Surrogate Modeling Strategy Research Articles

A microstructure-based graph neural network for accelerating multiscale simulations

Simulating the mechanical response of advanced materials can be done more accurately using concurrent multiscale models than with single-scale simulations. However, the computational costs stand in the way of the practical application of this approach. The costs originate from microscale Finite Element (FE) models that must be solved at every macroscopic integration point. A plethora of surrogate modeling strategies attempt to alleviate this cost by learning to predict macroscopic stresses from macroscopic strains, completely replacing the microscale models. In this work, we introduce an alternative surrogate modeling strategy that allows for keeping the multiscale nature of the problem, allowing it to be used interchangeably with an FE solver for any time step. Our surrogate provides all microscopic quantities, which are then homogenized to obtain macroscopic quantities of interest. We achieve this for an elasto-plastic material by predicting full-field microscopic strains using a graph neural network (GNN) while retaining the microscopic constitutive material model to obtain the stresses. This hybrid data-physics graph-based approach avoids the high dimensionality originating from predicting full-field responses while allowing non-locality to arise. In addition, this approach introduces beneficial inductive bias to the model by encoding microscopic geometrical features. By training the GNN on a variety of meshes, it learns to generalize to unseen meshes, allowing a single model to be used for a range of microstructures. The embedded microscopic constitutive model in the GNN implicitly tracks history-dependent variables and leads to improved accuracy. While the microscopic stresses are fully dependent on the microscopic strains, we found it crucial to include both microscopic strains and stresses in the loss function. We demonstrate for several challenging scenarios that the surrogate can predict complex macroscopic stress–strain paths. As the computation time of our method scales favorably with the number of elements in the microstructure compared to the FE method, our method can significantly accelerate FE2 simulations.

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.

Phase Diagrams of Alloys and Their Hydrides via On-Lattice Graph Neural Networks and Limited Training Data.

Efficient prediction of sampling-intensive thermodynamic properties is needed to evaluate material performance and permit high-throughput materials modeling for a diverse array of technology applications. To alleviate the prohibitive computational expense of high-throughput configurational sampling with density functional theory (DFT), surrogate modeling strategies like cluster expansion are many orders of magnitude more efficient but can be difficult to construct in systems with high compositional complexity. We therefore employ minimal-complexity graph neural network models that accurately predict and can even extrapolate to out-of-train distribution formation energies of DFT-relaxed structures from an ideal (unrelaxed) crystallographic representation. This enables the large-scale sampling necessary for various thermodynamic property predictions that may otherwise be intractable and can be achieved with small training data sets. Two exemplars, optimizing the thermodynamic stability of low-density high-entropy alloys and modulating the plateau pressure of hydrogen in metal alloys, demonstrate the power of this approach, which can be extended to a variety of materials discovery and modeling problems.

ANOVA-GP Modeling for High-Dimensional Bayesian Inverse Problems

Markov chain Monte Carlo (MCMC) stands out as an effective method for tackling Bayesian inverse problems. However, when dealing with computationally expensive forward models and high-dimensional parameter spaces, the challenge of repeated sampling becomes pronounced. A common strategy to address this challenge is to construct an inexpensive surrogate of the forward model, which cuts the computational cost of individual samples. While the Gaussian process (GP) is widely used as a surrogate modeling strategy, its applicability can be limited when dealing with high-dimensional input or output spaces. This paper presents a novel approach that combines the analysis of variance (ANOVA) decomposition method with Gaussian process regression to handle high-dimensional Bayesian inverse problems. Initially, the ANOVA method is employed to reduce the dimension of the parameter space, which decomposes the original high-dimensional problem into several low-dimensional sub-problems. Subsequently, principal component analysis (PCA) is utilized to reduce the dimension of the output space on each sub-problem. Finally, a Gaussian process model with a low-dimensional input and output is constructed for each sub-problem. In addition to this methodology, an adaptive ANOVA-GP-MCMC algorithm is proposed, which further enhances the adaptability and efficiency of the method in the Bayesian inversion setting. The accuracy and computational efficiency of the proposed approach are validated through numerical experiments. This innovative integration of ANOVA and Gaussian processes provides a promising solution to address challenges associated with high-dimensional parameter spaces and computationally expensive forward models in Bayesian inference.

