The authors propose GeoKAN, a Kolmogorov-Arnold Network variant that learns a coordinate transformation (a diagonal Riemannian metric) to warp the input space before applying basis functions. Instead of using fixed basis functions on the raw input coordinates, GeoKAN first stretches regions with rapid variation and compresses smoother regions, reallocating representational capacity where it's most needed. They test it on physics-informed learning and differential-equation problems where functions have sharp, localized features.
Main takeaways:
- Learns a geometry-adapted coordinate system (diagonal metric) before basis expansion, rather than using fixed Euclidean coordinates
- Stretches high-variation regions and compresses smooth regions, placing capacity where needed
- Offers three main variants (GeoKAN-NNMetric, GeoKAN-γ, LM-KAN) and basis-specific versions (RBF, Wavelet, Fourier)
- Well-suited to sharp, stiff, or strongly non-uniform regimes in scientific ML and differential equations