Advanced Physics-Informed Neural Network with Residuals for Solving Complex Integral Equations

Integral and integro-differential equations are foundational tools in many fields of science and engineering, modeling a wide range of phenomena from physics and biology to economics and engineering systems [1-3]. These equations describe processes that depend not only on local variables but also on historical or spatial factors, making them essential for understanding systems with memory effects, hereditary characteristics, and long-range interactions [4-7]. Despite their importance, solving integral and integro-differential equations is a challenging task due to the complexity of their integral operators, especially when extended to multi-dimensional or fractional forms [2, 8]. Classical numerical methods, such as finite difference [9, 10], finite element [11, 12], and spectral methods [13-15], have long been used to approximate solutions to these equations. However, these methods often suffer from several limitations.