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

arXiv: https://arxiv.org/abs/2501.16370 · HTML: https://mlsanity.com/paper/2501.16370/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 60 references that are in the map.
- First posted on arXiv 22 Jan 2025.

## Builds on (most cited first, 10 of 10)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Eighty Years of the Finite Element Method: Birth, Evolution, and Future](https://mlsanity.com/paper/2107.04960/) (2021)
- [PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems](https://mlsanity.com/paper/2409.01899/) (2024)

## Built on by (most cited first, 1 of 1)

- [ADAPT: Lightweight, Long-Range Machine Learning Force Fields Without Graphs](https://mlsanity.com/paper/2509.24115/) (2025)

## Abstract (arXiv)

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.