Gradient-enhanced Kriging Research Articles

Integrating small data and shape prior knowledge with gradient-enhanced Kriging through adaptive knowledge sampling

Integrating prior knowledge into surrogate models constitutes an advanced approach to tackling the challenges posed by small data. The existing gradient-enhanced Kriging (GEK) method can enhance accuracy by incorporating gradient points of shape prior knowledge. Nonetheless, our findings indicate that the strategy employed for setting gradient points significantly impacts accuracy, even when the number of gradient points remains constant. To enhance the efficient utilization of gradient information, we propose a novel Adaptive Knowledge Sampling (AKS) strategy with an innovative acquisition function that balances local exploitation and global exploration, effectively extracting gradient points from shape prior knowledge. The acquisition function consists of three parts: (1) The Local Information Filling Criterion (LIFC), designed to mitigate model error; (2) The Global Information Filling Criterion (GIFC), aimed at reducing model uncertainty in data-sparse areas; and (3) The Minimum Distance Constraint (MDC), implemented to prevent excessive clustering of gradient points. To strike a balance between model accuracy and computational cost, an adaptive stopping condition is introduced to regulate the number of gradient points sampled. The impact of hyperparameters in the AKS strategy was analyzed in detail, and the method’s performance was compared to existing approaches using six benchmark functions and a missile aerodynamic prediction case. The results demonstrate that AKS-GEK offers higher prediction accuracy and robustness when integrating small data with shape prior knowledge.

Analysis of Flight Loads during Symmetric Aircraft Maneuvers Based on the Gradient-Enhanced Kriging Model

The analysis of flight loads during symmetric aircraft maneuvers is an essential but computationally intensive task in aircraft design. The significant structural elastic deformation in modern aircraft further complicates this work, adding to the computational demands. Therefore, improving the analysis efficiency of flight loads during maneuvers is crucial for accelerating design interactions and shortening the development cycle. This study explores a method for analyzing flight loads in the time domain during maneuvers of elastic aircraft by introducing a database of high-precision rigid-body aerodynamic loads. Furthermore, it combines the gradient-enhanced Kriging model to efficiently predict elastic flight loads during longitudinal maneuvers. The results indicate that the proposed surrogate-based method has high fitting accuracy with significantly improved computational efficiency, providing a new approach for efficient analysis of flight loads during aircraft maneuvers.

A Story of Three Levels of Sophistication in SCF/KS-DFT Orbital Optimization Procedures.

In this work, three versions of self-consistent field/Kohn-Sham density functional theory (SCF/KS-DFT) orbital optimization are described and benchmarked. The methods are a modified version of the geometry version of the direct inversion in the iterative subspace approach (which we call r-GDIIS), the modified restricted step rational function optimization method (RS-RFO), and the novel subspace gradient-enhanced Kriging method combined with restricted variance optimization (S-GEK/RVO). The modifications introduced are aimed at improving the robustness and computational scaling of the procedures. In particular, the subspace approach in S-GEK/RVO allows the application to SCF/KS-DFT optimization of a machine learning technique that has proven to be successful in geometry optimizations. The performance of the three methods is benchmarked for a large number of small- to medium-sized organic molecules, at equilibrium structures and close to a transition state, and a second set of molecules containing closed- and open-shell transition metals. The results indicate the importance of the resetting technique in boosting the performance of the r-GDIIS procedure. Moreover, it is demonstrated that already at the inception of the subspace version of GEK to optimize SCF wave functions, it displays superior and robust convergence properties as compared to those of the standard state-of-the-art SCF/KS-DFT optimization methods.

Sliced Gradient-Enhanced Kriging for High-Dimensional Function Approximation

Sliced Gradient-Enhanced Kriging for High-Dimensional Function Approximation

Gradient enhanced gaussian process regression for constitutive modelling in finite strain hyperelasticity

