基于物理信息神经网络的多重积分求解方法

Solution of multiple integrals based on physical information neural networks

为探索多重积分的求解方法,提出了基于物理信息神经网络(physical information neural networks,PINN)的多重积分方程求解方法。首先,将多重积分方程转化为微分方程和边界条件;然后,设计神经网络结构、确定训练集、构造损失函数,利用PINN来逼近微分方程的解析解,并根据多重定积分将积分上下限代入解析解,即可求解出多重定积分方程;最后,将所提方法与蒙特卡罗法、数论网格法进行了对比。结果表明:3种方法皆可满足求解精度要求,但PINN法无需进行数学推导,求解过程更加简单。

In order to explore approaches for solving multiple integrals,a method was proposed to solve multiple integral equations based on physical information neural networks(PINN).Firstly,the multiple integral equations were transformed into differential equations and boundary conditions.Se-condly,by means of designing the neural network structure,determining the training set and constructing the loss function,the PINN were used to approximate the original function of the differential equations.And then,according to the multiple definite integrals,the multiple definite integral equations could be solved by substituting the upper and lower limits of the integrals into the original function.Finally,the proposed method was compared with Monte Carlo method and number theory grid method.The result shows that the three methods satisfy the requirements of solving accuracy,but the PINN method does not require mathematical derivation and the solving process is much simpler.