Optimal Tilt-Wing eVTOL Takeoff Trajectory Prediction Using Regression Generative Adversarial Networks

Electric vertical takeoff and landing (eVTOL) aircraft have attracted tremendous attention nowadays due to their flexible maneuverability, precise control, cost efficiency, and low noise. The optimal takeoff trajectory design is a key component of cost-effective and passenger-friendly eVTOL systems. However, conventional design optimization is typically computationally prohibitive due to the adoption of high-fidelity simulation models in an iterative manner. Machine learning (ML) allows rapid decision making; however, new ML surrogate modeling architectures and strategies are still desired to address large-scale problems. Therefore, we showcase a novel regression generative adversarial network (regGAN) surrogate for fast interactive optimal takeoff trajectory predictions of eVTOL aircraft. The regGAN leverages generative adversarial network architectures for regression tasks with a combined loss function of a mean squared error (MSE) loss and an adversarial binary cross-entropy (BC) loss. Moreover, we introduce a surrogate-based inverse mapping concept into eVTOL optimal trajectory designs for the first time. In particular, an inverse-mapping surrogate takes design requirements (including design constraints and flight condition parameters) as input and directly predicts optimal trajectory designs, with no need to run design optimizations once trained. We demonstrated the regGAN on optimal takeoff trajectory designs for the Airbus A3 Vahana. The results revealed that regGAN outperformed reference surrogate strategies, including multi-output Gaussian processes and conditional generative adversarial network surrogates, by matching simulation-based ground truth with 99.6% relative testing accuracy using 1000 training samples. A parametric study showed that a regGAN surrogate with an MSE weight of one and a BC weight of 0.01 consistently achieved over 99.5% accuracy (denoting negligible predictive errors) using 400 training samples, while other regGAN models require at least 800 samples.

Artificial neural network surrogate modeling for uncertainty quantification and structural optimization of reinforced concrete structures

AbstractOptimization approaches are important to design sustainable structures. In structural mechanics, different design objectives can be defined, for example, to minimize the required construction material or to maximize the structural durability. In this paper, the durability of a reinforced concrete (RC) structure is assessed by advanced finite element (FE) models to simulate the cracking behavior and the chloride transport process. The corrosion initiation time is used as durability measure to be maximized within an optimization approach, where the concrete cover is defined as design variable. The variability of structural loads and material parameters and unavoidable construction imprecision leads to a probabilistic reliability and durability assessment, where aleatory as well as epistemic uncertainties are quantified by random variables, intervals and probability‐boxes. The FE simulation models cannot directly be applied to structural analyses and optimizations with polymorphic uncertain parameters and design variables because of the high computational demand of the multi‐loop algorithm (Monte Carlo simulation, interval analysis, global optimization). In this paper, a new surrogate modeling strategy is presented, where artificial neural networks are trained sequentially to speed‐up the coupled mechanical and transport simulation FE models. The new approach is applied to the uncertainty quantification and the structural durability optimization of a RC structure.

SeismicNet: Physics-informed neural networks for seismic wave modeling in semi-infinite domain

Recently, there has been an increasing interest in leveraging physics-informed neural networks (PINNs) for modeling dynamical systems. However, limited studies have been conducted along this horizon on seismic wave modeling tasks. A critical challenge is that these geophysical problems are typically defined in large domains (i.e., semi-infinite), which leads to high computational costs. We present a new PINN model for seismic wave modeling in semi-infinite domain without the need for labeled data. Specifically, the absorbing boundary condition is introduced into the network as a soft regularizer for handling truncated boundaries. To scale up, we consider a sequential training strategy via temporal domain decomposition to improve the scalability of the network and solution accuracy. Moreover, we design a novel surrogate modeling strategy to account for parametric loading, which estimates the wave propagation in semi-infinite domain given the seismic loading at different locations. Various numerical experiments are implemented to evaluate the performance of the proposed PINN model in the context of forward modeling of seismic wave propagation. In particular, we use diverse material distributions to test the versatility of this approach. The results demonstrate excellent solution accuracy under distinctive scenarios.

