Assess whether interior-knot outer splines improve performance near layers and interfaces

Determine whether outer splines with interior knots in the local Kolmogorov–Arnold models improve the IGA–KAN post-process near boundary layers and material interfaces.

Background

The local models used in the paper employ single-span outer splines, whose span is exactly a polynomial space. Introducing interior knots would extend the local space beyond polynomials while preserving a linear least-squares fit. Preliminary tests on the quarter annulus showed changes of less than five percent in the errors, but the paper leaves unresolved whether this additional flexibility is beneficial for nonsmooth features such as layers and material interfaces.

References

Outer splines with interior knots, which leave the polynomial space while the fit stays linear, are the part of the Kolmogorov--Arnold form not used in the method (on the quarter annulus they change the errors by less than $5\%$, Remark~\ref{rem:space}); whether they help near layers and interfaces remains to be tested.

— IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs  (2610.06348 - Naraghi et al., 5 Oct 2026) in Section 8, “Conclusions and future work,” paragraph “Future work,” item 3