Gradient-enhanced Kriging (GE-Kriging) is a well-established surrogate modelling technique for approximating expensive computational models. However, it tends to get impractical for high-dimensional problems due to the large inherent correlation matrix and the associated high-dimensional hyper-parameter tuning problem. To address these issues, we propose a new method in this paper, called sliced GE-Kriging (SGE-Kriging) for reducing both the size of the correlation matrix and the number of hyper-parameters. Firstly, we perform a derivative-based global sensitivity analysis to detect the relative importance of each input variable with respect to model response. Then, we propose to split the training sample set into multiple slices, and invoke Bayes' theorem to approximate the full likelihood function via a sliced likelihood function, in which multiple small correlation matrices are utilized to describe the correlation of the sample set. Additionally, we replace the original high-dimensional hyper-parameter tuning problem with a low-dimensional counterpart by learning the relationship between the hyper-parameters and the global sensitivity indices. Finally, we validate SGE-Kriging by means of numerical experiments with several benchmarks problems. The results show that the SGE-Kriging model features an accuracy and robustness that is comparable to the standard one but comes at much less training costs. The benefits are most evident in high-dimensional problems.

This paper presents a surrogate modelling technique based on domain partitioning for Bayesian parameter inference of highly nonlinear engineering models. In order to alleviate the computational burden typically involved in Bayesian inference applications, a multielement Polynomial Chaos Expansion based Kriging metamodel is proposed. The developed surrogate model combines in a piecewise function an array of local Polynomial Chaos based Kriging metamodels constructed on a finite set of non-overlapping subdomains of the stochastic input space. Therewith, the presence of non-smoothness in the response of the forward model (e.g.~ nonlinearities and sparseness) can be reproduced by the proposed metamodel with minimum computational costs owing to its local adaptation capabilities. The model parameter inference is conducted through a Markov chain Monte Carlo approach comprising adaptive exploration and delayed rejection. The efficiency and accuracy of the proposed approach are validated through two case studies, including an analytical benchmark and a numerical case study. The latter relates the partial differential equation governing the hydrogen diffusion phenomenon of metallic materials in Thermal Desorption Spectroscopy tests.

Multi-fidelity Kriging model is a promising technique in surrogate-based design as it can balance the model accuracy and cost of sample preparation by fusing low- and high-fidelity data. However, the cost for building a multi-fidelity Kriging model increases significantly with the increase of the problem dimension. To attack this issue, an efficient Hierarchical Kriging modeling method is proposed. In building the low-fidelity model, the maximal information coefficient is utilized to calculate the relative value of the hyperparameter. With this, the maximum likelihood estimation problem for determining the hyperparameters is transformed as a one-dimension optimization problem, which can be solved in an efficient manner and thus improve the modeling efficiency significantly. A local search is involved further to exploit the search space of hyperparameters to improve the model accuracy. The high-fidelity model is built in a similar manner with the hyperparameter of the low-fidelity model served as the relative value of the hyperparameter for high-fidelity model. The performance of the proposed method is compared with the conventional tuning strategy, by testing them over ten analytic problems and an engineering problem of modeling the isentropic efficiency of a compressor rotor. The empirical results demonstrate that the modeling time of the proposed method is reduced significantly without sacrificing the model accuracy. For the modeling of the isentropic efficiency of the compressor rotor, the cost saving associated with the proposed method is about 90% compared with the conventional strategy. Meanwhile, the proposed method achieves higher accuracy.