Design and testing of a decision tree algorithm for early failure detection in steel truss bridges

This paper develops a methodology for damage identification in steel truss bridges that uses vibration-based monitoring data and a model-based decision tree algorithm. The methodology resorts to a calibrated FE model with an optimization-based parameter identification procedure to simulate and analyze all the potential damages that might affect the structure. The effect of environmental conditions on the modal parameters is also accounted for, which is modeled as structure stiffness variations using the Young’s modulus and forecasted using a surrogate modeling strategy. The feasibility of the methodology is demonstrated on a full-scale bridge in Vilagarcía de Arousa, Spain. The underlying hypotheses used in the algorithm implementation were validated, and the error ponderation and selection bound employed to detect and identify damage were optimized. The results show an average success rate of 95.0% and an average false positive rate of 1.0% in identifying damage indicating its robustness to be extrapolated to other case studies.

A multi-fidelity surrogate model for structural health monitoring exploiting model order reduction and artificial neural networks

Stochastic approaches to structural health monitoring (SHM) are often inevitably limited by computational constraints. For instance, for Markov chain Monte Carlo algorithms relying upon computationally expensive finite element models it is almost infeasible to sample the probability distribution of the structural state. To provide instead real-time procedures, this work proposes a non-intrusive surrogate modeling strategy, leveraging model order reduction and artificial neural networks. By relying upon a multi-fidelity (MF) framework, a composition of deep neural networks (DNNs) is devised to map damage and operational parameters onto time-dependent sensor recordings. Such an effective strategy is able to exploit datasets characterized by different fidelity levels without any prior assumption, allowing to blend a small high-fidelity (HF) dataset with a large low-fidelity (LF) dataset, ultimately alleviating the computational burden of supervised training while ensuring the accuracy of the approximated quantities of interest. The resulting surrogate model is made of an LF-DNN, which mimics sensor recordings in the undamaged condition, and of a long short-term memory HF-DNN, which adaptively refines the approximation with the effect of damage. An HF finite element model and an LF reduced order model are adopted offline to generate labeled training data of different fidelity, respectively in the presence or absence of a structural damage. Results relevant to an L-shaped cantilever beam and a portal frame railway bridge prove that the procedure efficiently provides remarkably accurate approximations, outperforming their single-fidelity counterparts. The capability of the MF-DNN to be exploited for SHM purposes is finally shown within an automated Bayesian procedure, aimed at updating the probability distribution of the structural state conditioned on sensor recordings, in the presence of operational variability and measurement noise.

A data-driven statistical-stochastic surrogate modeling strategy for complex nonlinear non-stationary dynamics

We propose a statistical-stochastic surrogate modeling approach to predict the response of the mean and variance statistics under various initial conditions and external forcing perturbations. The proposed modeling framework extends the purely statistical modeling approach that is practically limited to the homogeneous statistical regime for high-dimensional state variables. The new closure system allows one to overcome several practical issues that emerge in the non-homogeneous statistical regimes. First, the proposed ensemble modeling that couples the mean statistics and stochastic fluctuations naturally produces positive-definite covariance matrix estimation, which is a challenging issue that hampers the purely statistical modeling approaches. Second, the proposed closure model, which embeds a non-Markovian neural-network model for the unresolved fluxes such that the variance of the dynamics is consistent, overcomes the inherent instability of the stochastic fluctuation dynamics. Effectively, the proposed framework extends the classical stochastic parametric modeling paradigm for the unresolved dynamics to a semi-parametric parameterization with a residual Long-Short-Term-Memory neural network architecture. Third, based on empirical information metric, we provide an efficient and effective training procedure by fitting a loss function that measures the differences between response statistics. Supporting numerical examples are provided with the Lorenz-96 model, a system of ODEs that admits the characteristic of chaotic dynamics with both homogeneous and inhomogeneous statistical regimes. In the latter case, we will see the effectiveness of the statistical prediction even though the resolved Fourier modes corresponding to the leading mean energy and variance spectra do not coincide.

