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Green’s function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green’s function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain independent Green’s function, referred to as BIN-G. We evaluate the Green’s function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green’s function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green’s function for PDEs with variable coefficients. The learned Green’s function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients. 

In machine learning, we often face the situation where the event we are interested in has very few data points buried in a massive amount of data. This is typical in network monitoring, where data are streamed from sensing or measuring units continuously but most data are not for events. With imbalanced datasets, the classifiers tend to be biased in favor of the main class. Rare event detection has received much attention in machine learning, and yet it is still a challenging problem. In this paper, we propose a remedy for the standing problem. Weighting and sampling are two fundamental approaches to address the problem. We focus on the weighting method in this paper. We first propose a boosting-style algorithm to compute class weights, which is proved to have excellent theoretical property. Then we propose an adaptive algorithm, which is suitable for real-time applications. The adaptive nature of the two algorithms allows a controlled tradeoff between true positive rate and false positive rate and avoids excessive weight on the rare class, which leads to poor performance on the main class. Experiments on power grid data and some public datasets show that the proposed algorithms outperform the existing weighting and boosting methods, and that their superiority is more noticeable with noisy data.