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.

翻译：本文提出了一种新颖的神经网络架构——残差积分求解网络（RISN），用于求解包括一维、多维、常积分-微分、偏积分-微分、方程组及分数阶类型在内的广泛积分与积分-微分方程。RISN将残差连接与高斯求积、分数阶导数运算矩阵等高精度数值方法相结合，使其相比传统物理信息神经网络（PINN）实现了更高的精度与稳定性。残差连接有助于缓解梯度消失问题，使RISN能够处理更深层的网络和更复杂的核函数，尤其适用于多维问题。通过大量实验，我们证明RISN在不同类型方程求解中均持续优于PINN，获得了显著更低的平均绝对误差（MAE）。结果凸显了RISN在求解具有挑战性的积分与积分-微分问题时的鲁棒性与高效性，使其成为传统方法难以应对的实际应用场景中的有力工具。