PINNIES: An efficient physics-informed neural network framework for integral operator problems

This paper introduces an efficient tensor-vector product technique for the fast and accurate approximation of integral operators within physics-informed deep learning frameworks. Our approach leverages Kolmogorov-Arnold networks to evaluate problem dynamics at specific points, while employing Gaussian quadrature formulas to approximate the integral components, even in the presence of semi-infinite domains or singularities. We demonstrate the applicability of the proposed method to both Fredholm and Volterra integral operators, as well as to optimal control problems involving continuous time. Additionally, we outline how this approach can be extended to approximate fractional derivatives and integrals and propose a fast matrix-vector product algorithm for efficiently computing the fractional Caputo derivative. In the numerical section, we conduct comprehensive experiments on forward and inverse problems. For forward problems, we evaluate the performance of our method on over 50 diverse mathematical problems, including multi-dimensional integral equations, systems of integral equations, partial and fractional integro-differential equations, and various optimal control problems in delay, fractional, multi-dimensional, and nonlinear configurations. For inverse problems, we test our approach on several integral equations and fractional integro-differential problems. Finally, we introduce the pinnies Python package to facilitate the implementation and usability of the proposed method.本文提出一种高效张量-向量积（tensor-vector product）技术，用于在物理信息深度学习（physics-informed deep learning）框架内实现积分算子的快速精准近似。本方法借助柯尔莫哥洛夫-阿诺德网络（Kolmogorov-Arnold networks）在指定点处求解问题动力学特性，同时采用高斯求积公式（Gaussian quadrature formulas）对积分项进行近似，即便在存在半无限区域或奇点的场景下仍可有效运行。我们验证了所提方法可同时适用于弗雷德霍姆积分算子与沃尔泰拉积分算子，以及涉及连续时间的最优控制问题。此外，本文还阐述了该方法可被拓展用于近似分数阶导数与分数阶积分，并提出了一种快速矩阵-向量积算法，以高效计算分数阶卡普托导数（fractional Caputo derivative）。在数值实验章节中，我们针对正问题与反问题开展了全面的实验研究。针对正问题，我们在50余个多样化的数学问题上测试了本方法的性能，涵盖多维积分方程、积分方程组、偏积分微分方程与分数阶积分微分方程，以及时滞、分数阶、多维及非线性构型下的各类最优控制问题。针对反问题，我们在若干积分方程与分数阶积分微分问题上验证了本方法的效果。最后，本文推出了pinnies Python工具包，以简化所提方法的实现流程并提升其易用性。