KANtrol：一种用于求解多维和分数阶最优控制问题的物理信息 Kolmogorov-Arnold 网络框架
KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems
AI总结：
提出物理信息 KANtrol 框架，结合高斯求积、自动微分与分数阶矩阵-向量积离散化，求解连续时间、多维及分数阶最优控制问题，并在精度和效率上优于 MLP。
摘要：
本文提出 KANtrol 框架，利用 Kolmogorov-Arnold Networks（KAN）求解涉及连续时间变量的最优控制问题。我们阐述如何采用高斯求积近似问题中的积分部分，尤其是积分-微分状态方程。我们还展示如何利用自动微分计算整数阶动力学的精确导数；对于非整数阶的分数阶导数，则在 KAN 框架内采用矩阵-向量积离散化。我们处理了多维问题，包括二维热偏微分方程的最优控制。涵盖正问题和参数辨识问题的仿真结果表明，KANtrol 框架在精度和效率方面优于经典 MLP。
英文摘要：
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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