Reliability-based structural assessment of historical masonry arch bridges: The case study of Cernadela bridge

Nowadays, several historical masonry arch bridges present a deficient state of conservation due to degradation processes induced by natural or human actions. Usually, these constructions have significant economic, cultural, and heritage value. Therefore, they shall be thoroughly assessed to verify their structural integrity and safety condition. For this purpose, reliability-based structural assessments are typically performed. However, the associated outcomes (i.e., reliability index and probability of failure) highly rely on the accuracy of the structural parameters uncertainty quantification. This work presents a study regarding the influence of the scattering of the arches' thickness dimensions in the load-carrying capacity assessment of the Cernadela Bridge, a historical stone bridge located in Galicia, Spain. The study first involved a comprehensive experimental campaign to characterize the outer and inner bridge geometry through geomatic techniques, i.e., terrestrial laser scanning and ground penetrating radar. Subsequently, a limit analysis model was developed, considering only the arches' outer (visible) data. From this initial structural assessment, a reliability index of 2.38 was obtained. The influence of the uncertain structural parameters, both geometric features and material properties, in the collapse load was investigated through a global variance-based sensitivity analysis (i.e., Sobol' indices) complemented by a surrogate modeling strategy based on the Kriging approach. Finally, the measured inner geometry of the arches was introduced in the computational model through Bayesian inference procedures. Thus, two new structural assessments were performed: first, by considering the updated distributions of all arches thicknesses, and second, by considering only the updated distributions of the non-influential ones. Reliability indexes of 1.51 and 2.33 were derived, thus highlighting the importance of a proper uncertainty quantification process and the relevance of the sensitivity analysis outcomes to identify the critical parameters on the bridge mechanical response.

Surrogate Modeling Strategy for Urban Building Energy Simulation in Early-Stage Urban Design: A Case Study of Energy-Efficient Neighborhood Design in Seoul

This study focuses on developing an efficient surrogate modelling strategy for early-stage urban design, aiding in energy-efficient residential neighborhood design in Seoul. The methodology involves devising a design schema with important design parameters, generating 1,000 random designs via Latin Hypercube Sampling, and simulating each design’s energy use considering microclimate and shadow effects. Polynomial regression is used to develop surrogate models based on those energy simulation results, which is further validated with energy use measurements of real neighborhoods in Seoul. The developed surrogate model can provide quick evaluation of energy use with a moderate level of accuracy for early-stage energy efficient neighborhood design.

Dimensionality reduction-based extremum surrogate modeling strategy for transient reliability analysis of complex structures

To improve the transient reliability analysis of complex structures, dimensionality reduction (DR)-based extremum surrogate modeling strategy (DRESMS) is developed by absorbing the strengths of DR strategy and extremum response surface method (ERSM). In the proposed method, the DR is applied to find the embedding mapping the input data to high-dimensional space which has the potential to improve the transient reliability model, and the ERSM is used to build the regression relationship between input parameters and output response to establish a reliable surrogate model based on capturing dynamic extreme moments. The transient reliability analysis of turbine blisk is performed to verify the approximate accuracy and simulation performance of the proposed DRESMS with the assist of the comparison of methods. As seen in this study, (i) the reliability degree of turbine blisk fatigue life is 0.9939 when the allowable value of fatigue life is 1.7738×103 cycles (subject to 3 sigma levels), and (ii) the developed DRESMS holds advantages of high-modeling efficiency and low-fitting error, and (iii) the DRESMS holds excellent simulation efficiency and simulation precision. The efforts of this study provide the promising DRESMS to solve the transient probabilistic analysis of complex structures with multi-dimensional input parameters, which is significant in monitoring and maintaining complex structures and mechanical systems with the consideration of large-scale parameters.

