Multi-fidelity Gaussian process regression for prediction of random fields

We propose a new multi-fidelity Gaussian process regression (GPR) approach for prediction of random fields based on observations of surrogate models or hierarchies of surrogate models. Our method builds upon recent work on recursive Bayesian techniques, in particular recursive co-kriging, and extends it to vector-valued fields and various types of covariances, including separable and non-separable ones. The framework we propose is general and can be used to perform uncertainty propagation and quantification in model-based simulations, multi-fidelity data fusion, and surrogate-based optimization. We demonstrate the effectiveness of the proposed recursive GPR techniques through various examples. Specifically, we study the stochastic Burgers equation and the stochastic Oberbeck\u2013Boussinesq equations describing natural convection within a square enclosure. In both cases we find that the standard deviation of the Gaussian predictors as well as the absolute errors relative to benchmark stochastic solutions are very small, suggesting that the proposed multi-fidelity GPR approaches can yield highly accurate results

Gradient enhanced multi-fidelity regression with neural networks: application to turbulent flow reconstruction

A multi-fidelity regression model is proposed for combining multiple datasets with different fidelities, particularly abundant low-fidelity data and scarce high-fidelity observations. The model builds upon recent multi-fidelity frameworks based on neural networks, which employ two distinct networks for learning low- and high-fidelity data, and extends them by feeding the gradients information of low-fidelity data into the second network, while the gradients are computed using automatic differentiation with minimal computational overhead. The accuracy of the proposed framework is demonstrated through a variety of benchmark examples, and it is shown that the proposed model performs better than conventional multi-fidelity neural network models that do not use gradient information. Additionally, the proposed model is applied to the challenging case of turbulent flow reconstruction. In particular, we study the effectiveness of the model in reconstructing the instantaneous velocity field of the decaying of homogeneous isotropic turbulence given low-resolution/low-fidelity data as well as small amount of high-resolution/high-fidelity data. The results indicate that the proposed model is able to reconstruct turbulent field and capture small scale structures with good accuracy, making it suitable for more practical applications

Bayesian Quadrature for Multiple Related Integrals

Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to incomplete/finite information about the continuous mathematical problem being approximated. In this paper, we demonstrate that this paradigm can provide additional advantages, such as the possibility of transferring information between several numerical methods. This allows users to represent uncertainty in a more faithful manner and, as a by-product, provide increased numerical efficiency. We propose the first such numerical method by extending the well-known Bayesian quadrature algorithm to the case where we are interested in computing the integral of several related functions. We then prove convergence rates for the method in the well-specified and misspecified cases, and demonstrate its efficiency in the context of multi-fidelity models for complex engineering systems and a problem of global illumination in computer graphics.Comment: Proceedings of the 35th International Conference on Machine Learning (ICML), PMLR 80:5369-5378, 201

Multi-objective Bayesian Optimization of Super hydrophobic Coatings on Asphalt Concrete Surfaces

Conventional snow removal strategies add direct and indirect expenses to the economy through profit lost due to passenger delays costs, pavement durability issues, contaminating the water runoff, and so on. The use of superhydrophobic (super-water-repellent) coating methods is an alternative to conventional snow and ice removal practices for alleviating snow removal operations issues. As an integrated experimental and analytical study, this work focused on optimizing superhydrophobicity and skid resistance of hydrophobic coatings on asphalt concrete surfaces. A layer-by-layer (LBL) method was utilized for spray depositing polytetrafluoroethylene (PTFE) on an asphalt concrete at different spray times and variable dosages of PTFE. Water contact angle and coefficient of friction at the microtexture level were measured to evaluate superhydrophobicity and skid resistance of the coated asphalt concrete. The optimum dosage and spray time that maximized hydrophobicity and skid resistance of flexible pavement while minimizing cost were estimated using a multi-objective Bayesian optimization (BO) method that replaced the more costly experimental procedure of pavement testing with a cheap-to-evaluate surrogate model constructed based on kriging. In this method, the surrogate model is iteratively updated with new experimental data measured at proper input settings. The result of proposed optimization method showed that the super water repellency and coefficient of friction were not uniformly increased for all the specimens by increasing spray time and dosage. In addition, use of the proposed multi-objective BO method resulted in hydrophobicity and skid resistance being maximally augmented by approximately 23% PTFE dosage at a spray time of 5.5 s

Surrogate modelling and uncertainty quantification based on multi-fidelity deep neural network

To reduce training costs, several Deep neural networks (DNNs) that can learn from a small set of HF data and a sufficient number of low-fidelity (LF) data have been proposed. In these established neural networks, a parallel structure is commonly proposed to separately approximate the non-linear and linear correlation between the HF- and LF data. In this paper, a new architecture of multi-fidelity deep neural network (MF-DNN) was proposed where one subnetwork was built to approximate both the non-linear and linear correlation simultaneously. Rather than manually allocating the output weights for the paralleled linear and nonlinear correction networks, the proposed MF-DNN can autonomously learn arbitrary correlation. The prediction accuracy of the proposed MF-DNN was firstly demonstrated by approximating the 1-, 32- and 100-dimensional benchmark functions with either the linear or non-linear correlation. The surrogating modelling results revealed that MF-DNN exhibited excellent approximation capabilities for the test functions. Subsequently, the MF DNN was deployed to simulate the 1-, 32- and 100-dimensional aleatory uncertainty propagation progress with the influence of either the uniform or Gaussian distributions of input uncertainties. The uncertainty quantification (UQ) results validated that the MF-DNN efficiently predicted the probability density distributions of quantities of interest (QoI) as well as the statistical moments without significant compromise of accuracy. MF-DNN was also deployed to model the physical flow of turbine vane LS89. The distributions of isentropic Mach number were well-predicted by MF-DNN based on the 2D Euler flow field and few experimental measurement data points. The proposed MF-DNN should be promising in solving UQ and robust optimization problems in practical engineering applications with multi-fidelity data sources