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

• URL: http://arxiv.org/abs/2501.16370v1
• Date: Wed, 22 Jan 2025 19:47:03 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. We show that RISN consistently outperforms PINN, achieving significantly lower Mean Absolute Errors (MAE) across various types of equations. The results highlight RISN's robustness and efficiency in solving challenging integral and integro-differential problems.
• Score: 0.13499500088995461
• License:
• 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, and fractional types. RISN integrates residual connections with high-accurate 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 PINN, achieving significantly lower Mean Absolute Errors (MAE) across various types of equations. The 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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