Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) in scientific computing. While PINNs typically use multilayer perceptrons (MLPs) as their underlying architecture, recent advancements have explored alternative neural network structures. One such innovation is the Kolmogorov-Arnold Network (KAN), which has demonstrated benefits over traditional MLPs, including faster neural scaling and better interpretability. The application of KANs to physics-informed learning has led to the development of Physics-Informed KANs (PIKANs), enabling the use of KANs to solve PDEs. However, despite their advantages, KANs often suffer from slower training speeds, particularly in higher-dimensional problems where the number of collocation points grows exponentially with the dimensionality of the system. To address this challenge, we introduce Separable Physics-Informed Kolmogorov-Arnold Networks (SPIKANs). This novel architecture applies the principle of separation of variables to PIKANs, decomposing the problem such that each dimension is handled by an individual KAN. This approach drastically reduces the computational complexity of training without sacrificing accuracy, facilitating their application to higher-dimensional PDEs. Through a series of benchmark problems, we demonstrate the effectiveness of SPIKANs, showcasing their superior scalability and performance compared to PIKANs and highlighting their potential for solving complex, high-dimensional PDEs in scientific computing.

翻译：物理信息神经网络（PINNs）已成为科学计算中求解偏微分方程（PDEs）的一种有前景的方法。尽管PINNs通常使用多层感知机（MLPs）作为其基础架构，但近期的研究进展已探索了替代的神经网络结构。其中一项创新是Kolmogorov-Arnold网络（KAN），它已展现出优于传统MLPs的特性，包括更快的神经缩放速度和更好的可解释性。将KAN应用于物理信息学习催生了物理信息KAN（PIKANs），使得利用KAN求解PDEs成为可能。然而，尽管具有优势，KAN通常存在训练速度较慢的问题，尤其是在高维问题中，配置点的数量随系统维度的增加呈指数级增长。为应对这一挑战，我们提出了可分离物理信息Kolmogorov-Arnold网络（SPIKANs）。这一新颖架构将变量分离原理应用于PIKANs，通过分解问题使得每个维度由独立的KAN处理。该方法在不牺牲精度的情况下，大幅降低了训练的计算复杂度，从而促进了其在高维PDEs中的应用。通过一系列基准问题，我们验证了SPIKANs的有效性，展示了其相较于PIKANs更优的可扩展性和性能，并凸显了其在科学计算中求解复杂高维PDEs的潜力。