Systematic Study of Grid Adaptation Strategies in Kolmogorov–Arnold Networks

Kolmogorov--Arnold Networks (KANs) place B-spline knots uniformly across the input domain by default, an assumption inherited from the uniform data generation used in their original experiments. The \texttt{pykan} library includes a parameter for data-adaptive placement but freezes the grid after 50 training steps --- a design the authors explicitly note is out of scope for further study. We present the first systematic empirical comparison of four KAN grid strategies on non-uniform input distributions, evaluated across 20 random seeds, two grid sizes ($G \in \{3, 5\}$), and four synthetic datasets. Our proposed \emph{dynamic adaptation} strategy periodically repositions knots (every 150 epochs) at empirical quantiles of the current layer activations throughout training, using a warm-restart least-squares projection to preserve the learned function shape across knot moves. On the concentrated Gaussian benchmark, dynamic adaptation achieves a 91.3\% reduction in median RMSE over the uniform baseline ($G=3$, MW $U=333$, $p=0.0003$, BH $q=0.0013$, Cohen's $d=1.024$). An ablation study identifies the revert guard as the dominant mechanism: without it, adaptation collapses in 18/20 seeds ($G=3$); with it, collapse falls to 3/20. \texttt{pykan}'s existing mechanism collapses on all 20 seeds under concentrated distributions ($\text{std}=0.000$), confirming that adaptation beyond step 50 is necessary. S4 significantly outperforms a parameter-matched MLP baseline at both grid sizes ($p=0.0009$, BH $q=0.003$ at $G=3$; $p=0.022$, BH $q=0.047$ at $G=5$), confirming the benefit is specific to adaptive spline placement rather than model capacity. On California Housing --- where five of eight features are highly skewed --- S4 achieves a 7.6\% improvement over the uniform baseline (MW $p<0.001$, BH $q<0.001$) with substantially lower variance ($\text{std}=0.011$ vs.\ $0.111$ at $G=3$), and significantly outperforms the MLP baseline (BH $q<0.001$). Dynamic adaptation provides no statistically significant advantage --- and is significantly harmful at $G=3$ ($p=0.002$, BH $q=0.005$) --- on edge-concentrated distributions where uniform knot spacing already matches the data geometry, providing an honest negative control and a practical criterion for when to apply the method.