Deep learning has achieved wide success in solving Partial Differential Equations (PDEs), with particular strength in handling high dimensional problems and problems with irregular geometries. In this work, we report a residual-informed neural network (RINN) for PDEs. Compared to Physics-informed neural networks, RINN avoids computation of high order derivatives of the network, thus can significantly accelerate the training process. Meanwhile, we propose a non-uniform random walk to generate adaptive samples (Nurvas) for solving PDEs with low-regularity solutions. In Nurvas, the adaptive samples are obtained without additional computational cost and without an explicit representation of the desired probability density function.

胡丹, 上海交通大学数学科学学院教授, 博士生导师, 2017年教育部青年长江学者。主要从事血管与血流、生命科学中的稀有事件等问题的建模、模拟和分析和人工智能基础理论研究。主持/完成了国家重大研究计划重点支持项目、面上项目、上海市科技创新行动计划项目等。代表性工作发表于Phys. Rev. Lett., Nature Commun.和PLoS Biol.等顶级杂志, 其中关于血管适应性生长方面的工作被Nature选为年度工作亮点。