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.

Based on a single-pole Debye empirical formula, a multi frequency contrast source inversion (CSI) approach with a multiplicative regularization term is improved tentatively. The main improvement is a modification of inversion object from direct estimation of contrast between Debye scatterers and background medium to indirect reconstruction of four kinds of frequency-independent one-pole Debye model parameters, i.e., the optical relative permittivity, relative permittivity difference, static conductivity, and relaxation time. Two numerical examples confirmed preliminarily feasibility and robustness of the improved method.