用KAN网络求解高维和分数阶最优控制问题，精度效率双提升。
KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems
- 基于KAN网络与高斯积分逼近连续时间系统中的积分项
- 自动微分处理整数阶导数，矩阵向量积离散化分数阶导数
- 成功求解二维热方程等多维问题，优于传统MLP
本文提出KANtrol框架，利用Kolmogorov-Arnold网络（KANs）求解含连续时间变量的最优控制问题。通过高斯积分法近似问题中的积分部分，特别是积分-微分状态方程中的积分项。对于整数阶动力学，采用自动微分计算精确导数；对于非整数阶分数导数，则在KAN框架内使用矩阵-向量乘积进行离散化。该方法可处理多维问题，包括二维热偏微分方程的最优控制。仿真结果表明，无论是前向问题还是参数识别问题，KANtrol在准确性和效率上均优于经典MLP。
原文摘要 · Abstract (English)
In this paper, we introduce the KANtrol framework, which utilizes Kolmogorov-Arnold Networks (KANs) to solve optimal control problems involving continuous time variables. We explain how Gaussian quadrature can be employed to approximate the integral parts within the problem, particularly for integro-differential state equations. We also demonstrate how automatic differentiation is utilized to compute exact derivatives for integer-order dynamics, while for fractional derivatives of non-integer order, we employ matrix-vector product discretization within the KAN framework. We tackle multi-dimensional problems, including the optimal control of a 2D heat partial differential equation. The results of our simulations, which cover both forward and parameter identification problems, show that the KANtrol framework outperforms classical MLPs in terms of accuracy and efficiency.
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