RISN神经网络用残差结构解决复杂积分方程，精度和稳定性更优。
Advanced Physics-Informed Neural Network with Residuals for Solving Complex Integral Equations
- 引入残差连接与高精度数值方法融合，提升深层网络表现。
- 在多维、分数阶、振荡核等方程上，平均绝对误差显著更低。
- 适合需要高精度求解的科学计算与工程应用，尤其传统方法难解的场景。
本文提出残差积分求解网络（RISN），一种用于求解广泛类型积分及积分微分方程的新神经网络架构，涵盖一维、多维、常微分与偏微分、系统型、分数阶及含振荡核的赫尔姆霍兹型积分方程。RISN将残差连接与高精度数值方法（如高斯求积、分数阶导数运算矩阵）结合，有效缓解梯度消失问题，支持更深网络与更复杂核函数，尤其在多维问题中表现优异。大量实验表明，RISN不仅优于传统物理信息神经网络（PINN），也显著超越辅助PINN（A-PINN）与自适应PINN（SA-PINN），在各类方程上均实现更低的平均绝对误差（MAE）。结果凸显其在求解挑战性积分与积分微分问题上的鲁棒性与高效性，为实际应用中传统方法难以处理的问题提供有效工具。
原文摘要 · Abstract (English)
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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