SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networks

TL;DR

SPIKANs introduce a separable architecture for physics-informed Kolmogorov-Arnold networks, significantly improving scalability and training efficiency for high-dimensional PDEs in scientific computing.

Contribution

This paper proposes SPIKANs, a novel separable architecture for PIKANs that reduces computational complexity and enhances performance in high-dimensional PDE solving.

Findings

SPIKANs outperform PIKANs in high-dimensional benchmarks.

Separable architecture reduces training time significantly.

Effective for complex, high-dimensional PDEs.

Abstract

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.