Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 11 1 2026 06 07 Advanced Physics-Informed Neural Network with Residuals for Solving Complex Integral Equations 153 173 2109 10.22128/ansne.2026.3261.1200 EN Mahdi Movahedian Moghaddam Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran 0000-0002-6513-5251 Kourosh Parand Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran; Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University Saeed Reza Kheradpisheh Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran Journal Article 2026 02 18 In this paper, we present the Residual Integral Solver Network (RISN), a novel neural network architecture designed to solve a wide range of integral and integro-differential equations, including one-dimensional, multi-dimensional, ordinary and partial integro-differential, systems, fractional types, and Helmholtz-type integral equations involving oscillatory kernels. RISN integrates residual connections with high-accuracy numerical methods such as Gaussian quadrature and fractional derivative operational matrices, enabling it to achieve higher accuracy and stability than traditional Physics-Informed Neural Networks (PINN). The residual connections help mitigate vanishing gradient issues, allowing RISN to handle deeper networks and more complex kernels, particularly in multi-dimensional problems. Through extensive experiments, we demonstrate that RISN consistently outperforms not only classical PINNs but also advanced variants such as Auxiliary PINN (A-PINN) and Self-Adaptive PINN (SA-PINN), achieving significantly lower Mean Absolute Errors (MAE) across various types of equations. These results highlight RISN’s robustness and efficiency in solving challenging integral and integro-differential problems, making it a valuable tool for real-world applications where traditional methods often struggle. https://ansne.du.ac.ir/article_2109_d49a926a5977e1492d7556602484fdb9.pdf