Tao Xiong

Tao Xiong, Jingmei Qiu

A class of high order asymptotic preserving (AP) schemes has been developed for the BGK equation in Xiong et. al. (2015) [37], which is based on the micro-macro formulation of the equation. The nodal discontinuous Galerkin (NDG) method with Lagrangian basis functions for spatial discretization and globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta (RK) scheme as time discretization are introduced with asymptotic preserving properties. However, it is only necessary to solve the kinetic equation when the hydrodynamic description breaks down. Motivated by the recent work in Filbet and Rey (2015) [23], it is more naturally to construct a hierarchy scheme under the NDG-IMEX framework without hybridization, as the formal analysis in [37] shows that when $ε$ is small, the NDG-IMEX scheme becomes a local discontinuous Galerkin (LDG) scheme for the compressible Navier-Stokes equations, and when $ε=0$ it is a discontinuous Galerkin (DG) scheme for the compressible Euler equations. Moveover, we propose to combine the kinetic regime with the hydrodynamic regime including both the compressible Euler and Navier-Stokes equations. Numerical experiments demonstrate very decent performance of the new approach. In our numerics, all three regimes are clearly divided, leading to great savings in terms of the computational cost.

Hui Li, Xing Fu, Ruofan Wu et al.

Deep learning provides a promising way to extract effective representations from raw data in an end-to-end fashion and has proven its effectiveness in various domains such as computer vision, natural language processing, etc. However, in domains such as content/product recommendation and risk management, where sequence of event data is the most used raw data form and experts derived features are more commonly used, deep learning models struggle to dominate the game. In this paper, we propose a symbolic testing framework that helps to answer the question of what kinds of expert-derived features could be learned by a neural network. Inspired by this testing framework, we introduce an efficient architecture named SHORING, which contains two components: \textit{event network} and \textit{sequence network}. The \textit{event} network learns arbitrarily yet efficiently high-order \textit{event-level} embeddings via a provable reparameterization trick, the \textit{sequence} network aggregates from sequence of \textit{event-level} embeddings. We argue that SHORING is capable of learning certain standard symbolic expressions which the standard multi-head self-attention network fails to learn, and conduct comprehensive experiments and ablation studies on four synthetic datasets and three real-world datasets. The results show that SHORING empirically outperforms the state-of-the-art methods.

Tao Xiong, Giovanni Russo, Jing-Mei Qiu

In this paper, we propose a mass conservative semi-Lagrangian finite difference scheme for multi-dimensional problems without dimensional splitting. The semi-Lagrangian scheme, based on tracing characteristics backward in time from grid points, does not necessarily conserve the total mass. To ensure mass conservation, we propose a conservative correction procedure based on a flux difference form. Such procedure guarantees local mass conservation, while introducing time step constraints for stability. We theoretically investigate such stability constraints from an ODE point of view by assuming exact evaluation of spatial differential operators and from the Fourier analysis for linear PDEs. The scheme is tested by classical two dimensional linear passive-transport problems, such as linear advection, rotation and swirling deformation. The scheme is applied to solve the nonlinear Vlasov-Poisson system using a a high order tracing mechanism proposed in [Qiu and Russo, 2016]. Such high order characteristics tracing scheme is generalized to the nonlinear guiding center Vlasov model and incompressible Euler system. The effectiveness of the proposed conservative semi-Lagrangian scheme is demonstrated numerically by our extensive numerical tests.