局部自适应物理信息神经网络在求解Burgers-Fisher方程中的应用  

Application of locally adaptive physics-informed neural networks in solving the Burgers-Fisher equation

作　　者：孙宇 朱海龙[1] SUN Yu;ZHU Hailong(School of Statistics and Applied Mathematics,Anhui University of Finance and Economics,Bengbu 233000,China)

机构地区：[1]安徽财经大学统计与应用数学学院,安徽蚌埠233000

出　　处：《山东理工大学学报(自然科学版)》2025年第4期47-53,共7页Journal of Shandong University of Technology:Natural Science Edition

基　　金：安徽省科研编制计划重点项目(2024AH050013)。

摘　　要：针对物理信息神经网络(PINNs)在训练效率和求解精度方面的局限性,提出了一种改进的PINNs模型。该模型通过在每个神经元中引入可调节参数,使神经网络能更灵活地逼近非线性不连续函数,将神经网络的求解误差降低了2~3个数量级,同时将PINNs的求解误差降低了约1个数量级,提升了求解Burgers-Fisher方程的精度;探讨了不同缩放因子对模型性能的影响,当选择适当的缩放因子时,局部自适应激活函数不仅能够提高PINNs模型的收敛速度,还能够在保持计算效率的同时,获得更高精度的数值解。Aiming at the limitations of physics-informed neural networks(PINNs)in terms of training efficiency and solving accuracy,an improved PINNs model is proposed.This model introduces adjustable parameters into each neuron which allows the neural network to flexibly approximate nonlinear discontinuous functions.Consequently,the solution error of the neural network is reduced by 2~3 orders of magnitude,and the PINNs′solution error is reduced by approximately one order of magnitude,significantly improving the accuracy in solving the Burgers-Fisher equation.This study also explores the effects of different scaling factors on model performance.When appropriate scaling factors are selected,the local adaptive activation functions not only improve the convergence rate of PINNs models,but also obtain higher precision numerical solutions while maintaining computational efficiency.

关 键 词：BURGERS-FISHER方程 物理信息神经网络 自适应激活函数 偏微分方程求解 

分 类 号：TP183[自动化与计算机技术—控制理论与控制工程]