发表机构
Sliced Health（Sliced Health）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对数据稀缺的科学机器学习场景，提出ER-KAN，通过三种设计提升噪声鲁棒性，在多任务基准中验证其性能，还引入噪声退化比作为评估指标。
AI 中文摘要
现有高效柯尔莫哥洛夫-阿诺德网络（efficient-KAN）相关研究，涵盖原始柯尔莫哥洛夫-阿诺德网络（KAN）的切比雪夫变体（ChebyKAN）、小波变体、径向基函数变体，几乎全部在干净数据上进行基准测试。我们发现这一测试选择掩盖了不同架构间的巨大性能差异：当训练数据被sigma=0.1的噪声污染时，ChebyKAN的测试均方误差（MSE，与干净真值对比）会上升10.6倍，而普通KAN上升7.9倍，标准多层感知机（MLP）上升1.7倍，我们提出的ER-KAN仅上升1.4倍。ER-KAN结合了三个针对噪声、数据稀缺场景的设计选择：同一层所有边共享高斯径向基函数（Gaussian RBF）基函数（提供局部性与高效参数化）、训练期间的课程噪声注入（显式学习噪声鲁棒性）、熵加权自适应正则化（防止小样本N时过拟合）。最终得到一个仅595个参数的网络，在中等噪声下与MLP精度相当，且随噪声增大时性能下降更平缓。我们在8个解析函数（样本量N∈{50,200,500}，噪声水平sigma∈{0,0.03,0.1}）上进行评估，在受物理信息神经网络（PINN）约束的阻尼简谐振子任务中，ER-KAN的解MSE比MLP低4.2倍；在伯格斯方程（Burgers' equation）PINN任务中，所有模型均未收敛——我们报告这一真实局限而非隐瞒。我们引入噪声退化比作为简单的补充指标，并建议将其作为高效-KAN论文的标准报告要求。
英文摘要
The efficient-KAN literature---covering Chebyshev, wavelet, and radial-basis-function variants of the original Kolmogorov-Arnold Network---has been benchmarked almost entirely on clean data. We show that this choice conceals a large capability difference between architectures: ChebyKAN's test MSE (evaluated against clean ground truth) increases by a factor of 10.6x when training data is corrupted with sigma=0.1 noise, versus 7.9x for vanilla KAN, 1.7x for a standard MLP, and just 1.4x for our proposed ER-KAN. ER-KAN combines three design choices targeting the noisy, data-scarce setting: shared Gaussian RBF bases across all edges in a layer (providing locality and efficient parameterisation), curriculum noise injection during training (explicitly teaching noise robustness), and entropy-weighted adaptive regularisation (preventing overfitting at small N). The result is a 595-parameter network that matches MLP accuracy at moderate noise while degrading far more gracefully as noise grows. We evaluate on eight analytic functions (N in {50, 200, 500}, sigma in {0, 0.03, 0.1}), on a damped harmonic oscillator physics-informed neural network where ER-KAN achieves 4.2x lower solution MSE than MLP, and on a Burgers' equation PINN where all models fail to converge---a genuine limitation we report rather than suppress. We introduce the noise degradation ratio as a simple complementary metric and recommend it become a standard reporting requirement for efficient-KAN papers.
Comments22 pages, 20 figures, 8 tables; code and data at https://github.com/harshillodhiya/ER-KANs Version 2, includes review feedback from industry leaders like Oleksandr Kuznetsov, Zhaoxiang
