This study is concerned with multivariate approximation by non-polynomial functions with internal shape parameters. The main topics of this presentation are two folds. First, interpolation by radial basis function (RBF) is considered. We especially discuss the convergence behavior of the RBF interpolants when the basis function is scaled to be increasingly flat. Moreover, we investigate the advantages of interpolation methods based on exponential polynomials. The second topic of this presentation is the approximation method based on sparse grids in $[0,1]^d \subset \RR^d$. The goal of sparse grid methods is to approximate high dimensional functions with good accuracy using as few grid points as possible. In this study, we present a new class of quasi-interpolation schemes for the approximation of multivariate functions on sparse grids. Each scheme in this class is based on shifts of kernels constructed from one-dimensional RBFs such as multiquadrics. The kernels are modified near the boundaries to prevent deterioration of the fidelity of the approximation. We show that our methods provide significantly better rates of approximation, compared to another quasi-interpolation scheme in the literature based on the Gaussian kernel using the multilevel technique. Some numerical results are presented to demonstrate the performance of the proposed schemes.

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Online: https://kaist.zoom.us/j/81807153144

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, however, are very complex and sophisticated, commonly requiring many trial-and-error and tricks. These result in a lack of robustness and interpretability, which are critical factors for scientific applications. This talk centers around mathematical approaches for SciML, promoting trustworthiness. The first part is about how to embed physics into neural networks (NNs). I will present a general framework for designing NNs that obey the first and second laws of thermodynamics. The framework not only provides flexible ways of leveraging available physics information but also results in expressive NN architectures. The second part is about the training of NNs, one of the biggest challenges in ML. I will present an efficient training method for NNs - Active Neuron Least Squares (ANLS). ANLS is developed from the insight gained from the analysis of gradient descent training.

In recent years, community detection has been an active research area in various fields including machine learning and statistics. While a plethora of works has been published over the past few years, most of the existing methods depend on a predetermined number of communities. Given the situation, determining the proper number of communities is directly related to the performance of these methods. Currently, there does not exist a golden rule for choosing the ideal number, and people usually rely on their background knowledge of the domain to make their choices. To address this issue, we propose a community detection method that is equipped with data-adaptive methods of finding the number of the underlying communities. Central to our method is fused l-1 penalty applied on an induced graph from the given data. The proposed method shows promising results.