基于深度学习的分数阶Nernst-Plank方程求解

Solution of fractional Nernst-Plank equation based on deep learning

采用分数阶物理信息网络(fPINN)求解时间分数阶Nernst-Plank方程,并对其解决时间分数阶N-P的正问题与反问题的准确性和有效性进行说明.在这基础上,分析离散化时间分数阶算子所导致的离散误差、采样误差、神经网络优化误差对最终求解的影响.同时,分析离散误差与取样误差的关系,并发现当固定离散误差后存在最好的训练点集大小使得求解误差最低.最后,展示神经网络求解反问题的准确性与效率.

This paper used fractional physics-informed neural networks(fPINN)to solve the time-fractional equation,and demonstrate its accuracy and effectiveness in solving the forward and inverse problems of time-fractional N-P.Moreover,the paper explain result by analyzing the three sources of numerical errors due to discretization,sampling and optimization.The paper also analyze relative between the discretization and sampling error.The paper find that there exists the best training point set size to minimize the solution error with fixed discretization error.Finally,the paper demonstrate the effectiveness of NN in solving inverse problems.