有限差分与人工神经网络方法求解偏微分方程解对比及定性分析

Comparison and Qualitative Analysis of Solutions of Partial Differential Equations by Finite Difference Method and Artificial Neural Network Method

摘要: 本文对人工神经网络方法(ANN)求解偏微分方程进行了研究，选择特定偏微分方程将求解结果与经典有限差分方法(FDM)进行了比对，最后得出在样本与训练次数选择恰当时，ANN方法可以得出相对较精确的数值解，但需要选取恰当权重与激活函数，并对现有激活函数提出了修正。具体通过Lagaris发表文献以复算计结果并进行了分析与探讨，更详细地揭示计算求解注意事项，最后结论得出权重训练次数后每次迭代梯度愈小与解析解越接近，而激活函数改变对计算结果影响较小。此外对双曲二维Burgers方程进行了稳态求解在100次训练过程后，但对非稳态及耗散损失还需后续进一步探讨研究。

Abstract: In this paper, the artificial neural network (ANN) method for solving partial differential equations is studied. The specific equation is selected and the solution accuracy is compared with the classical finite element FDM method. Finally, when the sample and training times are selected properly, the ANN method can get relatively high numerical solution, but it needs to select the appropriate weight and activation function, and modify the existing activation function. The specific implementation is to analyze and discuss the numerical solution of partial differential equation by using ANN through the early lagaris published literature recalculation calculation, and reveal more detailed calculation and solution precautions. Finally, it is concluded that the smaller the iteration gradient after weight training times is, the closer it is to the finite element result, and the change of activation function has little influence on the calculation result. In addition, the two-dimensional hyperbolic Burgers equation is solved in steady state, but the unsteady state and dissipation need further study.