Inverse distance weighting (IDW) is a simple method for multivariate interpolation but has poor prediction accuracy. In this article we show that the prediction accuracy of IDW can be substantially...

In many areas of science and engineering, discovering the governing differential equations from the noisy experimental data is an essential challenge. It is also a critical step in understanding the physical phenomena and prediction of the future behaviors of the systems. However, in many cases, it is expensive or time-consuming to collect experimental data. This article provides an active learning approach to estimate the unknown differential equations accurately with reduced experimental data size. We propose an adaptive design criterion combining the D-optimality and the maximin space-filling criterion. In contrast to active learning for other regression models, the D-optimality here requires the unknown solution of the differential equations and derivatives of the solution. We estimate the Gaussian process (GP) regression models from the available experimental data and use them as the surrogates of these unknown solution functions. The derivatives of the estimated GP models are derived and used to substitute the derivatives of the solution. Variable selection-based regression methods are used to learn the differential equations from the experimental data. Through multiple case studies, we demonstrate the proposed approach outperforms the D-optimality and the maximin space-filling design alone in terms of model accuracy and data economy. The Supplementary Materials contain all the Matlab codes and data and are available online.

In many science and engineering systems both quantitative and qualitative output observations are collected. If modeled separately the important relationship between the two types of responses is ignored. In this article, we propose a Bayesian hierarchical modeling framework to jointly model a continuous and a binary response. Compared with the existing methods, the Bayesian method overcomes two restrictions. First, it solves the problem in which the model size (specifically, the number of parameters to be estimated) exceeds the number of observations for the continuous response. We use one example to show how such a problem can easily occur if the design of the experiment is not proper; all the frequentist approaches would fail in this case. Second, the Bayesian model can provide statistical inference on the estimated parameters and predictions, whereas it is not clear how to obtain inference using the latest method proposed by Deng and Jin ( 2015 Deng, X. , and R. Jin. 2015. QQ models: Joint modeling for quantitative and qualitative quality responses in manufacturing systems. Technometrics 57 (3):320–331. [Taylor & Francis Online] [Google Scholar] ), which jointly models the two responses via constrained likelihood. We also develop a Gibbs sampling scheme to generate accurate estimation and prediction for the Bayesian hierarchical model. Both the simulation and the real case study are shown to illustrate the proposed method.