SPIKANs：可分离物理信息Kolmogorov-Arnold网络
SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networks
AI总结：
本文提出SPIKANs，通过将变量分离原理应用于PIKANs，使每个维度由独立KAN处理，降低高维PDE训练复杂度，在保持精度的同时提升可扩展性。
摘要：
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。虽然PINNs通常采用多层感知器（MLPs）作为其底层架构，但近期的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），它已被证明优于传统MLPs，包括更快的神经缩放和更好的可解释性。将KAN应用于物理信息学习催生了物理信息KANs（PIKANs）的发展，使得利用KANs求解PDEs成为可能。然而，尽管有这些优势，KANs常常面临训练速度较慢的问题，尤其是在高维问题中，配点数随系统维度呈指数增长。为应对这一挑战，我们引入了可分离物理信息Kolmogorov-Arnold网络（SPIKANs）。这一新颖架构将变量分离原理应用于PIKANs，将问题分解为每个维度由独立的KAN处理。该方法在不牺牲精度的前提下大幅降低了训练的计算复杂度，促进了其在高维PDEs中的应用。通过一系列基准问题，我们展示了SPIKANs的有效性，证明了其相较于PIKANs的卓越可扩展性和性能，并凸显了其在科学计算中求解复杂高维PDEs的潜力。
英文摘要：
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.
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