This paper introduces a metamodelling technique that leverages gradient-enhanced Gaussian process regression (also known as gradient-enhanced Kriging), effectively emulating the response of diverse hyperelastic strain energy densities. The approach adopted incorporates principal invariants as inputs for the surrogate of the strain energy density. This integration enables the surrogate to inherently enforce fundamental physical constraints, such as material frame indifference and material symmetry, right from the outset. The proposed approach provides accurate interpolation for energy and the first Piola–Kirchhoff stress tensor (e.g. first order derivatives with respect to inputs). The paper presents three notable innovations. Firstly, it introduces the utilisation of Gradient-Enhanced Kriging to approximate a diverse range of phenomenological models, encompassing numerous isotropic hyperelastic strain energies and a transversely isotropic potential. Secondly, this study marks the inaugural application of this technique for approximating the effective response of composite materials. This includes rank-one laminates, for which analytical solutions are feasible. However, it also encompasses more complex composite materials characterised by a Representative Volume Element (RVE) comprising an elastomeric matrix with a centred spherical inclusion. This extension opens the door for future application of this technique to various RVE types, facilitating efficient three-dimensional computational analyses at the macro-scale of such composite materials, significantly reducing computational time compared to FEM2. The third innovation, facilitated by the integration of these surrogate models into a 3D Finite Element computational framework, lies in the assessment of these models scenarios encompassing intricate cases of extreme twisting and more importantly, buckling instabilities in thin-walled structures, thereby highlighting both the practical applicability and robustness of the proposed approach.

A novel sampling method for adaptive gradient-enhanced Kriging

This paper presents a novel infill-sampling strategy for adaptive gradient-enhanced Kriging (AGEK) that delivers superior results on a limited budget. The primary innovation of this method is the adaptive use of gradient information, blurring the line between Kriging and gradient-enhanced Kriging. To construct a flexible AGEK model that automatically determines whether to incorporate gradients, our proposed method unfolds in three stages: (1) primary infill-sampling, (2) secondary infill-sampling, and (3) modeling time stages. In the first stage, the primary infill-sampling technique identifies potential sample point sites. In the second stage, the secondary infill-sampling process decides whether to obtain only the response or both the response and gradient at the selected sample point. During this stage, a newly defined pseudo expected improvement reduction, pseudo integrated uncertainty reduction, and weight functions are incorporated into the secondary infill-sampling criteria. In the third stage, we propose a strategy to manage instances where training time becomes overly demanding. Benchmark test results validate the excellent performance of the proposed method. Finally, in application to an engineering problem, our method outperforms conventional approaches by producing more accurate results within a limited computational budget.

Multiobjective optimization of injection molding parameters based on the GEK-MPDE method

Abstract In plastic injection molding (PIM), the process parameters determine the quality and productivity of molded parts. The traditional injection molding process analysis method mainly relies on production experience. It is lack of advanced and rationality and seriously increases production costs. In this paper, a hybrid multiobjective optimization method is proposed to minimize the warpage, volumetric shrinkage and cycle time. The method integrates orthogonal experimental design, numerical simulation, and the metamodel method with multiobjective optimization. The orthogonal experiment chooses seven parameters as the design variables to generate sampling data and determines key factors that affect product quality by the numerical simulation. A gradient-enhanced Kriging (GEK) surrogate model strategy is introduced to construct the response predictors to calculate objective responses in the global design space. Multipopulation differential evolution (MPDE) is conducted to locate the Pareto-optimal solutions, where the response predictors are taken as the fitness functions. This study shows that the proposed GEK-MPDE method can reduce warpage, volumetric shrinkage and cycle time by 5.7 %, 4.7 %, and 18.1 %, respectively. It helps plastic industry to realize collaborative scheduling of multiple tasks between different production lines by providing a low-cost and effective dynamic control method.

An Efficient Hybrid Multi-Objective Optimization Method Coupling Global Evolutionary and Local Gradient Searches for Solving Aerodynamic Optimization Problems

Aerodynamic shape optimization is frequently complicated and challenging due to the involvement of multiple objectives, large-scale decision variables, and expensive cost function evaluation. This paper presents a bilayer parallel hybrid algorithm framework coupling multi-objective local search and global evolution mechanism to improve the optimization efficiency and convergence accuracy in high-dimensional design space. Specifically, an efficient multi-objective hybrid algorithm (MOHA) and a gradient-based surrogate-assisted multi-objective hybrid algorithm (GS-MOHA) are developed under this framework. In MOHA, a novel multi-objective gradient operator is proposed to accelerate the exploration of the Pareto front, and it introduces new individuals to enhance the diversity of the population. Afterward, MOHA achieves a trade-off between exploitation and exploration by selecting elite individuals in the local search space during the evolutionary process. Furthermore, a surrogate-assisted hybrid algorithm based on the gradient-enhanced Kriging with the partial least squares(GEKPLS) approach is established to improve the engineering applicability of MOHA. The optimization results of benchmark functions demonstrate that MOHA is less constrained by dimensionality and can solve multi-objective optimization problems (MOPs) with up to 1000 decision variables. Compared to existing MOEAs, MOHA demonstrates notable enhancements in optimization efficiency and convergence accuracy, specifically achieving a remarkable 5–10 times increase in efficiency. In addition, the optimization efficiency of GS-MOHA is approximately five times that of MOEA/D-EGO and twice that of K-RVEA in the 30-dimensional test functions. Finally, the multi-objective optimization results of the airfoil shape design validate the effectiveness of the proposed algorithms and their potential for engineering applications.

