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Abstract 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. 

Abstract In this work, we develop a method namedTwinningfor partitioning a dataset into statistically similar twin sets.Twinningis based onSPlit, a recently proposed model‐independent method for optimally splitting a dataset into training and testing sets.Twinningis orders of magnitude faster than theSPlitalgorithm, which makes it applicable to Big Data problems such as data compression.Twinningcan also be used for generating multiple splits of a given dataset to aid divide‐and‐conquer procedures andk‐fold cross validation. 

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 solvent-free post-treatment process known as vapor phase infiltration (VPI) is used to engineer the organic solvent reverse osmosis (OSRO) performance of polymer of intrinsic microporosity 1 (PIM-1) membranes via infiltration of trimethylaluminum (TMA) metal-organic vapor. The infiltration of inorganic aluminum constituents hybridizes the pure polymer PIM-1 into an organic-inorganic material (AlOxHy/PIM-1) with enhanced chemical stability. A homogenous distribution of inorganic loading in PIM-1 is achieved due to the reaction-limited infiltration mechanism, and the OSRO performance is enhanced as a result. OSRO separations of ethanol/isooctane mixtures using these membranes are shown to be capable of breaking the azeotropic composition with a separation factor for ethanol over isooctane greater than 5 and an ethanol permeance of 0.1 Lm–2h–1bar–1. Thus, these organic-inorganic hybrid membranes created via VPI show promise as an alternative method for separating azeotropic liquid mixtures. 

Vapor phase infiltration (VPI) is a post-polymerization modification technique that infuses inorganics into polymers to create organic–inorganic hybrid materials with new properties. Much is yet to be understood about the chemical kinetics underlying the VPI process. The aim of this study is to create a greater understanding of the process kinetics that govern the infiltration of trimethyl aluminum (TMA) and TiCl 4 into PMMA to form inorganic-PMMA hybrid materials. To gain insight, this paper initially examines the predicted results for the spatiotemporal concentrations of inorganics computed from a recently posited reaction–diffusion model for VPI. This model provides insight on how the Damköhler number (reaction versus diffusion rates) and non-Fickian diffusional processes (hindering) that result from the material transforming from a polymer to a hybrid can affect the evolution of inorganic concentration depth profiles with time. Subsequently, experimental XPS depth profiles are collected for TMA and TiCl 4 infiltrated PMMA films at 90 °C and 135 °C. The functional behavior of these depth profiles at varying infiltration times are qualitatively compared to various computed predictions and conclusions are drawn about the mechanisms of each of these processes. TMA infiltration into PMMA appears to transition from a diffusion-limited process at low temperatures (90 °C) to a reaction-limited process at high temperatures (135 °C) for the film thicknesses investigated here (200 nm). While TMA appears to fully infiltrate these 200 nm PMMA films within a few hours, TiCl 4 infiltration into PMMA is considerably slower, with full saturation not occurring even after 2 days of precursor exposure. Infiltration at 90 °C is so slow that no clear conclusions about mechanism can be drawn; however, at 135 °C, the TiCl 4 infiltration into PMMA is clearly a reaction-limited process, with TiCl 4 permeating the entire thickness (at low concentrations) within only a few minutes, but inorganic loading continuously increasing in a uniform manner over a course of 2 days. Near-surface deviations from the uniform-loading expected for a reaction-limited process also suggest that diffusional hindering is high for TiCl 4 infiltration into PMMA. These results demonstrate a new, ex situ analysis approach for investigating the rate-limiting process mechanisms for vapor phase infiltration. 

In this work, the vapor-phase infiltration (VPI) of polyethylene terephthalate (PET) fabrics with trimethylaluminum (TMA) and coreaction with water vapor is explored as a function of limiting TMA reagent conditions versus excess TMA reagent conditions at two infiltration temperatures. TMA is found to sorb rapidly into PET fibers, with a significant pressure drop occurring within seconds of TMA exposure. When large quantities of polymer are placed within the chamber, minimal residual precursor remains at the end of the pressure drop. This rapid and complete sorption facilitates the control of inorganic loading by purposely delivering a limited quantity of the TMA reagent. The inorganic loading for this system scales linearly with a Precursor:C=O molar ratio of up to 0.35 at 140 °C and 0.5 at 80 °C. After this point, inorganic loading is constant irrespective of the amount of additional TMA reagent supplied. The SEM analysis of pyrolyzed hybrids indicates that this is likely due to the formation of an impermeable layer to subsequent infiltration as the core of the fibers remains uninfiltrated. The Precursor:C=O molar ratio in the subsaturation regime is found to tune the hybrid fabric morphology and material properties such as the optical properties of the fabric. Overall, this work demonstrates how a reagent-limited processing route can control the inorganic loading in VPI synthesized hybrid materials in a simpler manner than trying to control kinetics-driven methods. 

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 ).