Abstract:Kolmogorov-Arnold Networks (KANs) have recently demonstrated promising potential in scientific machine learning, partly due to their capacity for grid adaptation during training. However, existing adaptation strategies rely solely on input data density, failing to account for the geometric complexity of the target function or metrics calculated during network training. In this work, we propose a generalized framework that treats knot allocation as a density estimation task governed by Importance Density Functions (IDFs), allowing training dynamics to determine grid resolution. We introduce a curvature-based adaptation strategy and evaluate it across synthetic function fitting, regression on a subset of the Feynman dataset and different instances of the Helmholtz PDE, demonstrating that it significantly outperforms the standard input-based baseline. Specifically, our method yields average relative error reductions of 25.3% on synthetic functions, 9.4% on the Feynman dataset, and 23.3% on the PDE benchmark. Statistical significance is confirmed via Wilcoxon signed-rank tests, establishing curvature-based adaptation as a robust and computationally efficient alternative for KAN training.
| Comments: | Accepted in IJCNN 2026 |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2601.18672 [cs.LG] |
| (or arXiv:2601.18672v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2601.18672 arXiv-issued DOI via DataCite |
From: Spyros Rigas [view email]
[v1]
Mon, 26 Jan 2026 16:49:06 UTC (140 KB)
[v2]
Mon, 30 Mar 2026 12:41:10 UTC (140 KB)
[v3]
Wed, 22 Apr 2026 20:17:03 UTC (140 KB)
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