Effect of spline-boundary behavior and normalization

Determine how the vanishing of the cubic B-spline basis at inputs clamped to exactly +1, together with layer normalization placing inputs outside the spline domain, affects the accuracy of the Parallel BesselKAN architecture.

Background

The implemented cubic B-spline basis is defined on [-1,1] and vanishes for inputs clamped to exactly +1 because of the endpoint-handling implementation. At the same time, layer normalization can produce values outside the nominal spline interval, although the spline path subsequently clips inputs to [-1,1].

The paper identifies these interactions as potential sources of model behavior but reports that their effect on predictive accuracy was not evaluated. A dedicated study is therefore needed to determine whether the endpoint and out-of-domain handling materially changes performance.

References

The spline basis vanishes for inputs clamped to exactly $+1$ (Section~\ref{sec:bspline}) and layer normalization places many inputs outside $[-1,1]$; the effect on accuracy was not measured.

— Hybrid Parallel Bessel KAN Model: a Kolmogorov Arnold Network: integrated framework for developing a novel architecture to reconstruct the fused-basis approach via a parameter free quadratic interaction  (2610.17779 - Pathak et al., 13 Oct 2026) in Section 6.5, “Limitations”