An efficient gradient-enhanced kriging modeling method assisted by fast kriging for high-dimension problems

PurposeThe modeling cost of the gradient-enhanced kriging (GEK) method is prohibitive for high-dimensional problems. This study aims to develop an efficient modeling strategy for the GEK method.Design/methodology/approachA two-step tuning strategy is proposed for the construction of the GEK model. First, an auxiliary kriging is built efficiently. Then, the hyperparameter of the kriging model is served as a good initial guess to that of the GEK model, and a local optimal search is further used to explore the search space of hyperparameter to guarantee the accuracy of the GEK model. In the construction of the auxiliary kriging, the maximal information coefficient is adopted to estimate the relative magnitude of the hyperparameter, which is used to transform the high-dimension maximum likelihood estimation problem into a one-dimensional optimization. The tuning problem of the auxiliary kriging becomes independent of the dimension. Therefore, the modeling efficiency can be improved significantly.FindingsThe performance of the proposed method is studied with analytic problems ranging from 10D to 50D and an 18D aerodynamic airfoil example. It is further compared with two efficient GEK modeling methods. The empirical experiments show that the proposed model can significantly improve the modeling efficiency without sacrificing accuracy compared with other efficient modeling methods.Originality/valueThis paper developed an efficient modeling strategy for GEK and demonstrated the effectiveness of the proposed method in modeling high-dimension problems.

Gradient-Enhanced Kriging for High-Dimensional Bayesian Optimization with Linear Embedding

This paper explores the application of gradient-enhanced (GE) kriging for Bayesian optimization (BO) problems with a high-dimensional parameter space. We utilize the active subspace method to embed the original parameter space in a low-dimensional subspace. The active subspace is detected by analyzing the spectrum of the empirical second-moment matrix of the gradients of the response function. By mapping the training data onto their respective subspace, the objective function and the constraint functions are efficiently approximated with low-dimensional GE-kriging models. In each cycle of the BO procedure, a new point is found by maximizing the constrained expected improvement function within a low-dimensional polytope, and it is mapped back to the original space for model evaluation. In this way, the computational costs are significantly reduced when compared with standard GE-kriging. We illustrate and assess the proposed approach with numerical experiments ranging from academic benchmark problems to aerodynamic engineering applications. The experimental results show that the proposed method is promising for optimizing high-dimensional expensive objective functions, especially for problems that exhibit a clear low-dimensional active subspace.

MOEA/D with gradient-enhanced kriging for expensive multiobjective optimization

MOEA/D with gradient-enhanced kriging for expensive multiobjective optimization

Adjoint-Based Boundary Condition Sensitivity Analysis

Turbine performance in modern aeroengines is heavily affected by the nonuniform temperature distribution generated in the combustor. The conventional method to study the effect of nonuniform boundary conditions (BCs) is to compare flowfields and performance metrics with the nonuniformity at several typical positions. A much more efficient and intuitive method to analyze the BC sensitivity is proposed in this paper. An inlet guide vane is analyzed as a demonstration. The sensitivity analysis is achieved by pointwise derivatives of objectives with respect to inlet BCs, which are obtained by discrete adjoint method. Furthermore, to study the variation of gradients, a random sampling method that is based on the Gaussian process is proposed to explore the high-dimensional parameter space. When the gradient is invariant within the parameter space, the relationship between objectives and BCs is linear, and the effect of any perturbation of BCs can be directly predicted. In this case, the computational cost of the sensitivity analysis is drastically reduced to one computational fluid dynamics and one adjoint simulation. Interestingly, some gradients are nearly invariant across the parameter space. When the gradient varies, the active subspace method is introduced to reduce the number of dimensions. Then, several models, including the kriging, gradient-enhanced kriging (GEK), and Taylor expansions, are compared to predict objectives. The ordinary kriging model is recommended for its accuracy and stability. The GEK model can also predict the gradient and is a good choice for small BC variations. The method presented here offers a paradigm that extends the gradient-based sensitivity analysis to the global sense, which is suitable for high-dimensional problems such as BCs and spatially varying model parameters.

