In continuum topology optimization (TO), two essential procedures are involved: structural analysis through the solution of partial differential equations (PDEs) and the subsequent update of design variables. Both procedures can be addressed by training neural networks using the corresponding physical information. Accordingly, Physics-Informed Neural Network (PINN)-based algorithms have been developed for TO. However, PINN-based methods suffer from several notable limitations, including high computational cost, spectral bias, and limited adaptability in solving PDEs.To overcome these challenges, this study proposes a novel algorithm that incorporates two Higher-Order ReLU-based Kolmogorov-Arnold Networks (HRKANs). Specifically, a displacement-informed HRKAN (d-HRKAN) is designed to predict PDE solutions, while a sensitivity-informed HRKAN (s-HRKAN) is developed to perform sensitivity analysis for updating design variables. For convenience, the proposed approach is referred to as the Dual Physics-Informed Kolmogorov-Arnold Networks-based Topology Optimization (DPIKAN-TO) method. By leveraging learnable activation functions, the proposed neural networks can accurately approximate the responses of complex structural systems. Moreover, compared with conventional PINN-based methods, DPIKAN-TO demonstrates significantly improved computational efficiency and reduced computational cost. Numerical examples show that DPIKAN-TO can successfully identify optimal material layouts for linear structures, compliant mechanisms, and fluid-solid coupled systems. Furthermore, owing to the use of learnable activation functions, the proposed framework can be readily extended to structural optimization problems governed by new types of PDEs.

翻译：在连续体拓扑优化（TO）中，涉及两个关键步骤：通过求解偏微分方程（PDE）进行结构分析，以及随后对设计变量的更新。这两个步骤均可通过利用相应物理信息训练神经网络来实现。为此，已开发出基于物理信息神经网络（PINN）的TO算法。然而，基于PINN的方法存在若干显著局限，包括计算成本高、频谱偏差以及求解PDE时适应性有限。为克服这些挑战，本研究提出一种新颖算法，该算法集成了两个基于高阶ReLU的Kolmogorov-Arnold网络（HRKAN）。具体而言，设计了一个位移驱动的HRKAN（d-HRKAN）用于预测PDE解，同时开发了一个灵敏度驱动的HRKAN（s-HRKAN）用于执行灵敏度分析以更新设计变量。为方便起见，所提方法被称为基于双物理信息Kolmogorov-Arnold网络的拓扑优化（DPIKAN-TO）方法。通过利用可学习激活函数，所提出的神经网络能够精确逼近复杂结构系统的响应。此外，与传统的基于PINN的方法相比，DPIKAN-TO显著提高了计算效率并降低了计算成本。数值算例表明，DPIKAN-TO能够成功识别线性结构、柔顺机构及流固耦合系统的最优材料布局。而且，由于采用了可学习激活函数，所提框架可轻松扩展至由新型PDE控制的结构优化问题。