Surrogate-assisted multi-objective particle swarm optimization for the operation of CO2 capture using VPSA

Multi-objective evolutionary algorithms (MOEAs) have received increasing attention over the past few decades. However, when applying MOEAs to solve computationally expensive real-world applications, they are often criticized due to the large number of function evaluations required. To this end, we proposed a method, called MOPSONN-EGO, for dealing with expensive multi-objective problems. MOPSONN-EGO uses Kriging models to approximate the objective function of expensive problems. The recently proposed MOPSONN algorithm is used to search the landscape of the Kriging models. The cheap-to-evaluate expected improvement matrix is adopted to select infill samples to update the Kriging models. Empirical results on several benchmark problems show the competitive performance of MOPSONN-EGO. Moreover, MOPSONN-EGO is applied to solve the multi-objective optimization problem of the Vacuum pressure swing adsorption (VPSA) for carbon dioxide capture. The simulation results are validated against the expensive model using normalized root mean square error (NRMSE) and compared to the One-shot surrogate modeling strategy. The results reveal the significant efficiency of the proposed algorithm in approximating the real Pareto front of VPSA using only a limited number of detailed model evaluations. Moreover, the obtained best values for each objective with their corresponding decision variables values are reported.

Optimization of thin electric propeller using physics-based surrogate model with space mapping

The propulsion system greatly effects the performance of small electric powered Unmanned Aerial Vehicles (UAV). The propeller is one of the key elements of an electric propulsion system. The present research aims to optimize the propeller shape by utilizing the physics-based surrogate modeling strategy along with a space mapping method. A quasi-global construction is made that perceptively links a low-fidelity (coarse) model and a high-fidelity (fine) model, thus avoiding the direct optimization of the fine model. The space mapping utilizes a parameter extraction process that assist the surrogate model to locally realign itself with the fine model. For coarse model, an open source propeller analyses and design program (QPROP) is used. This program uses an extension of the classical blade element formulation. The fine model is based on the computational fluid dynamics analysis utilizing the finite-volume discretization of the Reynolds-averaged Navier–Stokes equations. The computed results have been verified by comparing them with the available propeller performance data. After a successful validation, geometric parameters for the propeller are varied to study their effect on performance and carry out parametric sensitivity analysis. The selected design variables control the section airfoil shape and the propeller geometry. The optimized design has been achieved with few iterations of the fine model reducing the overall computational cost of optimization process. The results indicate that the space mapping surrogate modeling is an effective strategy for propeller shape optimization.

Diffusion maps-aided Neural Networks for the solution of parametrized PDEs

This work introduces a surrogate modeling strategy, based on diffusion maps manifold learning and artificial neural networks. On this basis, a numerical procedure is developed for cost-efficient predictions of a complex system’s response modeled by parametrized partial differential equations. The idea is to utilize a collection of solution snapshots, obtained by solving the partial differential equation for a small number of parameter values, in order to establish an efficient yet accurate mapping from the problem’s parametric space to its solution space. In common practice, solving the partial differential equations in the framework of the finite element method leads to high-dimensional data sets, which are a major challenge for machine learning algorithms to handle (curse of dimensionality). To overcome this problem, the proposed method exploits the dimensionality reduction properties of the diffusion maps algorithm in order to obtain a meaningful low-dimensional representation of the solution data set. With this approach, a reduced set of ‘hyperparameters’ is obtained, namely, the diffusion map coordinates, that characterize the high-dimensional solution vectors. Using this reduced representation, a feed-forward neural network is efficiently trained that maps the problem’s parameter values to their images in the low-dimensional diffusion maps space. This approach offers two advantages. The obvious one is that it reduces the computational resources and CPU time needed to train the neural network, which can be prohibitive for high dimensional problems. The second advantage is that training the neural network on the diffusion map coordinates, essentially translates to using the diffusion maps distance in the MSE loss function of the network. Compared to the Euclidean distance in the ambient space, the diffusion distance gives a better approximation of the distance between two points on the solution manifold, which leads to a more accurate network. Lastly, a mapping is developed based on the Laplacian pyramids scheme in order to convert points from the diffusion maps space back to the solution space. The composition of the neural network with the Laplacian pyramid scheme, substitutes the process of solving the partial differential equation under consideration, and is capable of providing the additional system solutions at a very low cost and high accuracy.

