LeanKAN: A Parameter-Lean Kolmogorov-Arnold Network Layer with Improved Memory Efficiency and Convergence Behavior

Abstract

The recently proposed Kolmogorov-Arnold network (KAN) is a promising alternative to multi-layer perceptrons (MLPs) for data-driven modeling. While original KAN layers were only capable of representing the addition operator, the recently-proposed MultKAN layer combines addition and multiplication subnodes in an effort to improve representation performance. Here, we find that MultKAN layers suffer from a few key drawbacks including limited applicability in output layers, bulky parameterizations with extraneous activations, and the inclusion of complex hyperparameters. To address these issues, we propose LeanKANs, a direct and modular replacement for MultKAN and traditional AddKAN layers. LeanKANs address these three drawbacks of MultKAN through general applicability as output layers, significantly reduced parameter counts for a given network structure, and a smaller set of hyperparameters. As a one-to-one layer replacement for standard AddKAN and MultKAN layers, LeanKAN is able to provide these benefits to traditional KAN learning problems as well as augmented KAN structures in which it serves as the backbone, such as KAN Ordinary Differential Equations (KAN-ODEs) or Deep Operator KANs (DeepOKAN). We demonstrate LeanKAN's simplicity and efficiency in a series of demonstrations carried out across a standard KAN toy problem as well as ordinary and partial differential equations learned via KAN-ODEs, where we find that its sparser parameterization and compact structure serve to increase its expressivity and learning capability, leading it to outperform similar and even much larger MultKANs in various tasks.

21 pages, 7 figures, and 4 tables. Updated after acceptance to journal

References in corpus (6)

• Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks
• KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics
• Kolmogorov-Arnold Networks are Radial Basis Function Networks
• Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems
• Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN
• Physics-informed Kolmogorov-Arnold Network with Chebyshev Polynomials for Fluid Mechanics

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