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

TL;DR

This paper introduces RISN, a novel neural network architecture that combines residual connections with numerical methods to solve complex integral equations more accurately and stably than existing PINNs.

Contribution

The paper presents RISN, a new physics-informed neural network that integrates residuals with numerical techniques to improve accuracy and stability in solving diverse integral equations.

Findings

RISN outperforms classical PINNs and variants in accuracy.

RISN achieves lower MAE across various integral equations.

Residual connections enhance stability and depth handling.

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.