计算机实验的主要目标是通过分散的评估重建计算机代码给出的功能。当输入维度相对较高时，传统的各向同性高斯过程模型受到维数的诅咒。具有加性相关功能的高斯过程模型可扩展到维度，但是它们仅适用于加性功能，因此更具限制性。在这项工作中，我们考虑了一种投影追求模型，其中非参数部分是由加性高斯过程回归驱动的。我们选择高于原始输入尺寸的添加函数的维度，并将此策略称为“维度扩展”。我们表明尺寸扩展可以帮助近似更复杂的功能。根据最大似然估计，提出了一种用于模型训练的梯度下降算法。仿真研究表明，所提出的方法的表现优于传统的高斯工艺模型。补充材料可在线获得。

我们开发了一个计算程序，以估计具有附加噪声的半摩托车高斯过程回归模型的协方差超参数。也就是说，提出的方法可用于有效估计相关误差的方差，以及基于最大化边际似然函数的噪声方差。我们的方法涉及适当地降低超参数空间的维度，以简化单变量的根发现问题的估计过程。此外，我们得出了边际似然函数及其衍生物的边界和渐近线，这对于缩小高参数搜索的初始范围很有用。使用数值示例，我们证明了与传统参数优化相比，提出方法的计算优势和鲁棒性。

Partial differential equations (PDEs) are widely used for description of physical and engineering phenomena. Some key parameters involved in PDEs, which represents certain physical properties with important scientific interpretations, are difficult or even impossible to be measured directly. Estimation of these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations of numerical solution of PDE through algorithms such as finite element method, which can be time-consuming especially for nonlinear PDEs. In this paper, we propose a novel method for estimating unknown parameters in PDEs, called PDE-Informed Gaussian Process Inference (PIGPI). Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transfers the nonlinear PDE into an equivalent PDE system linear in all derivatives that our PIGPI can handle. PIGPI can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. The method completely bypasses the numerical solver for PDE, thus achieving drastic savings in computation time, especially for nonlinear PDEs. Moreover, the PIGPI method can give the uncertainty quantification for both the unknown parameters and the PDE solution. The proposed method is demonstrated by several application examples from different areas.

In many engineering optimization problems, the number of function evaluations is severely limited by time or cost. These problems pose a special challenge to the field of global optimization, since existing methods often require more function evaluations than can be comfortably afforded. One way to address this challenge is to fit response surfaces to data collected by evaluating the objective and constraint functions at a few points. These surfaces can then be used for visualization, tradeoff analysis, and optimization. In this paper, we introduce the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering. We then show how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule. The key to using response surfaces for global optimization lies in balancing the need to exploit the approximating surface (by sampling where it is minimized) with the need to improve the approximation (by sampling where prediction error may be high). Striking this balance requires solving certain auxiliary problems which have previously been considered intractable, but we show how these computational obstacles can be overcome.

替代模型用于减轻工程任务中的计算负担，这些计算负担需要重复评估计算要求的物理系统模型，例如不确定性的有效传播。对于显示出非常非线性依赖其输入参数的模型，标准的替代技术（例如多项式混沌膨胀）不足以获得原始模型响应的准确表示。通过应用有理近似，对于通过有理函数准确描述的模型可以有效地降低近似误差。具体而言，我们的目标是近似复杂值模型。获得替代系数的一种常见方法是最小化模型和替代物之间的基于样本的误差，从最小二乘意义上讲。为了获得原始模型的准确表示并避免过度拟合，样品集的量是扩展中多项式项数的两到三倍。对于需要高多项式程度或在其输入参数方面具有高维度的模型，该数字通常超过负担得起的计算成本。为了克服这个问题，我们将稀疏的贝叶斯学习方法应用于理性近似。通过特定的先前分布结构，在替代模型的系数中诱导稀疏性。分母的多项式系数以及问题的超参数是通过类型-II-Maximim-Maximim类似方法来确定的。我们应用了准牛顿梯度散发算法，以找到最佳的分母系数，并通过应用$ \ mathbb {cr} $ -Colculus来得出所需的梯度。

The saddle point (SP) calculation is a grand challenge for computationally intensive energy function in computational chemistry area, where the saddle point may represent the transition state (TS). The traditional methods need to evaluate the gradients of the energy function at a very large number of locations. To reduce the number of expensive computations of the true gradients, we propose an active learning framework consisting of a statistical surrogate model, Gaussian process regression (GPR) for the energy function, and a single-walker dynamics method, gentle accent dynamics (GAD), for the saddle-type transition states. SP is detected by the GAD applied to the GPR surrogate for the gradient vector and the Hessian matrix. Our key ingredient for efficiency improvements is an active learning method which sequentially designs the most informative locations and takes evaluations of the original model at these locations to train GPR. We formulate this active learning task as the optimal experimental design problem and propose a very efficient sample-based sub-optimal criterion to construct the optimal locations. We show that the new method significantly decreases the required number of energy or force evaluations of the original model.

