线性积分方程原函数变换深度学习求解方法

Deep Learning Method for Solving Linear Integral Equations Through Primitive Function Transformation

针对深度学习数值计算方法求解积分方程,提出求解线性积分方程的原函数变换深度学习方法,通过被积函数的原函数变换,将积分方程转化为纯粹的微分方程,并给出原函数定解条件确定方法,以及相应的神经网络损失函数生成方式。通过深度学习使得神经网络函数逼近原函数后,将原函数求导并根据积分核的形式进行逆变换,最终得到积分方程未知函数的数值解。通过多种典型算例数值实验证明,本文方法具有良好的精度与适用性,数值计算结果具有连续性的优点。

Due to factors such as limited integral terms and approximations,solving integral equations using classical numerical methods is often more challenging than solving differential equations.This paper proposes a theory of solving linear integral equations through the transformation of primitive functions using deep learning.By transforming the integrand into a primitive function,the integral equation is converted into a purely differential equation.The paper also provides a method for determining the initial conditions of the primitive function and a technique for generating the neural network loss function.After approximating the primitive function using deep learning with neural networks,the derivative of the primitive function is calculated and transformed according to the form of the integral kernel,ultimately obtaining the numerical solution of the unknown function in the integral equation.Through numerical experiments on various typical examples,the paper demonstrates that the proposed theory and key techniques exhibit good accuracy and applicability,thereby opening up new technical approaches for the numerical solution of linear integral equations.