Compressed Machine Learning Models for the Uncertainty Quantification of Power Distribution Networks

Today’s spread of power distribution networks, with the installation of a significant number of renewable generators that depend on environmental conditions and on users’ consumption profiles, requires sophisticated models for monitoring the power flow, regulating the electricity market, and assessing the reliability of power grids. Such models cannot avoid taking into account the variability that is inherent to the electrical system and users’ behavior. In this paper, we present a solution for the generation of a compressed surrogate model of the electrical state of a realistic power network that is subject to a large number (on the order of a few hundreds) of uncertain parameters representing the power injected by distributed renewable sources or absorbed by users with different consumption profiles. Specifically, principal component analysis is combined with two state-of-the-art surrogate modeling strategies for uncertainty quantification, namely, the least-squares support vector machine, which is a nonparametric regression belonging to the class of machine learning methods, and the widely adopted polynomial chaos expansion. Such methods allow providing compact and efficient surrogate models capable of predicting the statistical behavior of all nodal voltages within the network as functions of its stochastic parameters. The IEEE 8500-node test feeder benchmark with 450 and 900 uncertain parameters is considered as a validation example in this study. The feasibility and strength of the proposed method are verified through a systematic assessment of its performance in terms of accuracy, efficiency, and convergence, based on reference simulations obtained via classical Monte Carlo analysis.

Moving extremum surrogate modeling strategy for dynamic reliability estimation of turbine blisk with multi-physics fields

To improve the dynamic reliability analysis of complex structures like turbine blisk, moving extremum surrogate modeling strategy (MESMS) is proposed in respect of multi-physics coupling with various dynamics/uncertainties. In this strategy, extremum thought is adopted to handle the dynamic process of input parameters and output response, and the importance sampling (IS) method is utilized to extract efficient samples and improve the efficiency of dynamic reliability estimation, and moving least square (MLS) method is used to select good samples from training samples with local compact support region to establish a precise surrogate model. The dynamic reliability analysis of turbine blisk radial deformation with fluid-thermo-structural interaction is performed to validate the developed MESMS in approximate precision and simulation performance by comparing to other methods. As shown in this study, (i) the reliability degree of turbine blisk is 0.9975 when the allowable value of radial deformation is 2.6856 × 10−3 m subject to 3 sigma levels; (ii) the proposed MESMS processes high modeling accuracy and efficiency due to small error; (iii) the MESMS holds high simulation performance in efficiency and accuracy owing to outstanding computational consumption and simulation. These results demonstrate that the MESMS is effective and applicable in structural reliability estimation regarding dynamics and uncertainties. The efforts of this paper provide a useful insight for performing the reliability-based design optimization of complex structures besides turbine blisk.

A novel surrogate modeling strategy of the mechanical properties of 3D braided composites

In this paper, a surrogate-based modeling methodology is developed and presented to predict the elastic properties of three dimensional (3D) four-directional braided composites. Using this approach, the prediction process becomes feasible with only a limited number of training points. The surrogate models constructed using Finite Element (FE) method and Diffuse Approximation, reduce the computational time and cost for preparing experimental samples. In the FE model, multiscale method is applied to couple the computations of elastic properties at microscale and mesoscale. Subsequently, a parametric study is performed to analyze the effects of the three design parameters on the elastic properties. Satisfactory results are obtained via the surrogate-based modeling predictions, which are compared with the experimental measurements. Moreover, the predictions obtained from surrogate models concur well with the FE predictions. This study orients a new direction for predicting the mechanical properties based on surrogate models which can effectively reduce the sample preparation cost and computational efforts.

ANOVA Gaussian process modeling for high-dimensional stochastic computational models

In this paper we present a novel analysis of variance Gaussian process (ANOVA-GP) emulator for models governed by partial differential equations (PDEs) with high-dimensional random inputs. The Gaussian process (GP) is a widely used surrogate modeling strategy, but it can become invalid when the inputs are high-dimensional. In this new ANOVA-GP strategy, high-dimensional inputs are decomposed into unions of local low-dimensional inputs, and principal component analysis (PCA) is applied to provide dimension reduction for each ANOVA term. We then systematically build local GP models for PCA coefficients based on ANOVA decomposition to provide an emulator for the overall high-dimensional problem. We present a general mathematical framework of ANOVA-GP, validate its accuracy and demonstrate its efficiency with numerical experiments.