有限差分法和PINN法求解微分方程的探讨

Discussion on Solving Differential Equations by Finite Difference Method and PINN Method

摘要: 在工程实际中的许多问题最终都可以转化为微分方程。由于一些微分方程复杂性，这些方程求解通常具有一定的难度。随着计算机的迅速发展，使得这些方程可以数值求解。如何设计高效的微分方程数值解法尤其重要。微分方程数值解法通常包括有限差分、有限元、有限体积等。近年来基于深度学习的微分方程求解方法十分火热。本文对内嵌物理信息神经网络(PINN)方法进行探讨。我们用传统的有限差分法和PINN法对常微分两点边值问题和偏微分方程中的一类热传导方程进行数值求解，对比分析两种数值解法的优缺点。从数值实验结果中可以看出用PINN相对于传统有限差分法求解微分方程具有更好的精度和效率。

Abstract: Many problems in engineering practice can ultimately be transformed into differential equations. Due to the complexity of some differential equations, solving these equations usually has some dif-ficulty. With the rapid development of computers, these equations can be numerically solved. How to design efficient numerical solutions for differential equations is particularly important. The nu-merical solution of differential equations usually includes finite difference, finite element, finite volume, etc. In recent years, differential equation solving methods based on deep learning have be-come very popular. This article explores the embedded Physics-Informed Neural Networks (PINN) method. We use the traditional finite difference method and the PINN method to numerically solve the ordinary differential two-point boundary value problem and a kind of heat conduction equation in the partial differential equation, and compare the advantages and disadvantages of the two nu-merical methods. From the numerical experiment results, it can be seen that PINN has better accu-racy and efficiency than the traditional finite difference method in solving differential equations.