arXiv：cs.LG· Gregorio P\'erez-Bernal, Oscar Rinc\'on-Carde\~no, Silvana Montoya-Noguera, Nicol\'as Guar\'in-Zapata·· 9 小时前AI 评分28
无界域中采样策略与物理信息 Kolmogorov–Arnold 网络的评估
Evaluation of Sampling Strategies and Physics-Informed Kolmogorov--Arnold Networks in Unbounded Domains
AI 导读
一项基准研究评估了无界域逆 PDE 问题中物理信息学习的采样策略与网络结构，测试了均匀、高斯和指数分布采样，以及 MLP 与 KAN 在 PINN 框架下的表现。结果显示，通过显式分布参数控制的高斯和指数采样能提升远场重建精度，并可直接嵌入问题先验知识。PIKAN 以更少神经元达到或超过 MLP 版 PINN 的精度，但其样条公式增加了训练、推理与导数计算的开销。
正文
Abstract:Physics-informed neural networks (PINNs) have emerged as an effective approach for solving partial differential equations (PDEs) by incorporating physical laws into the learning process. However, their application to infinite and semi-infinite domains remains challenging due to the difficulty of representing unbounded regions with a finite number of training points. This work evaluates physics-informed learning strategies for inverse PDE problems in unbounded domains, focusing on the influence of sampling strategies based on uniform, Gaussian, and exponential distributions, and on the use of conventional multilayer perceptrons (MLPs) and Kolmogorov--Arnold Networks (KANs) within the PINN framework. The proposed benchmark considers manufactured inverse problems on infinite and semi-infinite domains, enabling quantitative assessment of reconstruction accuracy and computational efficiency. Results show that Gaussian and exponential sampling, controlled through explicit distribution parameters, improve reconstruction accuracy in the far field and offer a direct mechanism for embedding problem-specific prior knowledge into training. PIKANs, whose edge-based functional representation is grounded in the Kolmogorov--Arnold representation theorem, match or exceed the accuracy of MLP-based PINNs while requiring fewer neurons, though their spline-based formulation increases the computational cost of training, inference, and derivative evaluation.
| Comments: | 14 pages, 7 figures |
| Subjects: | Machine Learning (cs.LG); Mathematical Physics (math-ph) |
| Cite as: | arXiv:2512.12074 [cs.LG] |
| (or arXiv:2512.12074v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2512.12074 arXiv-issued DOI via DataCite |
Submission history
From: Gregorio Pérez Bernal [view email]
[v1]
Fri, 12 Dec 2025 22:44:46 UTC (14,127 KB)
[v2]
Thu, 1 Oct 2026 19:39:13 UTC (1,472 KB)
来源：arXiv：cs.LG · arxiv.org