In this work we develop a neural network for the numerical simulation of time-dependent linear transport equations with diffusive scaling and uncertainties. The goal of the network is to resolve the computational challenges of curse-of-dimensionality and multiple scales of the problem. We first show that a standard Physics-Informed Neural Network (PINN) fails to capture the multiscale nature of the problem, hence justifies the need to use Asymptotic-Preserving Neural Networks (APNNs). We show that not all classical AP formulations are fit for the neural network approach. We construct a micro-macro decomposition based neural network, and also build in a mass conservation mechanism into the loss function, in order to capture the dynamic and multiscale nature of the solutions. Numerical examples are used to demonstrate the effectiveness of this APNNs. 

马征博士，上海交通大学数学科学学院副教授，2012年与2017年分别本科、博士毕业于上海交通大学；2017-2020年美国普渡大学数学系Golomb访问助理教授，2020年9月入职上海交通大学数学科学学院。主要研究方向：机器学习在科学计算中的应用，动理学方程的快速数值算法，机器学习的数学理论。目前发表与预发表学术论文十余篇（PNAS, JCP, RIMS等)。入选RIMS五年来最佳论文、上海交通优秀博士毕业生等。