ER-KANs: Efficient and Robust Kolmogorov-Arnold Networks for Data-Scarce Scientific Machine Learning

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.