多保真建模和校准是在工程设计中普遍出现的数据融合任务。在本文中，我们介绍了一种基于潜在地图高斯过程（LMGPS）的新方法，可实现高效准确的数据融合。在我们的方法中，我们将数据融合转换为潜在的空间学习问题，其中自动学习不同数据源之间的关系。这种转换赋予我们的方法具有有吸引力的优点，例如提高准确性，降低成本，灵活性，共同熔断任何数量的数据源，以及可视化数据源之间的相关性。该可视化允许用户通过拟合LMGP仅拟合到具有良好相关的数据源的子集的子集来检测模型形式误差或确定用于高保真仿真的最佳策略。我们还开发了一种新的内核功能，使LMGPS能够不仅构建概率的多保真代理，而且还具有高精度和一致性的估计参数。与现有技术相比，我们的方法的实施和使用易于更简单，更不容易出现数值问题。我们通过在广泛的示例中比较其对竞争方法的性能来证明基于LMGP的数据融合的好处。

Estimating the probability of failure for complex real-world systems using high-fidelity computational models is often prohibitively expensive, especially when the probability is small. Exploiting low-fidelity models can make this process more feasible, but merging information from multiple low-fidelity and high-fidelity models poses several challenges. This paper presents a robust multi-fidelity surrogate modeling strategy in which the multi-fidelity surrogate is assembled using an active learning strategy using an on-the-fly model adequacy assessment set within a subset simulation framework for efficient reliability analysis. The multi-fidelity surrogate is assembled by first applying a Gaussian process correction to each low-fidelity model and assigning a model probability based on the model's local predictive accuracy and cost. Three strategies are proposed to fuse these individual surrogates into an overall surrogate model based on model averaging and deterministic/stochastic model selection. The strategies also dictate which model evaluations are necessary. No assumptions are made about the relationships between low-fidelity models, while the high-fidelity model is assumed to be the most accurate and most computationally expensive model. Through two analytical and two numerical case studies, including a case study evaluating the failure probability of Tristructural isotropic-coated (TRISO) nuclear fuels, the algorithm is shown to be highly accurate while drastically reducing the number of high-fidelity model calls (and hence computational cost).

本文介绍了在高斯过程回归/克里格替代建模技术中选择/设计内核的算法。我们在临时功能空间中采用内核方法解决方案的设置，即繁殖内核希尔伯特空间（RKHS），以解决在观察到它的观察值的情况下近似定期目标函数的问题，即监督学习。第一类算法是内核流，该算法是在机器学习中的分类中引入的。它可以看作是一个交叉验证过程，因此选择了“最佳”内核，从而最小化了通过删除数据集的某些部分（通常为一半）而产生的准确性损失。第二类算法称为光谱内核脊回归，旨在选择“最佳”核，以便在相关的RKHS中，要近似的函数的范围很小。在Mercer定理框架内，我们就目标函数的主要特征来获得该“最佳”内核的明确结构。从数据中学习内核的两种方法均通过有关合成测试功能的数值示例，以及在湍流建模验证二维机翼的湍流模型验证中的经典测试用例。

