinder, demonstrating that the developed framework requires significantly less computation.

Digital twin-based structural health monitoring by combining measurement and computational data: An aircraft wing example

Digital twin is a concept that utilizes digital technologies to mirror the real-time states of physical assets and extract the hidden yet valuable information of physical assets for optimization, decision-making or scheduling. By combining measurement and computational data, this paper presents a digital twin-based structural health monitoring framework of physical assets. The process for building the measurement-computation combined digital twin (MCC-DT) involves four steps. First, an artificial intelligence-driven load identification method combining measurement and computational data is employed to recognize the loads applied on physical assets. Two approaches were proposed to realize load identification, based on single fidelity surrogate models and deep learning techniques, respectively. Second, multi-fidelity surrogate (MFS) models are applied to improve the accuracy in the MCC-DT. Two routes for implementing the MFS models are introduced and the advantages and shortcomings of both are analyzed. Third, an online rainflow counting algorithm is developed to calculate the degradation of the physical assets. The main advantage of the algorithm is that it can provide a near real-time estimation for the damage accumulated of physical assets. Finally, the data generated from the first three steps can be fused into a three-dimensional scene using Web graphics library to provide an intuitive view of the MCC-DT. To describe the implementation details of the framework and verify its applicability and effectiveness, the MCC-DT was established using an aircraft model as an example.