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

doi:10.19596/j.cnki.1001-246x.8813

Abstract

Abstract:

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.

Key words: deep learning, linear integral equation, degenerate kernel, transformation of primitive functions, loss function, numerical validation

CLC Number:

TL329

Dong LIU, Qilong CHEN, Xueqiang WANG. Deep Learning Method for Solving Linear Integral Equations Through Primitive Function Transformation[J]. Chinese Journal of Computational Physics, 2024, 41(5): 651-662.

Figures/Tables 11

Fig.1 Iterative solution process of multi-integral term original function transformation

Table 1 Loss function, sample generation method and numerical operation results of the first kind of Volterra equation

损失函数类型	约束形式	样本数量	样本生成方式	损失函数权重	σ_MSE/10^-8	σ_{MSE, max}/10^-6	Loss/10^-8
控制方程F_f－loss	方程(16)	5 000	随机分布	P_f= 1	2.169 4	6.105 1	2.239 1
原函数定解约束F_d－loss	t=0, F₀(x, 0) = 0	150	固定	P_d =50

Fig.2 Numerical results of the first kind Volterra equation (a) scatterplot of the original function distribution of ascending order; (b) scatterplot of distribution of numerical solution of equation; (c) Fall-loss trend graph with time (first 50 times)

Table 2 Loss function, sample generation method and numerical operation results of the second type of Fredholm equation

损失函数类型	约束形式	样本数量	样本生成方式	损失函数权重	σ_MSE/10^-8	σ_{MSE, max}/10^-7	Loss/10^-8
控制方程F_f－loss	方程(9)	5 000	随机分布	1	1.251 5	7.841 6	2.683 0
原函数确定解约束F_d－loss	t=0, F₀(x, 0)=0	150	固定	50

Fig.3 Numerical results ot the second kind Fredholm equation (a) scatterplot of the original function distribution of ascending order; (b) scatterplot of distribution of the numerical solution of the equation; (c) Fall-loss trend graph with time (first 50 times)

Table 3 Loss function, sample generation method and numerical operation results of degenerate kernel Fredholm equation

损失函数来源	约束形式	样本数量	样本生成方式	损失函数权重	σ_MSE/10^-8	σ_{MSE, max}/10^-6	Loss/10^-9
控制方程F_f-loss－1; F_f-loss－2	方程(17)、(18)	1 000	随机分布	1	6.006 3(图 4(a))	6.284 1(图 4(a))	28.289(图 4(a))
原函数定解约束F_d－loss－1; F_d－loss－2	t₁=0, F₀₁(0)=0;t₂=0, F₀₂(0)=0	1	固定	30	4.906 7(图 4(b))	9.261 8(图 4(b))	7.335 6(图 4(b))

Fig.4 Numerical results of degenerate kernel Fredholm equation (a) distribution scatter diagram of the original function of integral item 1 and numerical solution of equation; (b) distribution scatter diagram of the original function of integral item 2 and numerical solution of equation; (c) Fall-Loss-1 trend graph with time (first 50 times); (d) Fall-Loss-2 trend graph with time (first 50 times)

Table 4 Loss function, sample generation method and numerical operation results of linear Volterra-Fredholm equation

损失函数来源	约束形式	样本数量	样本生成方式	损失函数权重	σ_MSE/10^-8	σ_{MSE, max}/10^-6	Loss/10^-8
控制方程F_f-loss－1；F_f-loss－2	方程(17)、(18)	1 000	随机分布	1	6.503 8(图 5(a))	4.307 3(图 5(a))	1.965 1(图 5(a))
原函数定解约束F_d－loss－1；F_d－loss－2	t₁=0, F₀₁(0)=0;t₂=0, F₀₂(0)=0	1	固定	30	9.735 7(图 5(b))	5.995 4(图 5(b))	2.510 0(图 5(b))

Fig.5 Numerical results of the linear Volterra-Fredholm equation (a) distribution scatter diagram of the original function of integral item 1 and numerical solution of equation; (b) distribution scatter diagram of the original function of integral item 2 and numerical solution of equation; (c) Fall-loss-1 trend graph with time (first 50 times); (d) Fall-loss-2 trend graph with time (first 50 times)

Table 5 Loss function, sample generation method and numerical operation results of delay equation

损失函数来源	约束形式	样本数量	样本生成方式	损失函数权重	σ_MSE/10^-8	σ_{MSE, max}/10^-6	Loss/10^-8
控制方程F_f－loss－1；F_f－loss－2	方程(17)、(18)	1000	随机分布	1	8.348 3(图 6(a))	2.439 4(图 6(a))	3.008 5(图 6(a))
原函数定解约束F_d－loss－1；F_d－loss－2	t₁=0, F₀₁(0)=0;t₂=0, F₀₂(0)=0	1	固定	30	7.214 4(图 6(b))	2.154 8(图 6(b))	2.747 2(图 6(b))

Fig.6 Numerical results of time-delay equation (a) distribution scatter diagram of the original function of integral item 1 and numerical solution of equation; (b) distribution scatter diagram of the original function of integral item 2 and numerical solution of equation; (c) Fall-Loss-1 trend graph with time (first 50 times); (d)Fall-Loss-2 trend graph with time (first 50 times)

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