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

2026-05-05 · Source: cs.NE updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, long

Summary

The P1-KAN network is a new Kolmogorov-Arnold Network (KAN) designed for approximating potentially irregular, high-dimensional functions. It addresses limitations of previous KAN implementations, particularly the computational expense of spline approximations and issues with layer output grids. Unlike the original spline-based KANs and ReLU-KANs, P1-KAN uses a P1 finite element method for discretizing 1D functions and explicitly defines the support of layer functions, avoiding complex grid adaptations. Numerical results demonstrate that P1-KAN outperforms multilayer perceptrons (MLPs) in accuracy and convergence speed. While P1-KAN is 1.5 to 2 times slower than ReLU-KAN on an Intel i7-11850H processor, it consistently achieves higher accuracy, especially for highly irregular functions and in higher dimensions (up to 13 for regular functions and 5 for irregular functions), where ReLU-KAN and MLPs often fail or diverge.

Key takeaway

For AI Engineers working on high-dimensional function approximation, especially with irregular data or in stochastic optimization, P1-KAN offers a more robust and accurate alternative to traditional MLPs and even ReLU-KANs. You should consider implementing P1-KAN, particularly when previous KAN architectures show convergence issues or insufficient accuracy, noting its higher computational cost compared to ReLU-KAN but superior performance in challenging scenarios.

Key insights

P1-KAN offers superior accuracy and stability for high-dimensional function approximation compared to MLPs and ReLU-KANs.

Principles

• Explicitly define layer function support.
• P1 finite elements improve approximation accuracy.
• Smaller mesh size (M) can be optimal in high dimensions.

Method

P1-KAN discretizes 1D functions using a P1 finite element method, with trainable variables for coefficients and mesh vertices. Layers are stacked without grid adaptation, taking input values and a grid, and outputting values and a new lattice.

In practice

• Use P1-KAN for irregular, high-dimensional function approximation.
• Consider P1-KAN when ReLU-KAN exhibits divergence.
• Optimize P1-KAN with a small mesh size (M) for complex functions.

Topics

• Kolmogorov-Arnold Networks
• P1-KAN
• ReLU-KAN
• Function Approximation
• Finite Element Method

Best for: AI Engineer, AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.NE updates on arXiv.org.