本文为工程产品的计算模型或仅返回分类信息的过程提供了一种新的高效和健壮方法，用于罕见事件概率估计，例如成功或失败。对于此类模型，大多数用于估计故障概率的方法，这些方法使用结果的数值来计算梯度或估计与故障表面的接近度。即使性能函数不仅提供了二进制输出，系统的状态也可能是连续输入变量域中定义的不平滑函数，甚至是不连续的函数。在这些情况下，基于经典的梯度方法通常会失败。我们提出了一种简单而有效的算法，该算法可以从随机变量的输入域进行顺序自适应选择点，以扩展和完善简单的基于距离的替代模型。可以在连续采样的任何阶段完成两个不同的任务：（i）估计失败概率，以及（ii）如果需要进一步改进，则选择最佳的候选者进行后续模型评估。选择用于模型评估的下一个点的建议标准最大化了使用候选者分类的预期概率。因此，全球探索与本地剥削之间的完美平衡是自动维持的。该方法可以估计多种故障类型的概率。此外，当可以使用模型评估的数值来构建平滑的替代物时，该算法可以容纳此信息以提高估计概率的准确性。最后，我们定义了一种新的简单但一般的几何测量，这些测量是对稀有事实概率对单个变量的全局敏感性的定义，该度量是作为所提出算法的副产品获得的。

在不断努力提高产品质量和降低运营成本中，越来越多地部署计算建模以确定产品设计或配置的可行性。通过本地模型代理这些计算机实验的建模，仅考虑短程交互，诱导稀疏性，可以解决复杂输入输出关系的巨大分析。然而，缩小到地方规模的重点意味着必须一遍又一遍地重新学习全球趋势。在本文中，我们提出了一种框架，用于将来自全局敏感性分析的信息纳入代理模型作为输入旋转和重新扫描预处理步骤。我们讨论了基于内核回归的几个敏感性分析方法的关系在描述它们如何产生输入变量的转换之前。具体而言，我们执行输入扭曲，使得“翘曲模拟器”对所有输入方向同样敏感，释放本地模型以专注于本地动态。观测数据和基准测试功能的数值实验，包括来自汽车行业的高维计算机模拟器，提供了实证验证。

我们考虑了使用显微镜或X射线散射技术产生的图像数据自组装的模型的贝叶斯校准。为了说明BCP平衡结构中的随机远程疾病，我们引入了辅助变量以表示这种不确定性。然而，这些变量导致了高维图像数据的综合可能性，通常可以评估。我们使用基于测量运输的可能性方法以及图像数据的摘要统计数据来解决这一具有挑战性的贝叶斯推理问题。我们还表明，可以计算出有关模型参数的数据中的预期信息收益（EIG），而无需额外的成本。最后，我们介绍了基于二嵌段共聚物薄膜自组装和自上而下显微镜表征的ohta-kawasaki模型的数值案例研究。为了进行校准，我们介绍了一些基于域的能量和傅立叶的摘要统计数据，并使用EIG量化了它们的信息性。我们证明了拟议方法研究数据损坏和实验设计对校准结果的影响的力量。

We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finite dimensional Gaussian process (GP) regression approach under no-arbitrage constraints based on prices, and a neural net (NN) approach with penalization of arbitrages based on implied volatilities. We demonstrate the performance of these approaches relative to the SSVI industry standard. The GP approach is proven arbitrage-free, whereas arbitrages are only penalized under the SSVI and NN approaches. The GP approach obtains the best out-of-sample calibration error and provides uncertainty quantification.The NN approach yields a smoother local volatility and a better backtesting performance, as its training criterion incorporates a local volatility regularization term.