Sliced Gradient-Enhanced Kriging for High-Dimensional Function Approximation and Aerodynamic Modeling

Sliced Gradient-Enhanced Kriging for High-Dimensional Function Approximation and Aerodynamic Modeling

An optimization method for material sound absorption performance based on surrogate model

In some complex engineering design problems, the use of numerical simulation methods to solve the target value often consumes several hours or even longer, which limits the real-time response to the model. The surrogate model can solve the above-mentioned shortcomings because of its use of statistical ideas to link the design variables with the target value. Kriging model has been widely used in other fields due to its simple algorithm compilation and good stability of calculation results, but there is little research in the field of silencing structure optimization. In order to study the optimization efficiency and optimization effect of the surrogate model in the optimization design of the anechoic structure, combined with the surrogate model and the multi-point plus point criterion, a set of general optimization algorithm framework suitable for the surrogate model and the gradient-enhanced Kriging model (GEK) was developed. Based on this framework, the evolution of the sound absorption coefficient of the anechoic structure under three different working conditions (100–10 000 Hz, 100–1500 Hz, 100–10 000 Hz frequency under static pressure) was compared. The gradient enhancement Kriging model and the gradient optimization algorithm were compared and studied. The results show that under the assumption that the gradient of the objective function and the objective function have the same amount of calculation, the optimization times obtained by the Kriging model with gradient enhancement are better than those obtained by the Kriging model and the gradient optimization algorithm in most cases, and the optimization results of GEK and Kriging models are better than those of gradient optimization.

Reliability index function approximation based on adaptive double-loop Kriging for reliability-based design optimization

Reliability-based design optimization (RBDO) aims at minimizing general cost under the reliability constraints by considering the inherent uncertainties in engineering. In this work, we develop a decoupled RBDO approach named RIFA-ADK which aims at reliability index function (RIF) approximation by adaptive double-loop Kriging. The proposed RIFA-ADK contains three main blocks, namely reliability analysis (Block 1), reliability index function approximation (Block 2) and optimization (Block 3). In RIFA-ADK, RIF is approximated by the outer loop adaptive gradient-enhanced Kriging (GEK) model which takes into account reliability sensitivity in addition to reliability index. The required reliability analysis in GEK is based on the inner loop adaptive Kriging model which focuses on approximating the performance function, and the required reliability sensitivity analysis in GEK is a post-processing of reliability analysis. Then the optimization can be proceeded using the cheap GEK model of RIF. In addition, an adaptive learning strategy which involves two stages of enrichment is also developed to improve the surrogate precision in the region of interest. Finally, four mathematical and practical engineering examples for RBDO are presented to illustrate the accuracy and the efficiency of the proposed RIFA-ADK decoupled approach.

Polynomial-Chaos–Kriging with Gradient Information for Surrogate Modeling in Aerodynamic Design

This paper proposes a variant of gradient-enhanced surrogate model based on polynomial-chaos–kriging (PCK) to assist aerodynamic design exploration. The main aim is to improve the accuracy of kriging when gradient information is available via the adjoint method. In contrast to the standard gradient-enhanced kriging (GEK), the gradient information is used to enrich both the trend function and the stochastic part, using the polynomial-chaos expansion (PCE) as the trend function. The optimal polynomial terms are selected by scanning through various polynomial orders in which the coefficients are calculated by means of the generalized least squares or the least angle regression algorithm. The performance of the gradient-enhanced PCK (GEPCK) was compared with GEK, gradient-enhanced PCE (GEPCE), and their nongradient counterparts on a set of analytical and a suite of aerodynamic problems consisting of aerodynamic design exploration of a blended-wing–body configuration, an inviscid transonic flow over an airfoil, and a low-fidelity aerostructural wing design. The results show notable improvements obtained by GEPCK in terms of performance and robustness compared with other gradient-enhanced surrogate models. Based on these results, it can be concluded that enhancing the trend part with gradient is highly beneficial in improving the accuracy of the kriging surrogate model.

