🤖 AI Summary
Conventional numerical methods struggle with high-dimensional, strongly singular, fractional-order integral and integro-differential equations. Method: This paper proposes the Residual Integral Solver Network (RISN), a novel physics-informed neural network that integrates residual connections with high-accuracy numerical operators—including Gaussian quadrature kernels and fractional-order derivative matrices—while incorporating physical constraints, automatic differentiation, and joint loss optimization. Contribution/Results: RISN significantly enhances network depth adaptability and modeling capability for complex integral kernels. Across diverse benchmark problems—including one- and multi-dimensional, ordinary and partial, coupled, and fractional-order equations—RISN achieves 40–78% lower mean absolute error (MAE) than standard PINNs. It demonstrates superior stability and generalization, particularly for high-dimensional, strongly singular problems where classical numerical solvers fail.
📝 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.