Geometric Kolmogorov--Arnold Network (GeoKAN)
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913099366268928 |
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| author | Sen, Abhijit Parida, Bikram Keshari Maiti, Giridas Arya, Mahima Bondar, Denys I. |
| author_facet | Sen, Abhijit Parida, Bikram Keshari Maiti, Giridas Arya, Mahima Bondar, Denys I. |
| contents | We introduce Geometric Kolmogorov--Arnold Networks (GeoKANs), a family of geometry-aware KAN-type models in which approximation is carried out in learned, geometry-adapted coordinates rather than in fixed Euclidean input coordinates. GeoKAN achieves this by learning a diagonal Riemannian metric that warps the input before basis expansion and feature mixing. The learned metric provides a geometric inductive bias through local length scaling and volume distortion, and in physics-informed settings it also affects the differential structure seen by the model. Within this framework, we develop three main variants, namely GeoKAN-NNMetric, GeoKAN-$γ$, and LM-KAN. For LM-KAN, we further consider three basis-specific versions, LM-KAN-RBF, LM-KAN-Wav, and LM-KAN-Fourier. These variants allow us to study geometry-aware KAN models both as general function approximators and as surrogates in physics-informed learning. By stretching regions with rapid variation and compressing smoother regions, GeoKAN reallocates representational resolution in a task-dependent manner, allowing the model to place capacity where it is most needed. As a result, GeoKAN is well suited to sharp, stiff, localized, and strongly non-uniform regimes arising in scientific machine learning and differential-equation problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06740 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric Kolmogorov--Arnold Network (GeoKAN) Sen, Abhijit Parida, Bikram Keshari Maiti, Giridas Arya, Mahima Bondar, Denys I. Machine Learning Artificial Intelligence We introduce Geometric Kolmogorov--Arnold Networks (GeoKANs), a family of geometry-aware KAN-type models in which approximation is carried out in learned, geometry-adapted coordinates rather than in fixed Euclidean input coordinates. GeoKAN achieves this by learning a diagonal Riemannian metric that warps the input before basis expansion and feature mixing. The learned metric provides a geometric inductive bias through local length scaling and volume distortion, and in physics-informed settings it also affects the differential structure seen by the model. Within this framework, we develop three main variants, namely GeoKAN-NNMetric, GeoKAN-$γ$, and LM-KAN. For LM-KAN, we further consider three basis-specific versions, LM-KAN-RBF, LM-KAN-Wav, and LM-KAN-Fourier. These variants allow us to study geometry-aware KAN models both as general function approximators and as surrogates in physics-informed learning. By stretching regions with rapid variation and compressing smoother regions, GeoKAN reallocates representational resolution in a task-dependent manner, allowing the model to place capacity where it is most needed. As a result, GeoKAN is well suited to sharp, stiff, localized, and strongly non-uniform regimes arising in scientific machine learning and differential-equation problems. |
| title | Geometric Kolmogorov--Arnold Network (GeoKAN) |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2605.06740 |