Yunyang Xiong

Hongwei Lin, Yunyang Xiong, Xiao Wang et al.

In this paper, we present the isogeometric least-squares collocation (IGA-L) method, which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense. The number of collocation points employed in IGA-L can be larger than that of the unknowns. Theoretical analysis and numerical examples presented in this paper show the superiority of IGA-L over state-of-the-art collocation methods. First, a small increase in the number of collocation points in IGA-L leads to a large improvement in the accuracy of its numerical solution. Second, IGA-L method is more flexible and more stable, because the number of collocation points in IGA-L is variable. Third, IGA-L is convergent in some cases of singular parameterization. Moreover, the consistency and convergence analysis are also developed in this paper.

Hao Henry Zhou, Yunyang Xiong, Vikas Singh

We provide simple schemes to build Bayesian Neural Networks (BNNs), block by block, inspired by a recent idea of computation skeletons. We show how by adjusting the types of blocks that are used within the computation skeleton, we can identify interesting relationships with Deep Gaussian Processes (DGPs), deep kernel learning (DKL), random features type approximation and other topics. We give strategies to approximate the posterior via doubly stochastic variational inference for such models which yield uncertainty estimates. We give a detailed theoretical analysis and point out extensions that may be of independent interest. As a special case, we instantiate our procedure to define a Bayesian {\em additive} Neural network -- a promising strategy to identify statistical interactions and has direct benefits for obtaining interpretable models.