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In this work, we propose a novel framework for large-scale Gaussian process (GP) modeling. Contrary to the global, and local approximations proposed in the literature to address the computational bottleneck with exact GP modeling, we employ a combined global-local approach in building the approximation. Our framework uses a subset-of-data approach where the subset is a union of a set of global points designed to capture the global trend in the data, and a set of local points specific to a given testing location to capture the local trend around the testing location. The correlation function is also modeled as a combination of a global, and a local kernel. The predictive performance of our framework, which we refer to as TwinGP, is comparable to the state-of-the-art GP modeling methods, but at a fraction of their computational cost. 

A central problem of materials science is to determine whether a hypothetical material is stable without being synthesized, which is mathematically equivalent to a global optimization problem on a highly nonlinear and multimodal potential energy surface (PES). This optimization problem poses multiple outstanding challenges, including the exceedingly high dimensionality of the PES, and that PES must be constructed from a reliable, sophisticated, parameters-free, and thus very expensive computational method, for which density functional theory (DFT) is an example. DFT is a quantum mechanics-based method that can predict, among other things, the total potential energy of a given configuration of atoms. DFT, although accurate, is computationally expensive. In this work, we propose a novel expansion-exploration-exploitation framework to find the global minimum of the PES. Starting from a few atomic configurations, this “known” space is expanded to construct a big candidate set. The expansion begins in a nonadaptive manner, where new configurations are added without their potential energy being considered. A novel feature of this step is that it tends to generate a space-filling design without the knowledge of the boundaries of the domain space. If needed, the nonadaptive expansion of the space of configurations is followed by adaptive expansion, where “promising regions” of the domain space (those with low-energy configurations) are further expanded. Once a candidate set of configurations is obtained, it is simultaneously explored and exploited using Bayesian optimization to find the global minimum. The methodology is demonstrated using a problem of finding the most stable crystal structure of aluminum. History: Kwok Tsui served as the senior editor for this article. Funding: The authors acknowledge a U.S. National Science Foundation Grant DMREF-1921873 and XSEDE through Grant DMR170031. Data Ethics & Reproducibility Note: The code capsule is available on Code Ocean at https://codeocean.com/capsule/3366149/tree and in the e-Companion to this article (available at https://doi.org/10.1287/ijds.2023.0028 ). 

Acoustic metasurfaces are two-dimensional materials that impart non-trivial amplitude and phase shifts on incident acoustic waves at a predetermined frequency. While acoustic metasurfaces enable extraordinary wavefront engineering capabilities, they are not developed well enough to independently control the amplitude and phase of reflected and transmitted acoustic waves simultaneously, which are governed by their geometry. We aim to solve the inverse design problem of finding a geometry to achieve a specified set of acoustic properties. The geometry is modeled by discretizing the continuous space into a finite number of elements, where each element can either be filled with air or solid material. Full wave simulations are performed to obtain the acoustic properties for a given geometry. It is computationally infeasible to simulate all geometries. To address this challenge, we develop an experimental design-based algorithm to efficiently perform the simulations. The algorithm starts with a few geometries and adaptively adds geometries to the set, such that they fill the entire space of the desired acoustic properties using a small fraction of the possible geometries. We find that the geometry needs to have at least 7 × 7 elements to obtain any given acoustic property with a tolerance of 5.4% of its maximum range. This is achieved by simulating 24 000 geometries using the proposed algorithm, which is only [Formula: see text] of the 563 × 10 12 possible geometries. The method provides a general solution to the inverse design problem that can be extended to control more acoustic properties. 

It is common to split a dataset into training and testing sets before fitting a statistical or machine learning model. However, there is no clear guidance on how much data should be used for training and testing. In this article, we show that the optimal training/testing splitting ratio is , where is the number of parameters in a linear regression model that explains the data well.