Restricted-Variance Constrained, Reaction Path, and Transition State Molecular Optimizations Using Gradient-Enhanced Kriging

Gaussian process regression has recently been explored as an alternative to standard surrogate models in molecular equilibrium geometry optimization. In particular, the gradient-enhanced Kriging approach in association with internal coordinates, restricted-variance optimization, and an efficient and fast estimate of hyperparameters has demonstrated performance on par or better than standard methods. In this report, we extend the approach to constrained optimizations and transition states and benchmark it for a set of reactions. We compare the performance of the newly developed method with the standard techniques in the location of transition states and in constrained optimizations, both isolated and in the context of reaction path computation. The results show that the method outperforms the current standard in efficiency as well as in robustness.

Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics

Efficient Robust Design Optimization (RDO) strategies coupling a parsimonious uncertainty quantification (UQ) method with a surrogate-based multi-objective genetic algorithm (SMOGA) are investigated for a test problem in computational fluid dynamics (CFD), namely the inverse robust design of an expansion nozzle. The low-order statistics (mean and variance) of the stochastic cost function are computed through either a gradient-enhanced kriging (GEK) surrogate or through the less expensive, lower fidelity, first-order method of moments (MoM). Both the continuous (non-intrusive) and discrete (intrusive) adjoint methods are evaluated for computing the gradients required for GEK and MoM. In all cases, the results are assessed against a reference kriging UQ surrogate not using gradient information. Subsequently, the GEK and MoM UQ solvers are fused together to build a multi-fidelity surrogate with adaptive infill enrichment for the SMOGA optimizer. The resulting hybrid multi-fidelity SMOGA RDO strategy ensures a good tradeoff between cost and accuracy, thus representing an efficient approach for complex RDO problems.

Low-cost performance-driven modelling of compact microwave components with two-layer surrogates and gradient kriging

Utilization of electromagnetic (EM) simulation tools has become indispensable for reliable evaluation of microwave components. As the cost of an individual analysis may already be considerable, the computational overhead associated with EM-driven tasks that require massive simulations (e.g., optimization) may turn prohibitive. One of mitigation methods is the employment of equivalent network models. Yet, they are incapable of accounting for cross-coupling effects that occur in devices of complex geometries. Another option are fast replacement models (surrogates), especially the data-driven ones: readily available, generic and problem independent. Unfortunately, due to the curse of dimensionality, their applicability is limited to low-dimensional parameter spaces and narrow parameter ranges. From the utility perspective, however, the surrogate has to be valid over broad ranges of parameters and operating conditions. The recently reported performance-modeling techniques (especially nested-kriging) allow for rendering such surrogates even for complex devices. Key concept is to carry out the modeling process within a confined domain, being a subset of the parameter space that encompasses the designs of high-quality regarding the performance figures of choice. The goal of this work is to reduce the cost of reference design acquisition, which adds up to the total cost of constructing the surrogate. Toward this end, gradient-enhanced kriging is incorporated into the performance-driven modeling framework. The predictive power of the surrogates rendered using our approach by far exceeds that of the conventional methods and is comparable to the original nested kriging technique while requiring a significantly smaller number of reference designs (thus, the CPU cost). These features are demonstrated using a three-section transformer and a rat-race coupler.

Design of a low noise turbofan duct via an acoustic gradient-enhanced Kriging method

The Kriging models which are frequently used in aerodynamic shape optimization may become computationally inefficient when solving problems with large numbers of design variables. One solution to this problem would be the application of gradient-enhanced Kriging model. A gradient-enhanced Kriging and acoustic adjoint-method approach to duct acoustic problems is developed, aimed to improve the efficiency and accuracy of the existing Kriging approach at acoustic problems with many design parameters. To our knowledge, it is the first application of gradient-enhanced Kriging for duct acoustic problem. It employs a Kriging response surface in the parameter space, augmented with gradients obtained from the acoustic adjoint equations efficiently. The present paper aims at describing the potential of the gradient-enhanced Kriging method for low noise turbofan duct design. Prior to the optimization process, the implementation of the unsteady aeroacoustic adjoint method in shape optimization is validated by comparing the gradient values with that obtained by finite differences. In this work, the ordinary Kriging model and gradient-enhanced Kriging method are applied firstly to a benchmark functions and the results show that the additional gradient information can significantly enhance the accuracy of Kriging model. And then, the original Kriging-based, adjoint-based and the gradient-enhanced Kriging method are all used to model 50 variable duct acoustic problems, respectively. The test results show that this approach whose gradient information is introduced by using acoustic adjoint method developed from multimode LEE, named as acoustic gradient-enhanced Kriging, can significantly enhance the accuracy of Kriging models when the gradient data are available and thus provide an optimized low noise intake while maintaining the aerodynamic performance.