We present the GPry algorithm for fast Bayesian inference of general (non-Gaussian) posteriors with a moderate number of parameters. GPry does not need any pre-training, special hardware such as GPUs, and is intended as a drop-in replacement for traditional Monte Carlo methods for Bayesian inference. Our algorithm is based on generating a Gaussian Process surrogate model of the log-posterior, aided by a Support Vector Machine classifier that excludes extreme or non-finite values. An active learning scheme allows us to reduce the number of required posterior evaluations by two orders of magnitude compared to traditional Monte Carlo inference. Our algorithm allows for parallel evaluations of the posterior at optimal locations, further reducing wall-clock times. We significantly improve performance using properties of the posterior in our active learning scheme and for the definition of the GP prior. In particular we account for the expected dynamical range of the posterior in different dimensionalities. We test our model against a number of synthetic and cosmological examples. GPry outperforms traditional Monte Carlo methods when the evaluation time of the likelihood (or the calculation of theoretical observables) is of the order of seconds; for evaluation times of over a minute it can perform inference in days that would take months using traditional methods. GPry is distributed as an open source Python package (pip install gpry) and can also be found at https://github.com/jonaselgammal/GPry.

我们考虑使用昂贵的功能评估（也称为实验）的黑匣子多目标优化（MOO）的问题，其中目标是通过最小化实验的总资源成本来近似真正的帕累托解决方案。例如，在硬件设计优化中，我们需要使用昂贵的计算模拟找到权衡性能，能量和面积开销的设计。关键挑战是选择使用最小资源揭示高质量解决方案的实验顺序。在本文中，我们提出了一种基于输出空间熵（OSE）搜索原理来解决MOO问题的一般框架：选择最大化每单位资源成本的信息的实验，这是真正的帕累托前线所获得的信息。我们适当地实例化了OSE搜索的原理，以导出以下四个Moo问题设置的高效算法：1）最基本的EM单一保真设置，实验昂贵且准确; 2）处理EM黑匣子约束}在不执行实验的情况下无法进行评估; 3）离散的多保真设置，实验可以在消耗的资源量和评估准确度时变化; 4）EM连续保真设置，其中连续函数近似导致巨大的实验空间。不同综合和现实世界基准测试的实验表明，基于OSE搜索的算法在既有计算效率和MOO解决方案的准确性方面改进了最先进的方法。

In a fissile material, the inherent multiplicity of neutrons born through induced fissions leads to correlations in their detection statistics. The correlations between neutrons can be used to trace back some characteristics of the fissile material. This technique known as neutron noise analysis has applications in nuclear safeguards or waste identification. It provides a non-destructive examination method for an unknown fissile material. This is an example of an inverse problem where the cause is inferred from observations of the consequences. However, neutron correlation measurements are often noisy because of the stochastic nature of the underlying processes. This makes the resolution of the inverse problem more complex since the measurements are strongly dependent on the material characteristics. A minor change in the material properties can lead to very different outputs. Such an inverse problem is said to be ill-posed. For an ill-posed inverse problem the inverse uncertainty quantification is crucial. Indeed, seemingly low noise in the data can lead to strong uncertainties in the estimation of the material properties. Moreover, the analytical framework commonly used to describe neutron correlations relies on strong physical assumptions and is thus inherently biased. This paper addresses dual goals. Firstly, surrogate models are used to improve neutron correlations predictions and quantify the errors on those predictions. Then, the inverse uncertainty quantification is performed to include the impact of measurement error alongside the residual model bias.

Surrogate models have shown to be an extremely efficient aid in solving engineering problems that require repeated evaluations of an expensive computational model. They are built by sparsely evaluating the costly original model and have provided a way to solve otherwise intractable problems. A crucial aspect in surrogate modelling is the assumption of smoothness and regularity of the model to approximate. This assumption is however not always met in reality. For instance in civil or mechanical engineering, some models may present discontinuities or non-smoothness, e.g., in case of instability patterns such as buckling or snap-through. Building a single surrogate model capable of accounting for these fundamentally different behaviors or discontinuities is not an easy task. In this paper, we propose a three-stage approach for the approximation of non-smooth functions which combines clustering, classification and regression. The idea is to split the space following the localized behaviors or regimes of the system and build local surrogates that are eventually assembled. A sequence of well-known machine learning techniques are used: Dirichlet process mixtures models (DPMM), support vector machines and Gaussian process modelling. The approach is tested and validated on two analytical functions and a finite element model of a tensile membrane structure.