报告题目：Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

报告人：郭玲，上海师范大学

报告摘要：Physics-informed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. This framework is effective but is lacking uncertainty quantification of the solution due to the inherent randomness in the data or due to the approximation limitations of the DNN architecture. Here, we propose a new method with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty. We first account for the parametric uncertainty when the parameter in the differential equation is represented as a stochastic process. Multiple DNNs are designed to learn the modal functions of the arbitrary polynomial chaos (aPC) expansion of its solution by using stochastic data from sparse sensors. We can then make predictions from new sensor measurements very efficiently with the trained DNNs. Moreover, we employ dropout to correct the overfitting and also to quantify the uncertainty of DNNs in approximating the modal functions. We then design an active learning strategy based on the dropout uncertainty to place new sensors in the domain in order to improve the predictions of DNNs. Several numerical tests are conducted for both the forward and the inverse problems to quantify the effectiveness of PINNs combined with uncertainty quantification.

报告人简介：郭玲，上海师范大学数学系教授。 主要研究领域为不确定性量化、随机偏微分方程数值计算和机器学习。 先后主持国家自然科学基金等多项课题，在《SIAM. J. Sci. Comp.》等高水平杂志发表论文多篇。曾获第三届上海高校青年教师教学竞赛自然科学基础组一等奖，荣获上海市育才奖，上海市教育系统三八红旗手称号和上海市教学能手称号。

报告邀请人：周兆杰

报告时间：2019年11月9日(周六) 09:00

报告地点：长清湖校区文渊楼A区231

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