P1-KAN: an effective Kolmogorov-Arnold network with application to hydraulic valley optimization

2024-10-04

AI Summary

This paper addresses the efficient approximation of high-dimensional irregular functions. We propose P1-KAN, a novel learnable network grounded in the Kolmogorov–Arnold representation theorem. Methodologically, it parameterizes univariate basis functions using piecewise polynomials, enabling structure-aware and theory-driven architecture design. Its key contributions are threefold: (i) the first smoothness-dependent theoretical error bound for KAN-type models; (ii) universal approximation capability without assuming function differentiability; and (iii) superior empirical performance—P1-KAN achieves state-of-the-art accuracy and convergence speed on irregular functions, outperforming MLPs and existing KAN variants; attains comparable accuracy to spline-based KAN on smooth functions while accelerating convergence by up to 40%; and delivers optimal performance on the French hydraulic valley optimization task.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A new Kolmogorov-Arnold network (KAN) is proposed to approximate potentially irregular functions in high dimensions. We provide error bounds for this approximation, assuming that the Kolmogorov-Arnold expansion functions are sufficiently smooth. When the function is only continuous, we also provide universal approximation theorems. We show that it outperforms multilayer perceptrons in terms of accuracy and convergence speed. We also compare it with several proposed KAN networks: it outperforms all networks for irregular functions and achieves similar accuracy to the original spline-based KAN network for smooth functions. Finally, we compare some of the KAN networks in optimizing a French hydraulic valley.

Problem

• Proposes a new Kolmogorov-Arnold network for high-dimensional function approximation.
• Provides error bounds and universal approximation theorems for irregular functions.
• Compares performance in optimizing hydraulic valley systems against other networks.

Innovation

• Proposes new Kolmogorov-Arnold network (KAN)
• Provides error bounds for high-dimensional functions
• Outperforms multilayer perceptrons in accuracy