具有非线性边界Laplace方程的PINN求解方法  

PINN Method for Laplace Equation with Nonlinear Boundary Conditions

作　　者：闫郁郁 苏进[1] YAN Yuyu;SU Jin(School of Science,Xi’an Polytechnic University,Xi’an 710048,China)

机构地区：[1]西安工程大学理学院,西安710048

出　　处：《河南科学》2025年第3期313-320,共8页Henan Science

基　　金：国家自然科学基金项目(11601411)。

摘　　要：具有非线性边界条件的Laplace方程在电化学腐蚀领域中有着重要应用,传统PINN方法在求解过程中常面临预测结果不准确、精度较低等困难。针对该问题提出了一种边界多区域分段线性训练的PINN方法,该方法将非线性边界条件分段线性近似为Robin边界条件,并嵌入PINN中进行求解。数值实验结果表明,与传统PINN方法直接嵌入非线性边界求解相对比,该方法能够显著提高预测精度、减小误差。此外,进一步研究了算法中动态调整学习率、边界区间分段数量等因素对预测结果的影响。该方法为电化学腐蚀领域中非线性边界条件的Laplace方程提供了新的求解思路,具有较大的发展潜力和应用前景。The Laplace equation with nonlinear boundary conditions has important applications in the field of electrochemical corrosion.The traditional PINN method often faces the difficulties of inaccurate prediction results and low accuracy in the solution process.Aiming at this problem,a PINN method with piecewise linear training of multi-region boundary is proposed.The nonlinear boundary conditions are piecewise linearly approximated to Robin boundary conditions and embedded in PINN.Numerical experimental results show that compared with the traditional PINN method directly embedded in the nonlinear boundary solution,this method can significantly improve the prediction accuracy and reduce the error.The influence of dynamic adjustment learning rate and the number of boundary segments on the prediction results is further studied,which provides a new solution for the Laplace equation of nonlinear boundary conditions in the field of electrochemical corrosion,and has great application potential and practical significance.

关 键 词：LAPLACE方程 非线性边界 分段线性近似 PINN 

分 类 号：O242[理学—计算数学]