Multi-output regression problems have extensively arisen in modern engineering community. This article investigates the state-of-the-art multi-output Gaussian processes (MOGPs) that can transfer the knowledge across related outputs in order to improve prediction quality. We classify existing MOGPs into two main categories as (1) symmetric MOGPs that improve the predictions for all the outputs, and (2) asymmetric MOGPs, particularly the multi-fidelity MOGPs, that focus on the improvement of high fidelity output via the useful information transferred from related low fidelity outputs. We review existing symmetric/asymmetric MOGPs and analyze their characteristics, e.g., the covariance functions (separable or non-separable), the modeling process (integrated or decomposed), the information transfer (bidirectional or unidirectional), and the hyperparameter inference (joint or separate). Besides, we assess the performance of ten representative MOGPs thoroughly on eight examples in symmetric/asymmetric scenarios by considering, e.g., different training data (heterotopic or isotopic), different training sizes (small, moderate and large), different output correlations (low or high), and different output sizes (up to four outputs). Based on the qualitative and quantitative analysis, we give some recommendations regarding the usage of MOGPs and highlight potential research directions.

Remarks on multi-output Gaussian process regression

https://repository.tudelft.nl/islandora/object/uuid%3A4345ef85-e506-4390-9712-9fbaef011824/datastream/OBJ/download

Adaptive approaches in metamodel-based reliability analysis: A review

Review of finite element model updating methods for structural applications

In computational simulation, a high-fidelity (HF) model is generally more accurate than a low-fidelity (LF) model, while the latter is generally more computationally efficient than the former. To take advantages of both HF and LF models, a multi-fidelity surrogate model based on radial basis function (MFS-RBF) is developed in this paper by combining HF and LF models. To determine the scaling factor between HF and LF models, a correlation matrix is augmented by further integrating LF responses. The scaling factor and relevant basis function weights are then calculated by employing corresponding HF responses. MFS-RBF is compared with Co-Kriging model, multi-fidelity surrogate based on linear regression (LR-MFS) model, CoRBF model, and three single-fidelity surrogates. The impact of key factors, such as the cost ratio of LF to HF models and different combinations of HF and LF samples, is also investigated. The results show that (i) MFS-RBF presents a better accuracy and robustness than the three benchmark MFS models and single-fidelity surrogates in about 90% cases of this paper; (ii) MFS-RBF is less sensitive to the correlation between HF and LF models than the three MFS models; (iii) by fixing the total computational cost, the cost ratio of LF to HF models is suggested to be less than 0.2, and 10–80% of the total cost should be used for LF samples; (iv) the MFS-RBF model is able to save an average of 50 to 70% computational cost if HF and LF models are highly correlated.

A radial basis function-based multi-fidelity surrogate model: exploring correlation between high-fidelity and low-fidelity models

With the wide application of industrial robots in the field of precision machining, reliability analysis of positioning accuracy becomes increasingly important for industrial robots. Since the industrial robot is a complex nonlinear system, the traditional approximate reliability methods often produce unreliable results in analyzing its positioning accuracy. In order to study the positioning accuracy reliability of industrial robot more efficiently and accurately, a radial basis function network is used to construct the mapping relationship between the uncertain parameters and the position coordinates of the end-effector. Combining with the Monte Carlo simulation method, the positioning accuracy reliability is then evaluated. A novel hybrid learning algorithm for training radial basis function network, which integrates the clustering learning algorithm and the orthogonal least squares learning algorithm, is proposed in this article. Examples are presented to illustrate the high proficiency and reliability of the proposed method.

Hybrid Learning Algorithm of Radial Basis Function Networks for Reliability Analysis

A sequential surrogate method for reliability analysis based on radial basis function

By coupling the low-fidelity (LF) model with the high-fidelity (HF) samples, the variable-fidelity model (VFM) offers an efficient way to overcome the expensive computing challenge in multidisciplinary design optimization (MDO). In this paper, a cooperative radial basis function (Co-RBF) method for the VFM is proposed by modifying the basis function of RBF. The RBF method is constructed on the HF samples, while the Co-RBF method incorporates the entire information of the LF model with the HF samples. In Co-RBF, the LF model is regard as a basis function of Co-RBF and the HF samples are utilized to compute the Co-RBF model coefficients. Two numerical functions and three engineering problems are adopted to verify the proposed Co-RBF method. The predictive results of Co-RBF are compared with those of RBF and Co-Kriging, which show that the Co-RBF method improves the efficiency, accuracy and robustness of the existing VFMs.

A cooperative radial basis function method for variable-fidelity surrogate modeling

hp-Adaptive RPD Based Sequential Convex Programming for Reentry Trajectory Optimization

The inverse problem is a kind of engineering problem that estimates the input through the given output. In this paper, the given output described as interval uncertainty parameter is concerned, and the interval is formed by the interval midpoint and the interval radius. The two-step framework that estimates the midpoint and the radius separately is used. A novel nested optimization framework is proposed to estimate the input interval radius with more inputs than outputs. The nested framework has two loops: (i) the inner loop quantifies the lower and upper bounds of the output with given interval radiuses of inputs from the outer loop by two optimizations, and the results will be feedback to the outer loop as constraint values of outer loop; (ii) the outer loop maximizes the input interval radiuses to reduce the cost while meeting the constraints transformed from the given interval. The nested framework induces a high number of forward model computations in the loops and may lead to an unacceptable computational burden for most engineering applications. Therefore, the surrogate model is suggested. The Radial Basis Functions (RBF) surrogate model is used to relieve the computational burden. The effectiveness and the accuracy of the framework are verified through a mathematical example, a cantilever tube example and an airfoil example.

A surrogate model based nested optimization framework for inverse problem considering interval uncertainty

Research on Geometry Modeling Method Based on Three-dimensional CST Parameterization Technology

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