走向现代数学学术报告 - 张永超讲师（No. 834)

报告题目：Error analysis and numerical algorithm for PDE approximation with hidden-layer concatenated PINNs

报告时间：2025年10月31日 14:00

腾讯会议ID：987961017

报告人：张永超 讲师（西北大学）

邀请人：侯江勇 副教授

报告摘要：In this talk, we present the hidden-layer concatenated physics informed neural network (HLConcPINN) method, a modified block time marching strategy, and a physics informed approach for approximating partial differential equations (PDEs). We analyze the convergence properties and establish the error bounds of this method for two types of PDEs. We show that its approximation error of the solution can be effectively controlled by the training loss for dynamic simulations with long time horizons. The HLConcPINN method in principle allows an arbitrary number of hidden layers not smaller than two and any of the commonly-used smooth activation functions for the hidden layers beyond the first two, with theoretical guarantees. This generalizes several recent neural network techniques, which have theoretical guarantees but are confined to two hidden layers in the network architecture and the tanh activation function. Our theoretical analyses subsequently inform the formulation of appropriate training loss functions for these PDEs, leading to physics informed neural network (PINN) type computational algorithms that differ from the standard PINN formulation.

报告人简介：张永超, 西北大学数学学院讲师. 主要研究方向为偏微分方程数值解. 目前的主要工作包括自由流和多孔介质流等问题的间断 Galerkin 和杂交高阶间断方法的数值求解, 后验误差估计和相应的自适应算法设计, 以及神经网络求解偏微分方程. 部分相关的工作发表在 JCP, JSC 等杂志上. 获得国家自然科学青年基金、博士后基金等。