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

• URL: http://arxiv.org/abs/2501.16370v2
• Date: Thu, 01 May 2025 12:29:08 GMT
• Title: Advanced Physics-Informed Neural Network with Residuals for Solving Complex Integral Equations
• Authors: Mahdi Movahedian Moghaddam, Kourosh Parand, Saeed Reza Kheradpisheh,
• Abstract summary: RISN is a novel neural network architecture designed to solve a wide range of integral and integro-differential equations.<n>RISN integrates residual connections with high-accuracy numerical methods.<n>We demonstrate that RISN consistently outperforms classical PINNs.
• Score: 0.13499500088995461
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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