Adaptive Hermite Kolmogorov-Arnold Networks (AHKAN): Stable Geometric Neural Operators for Multi-Level Connectivity and Derivative-Informed Learning

Classical Kolmogorov-Arnold Networks (KANs) employ fixed-degree global bases (B-splines, RBFs) with abstract weights lacking interpretability. In this paper, we introduce Adaptive Hermite Kolmogorov-Arnold Networks (AHKAN), with trainable pseudo-derivative coefficients ( θ 0 : value, θ 1 : slope, θ 2 : curvature), spatially-adaptive polynomial orders, and partition-of-unity normalization. AHKAN reduces MSE by 2.1–5.0 × relative to cubic B-spline KAN on heterogeneous benchmarks, with θ 1 coefficients encoding gradient-like geometric structure—critical for physics-informed applications requiring derivative fidelity. Applied to naturalistic fMRI data ( n = 8 subjects, 451 timepoints per run, 7-Tesla), AHKAN demonstrates four neuroimaging capabilities absent in classical architectures: (i) Superior temporal reconstruction : + 25 % correlation improvement over B-spline KAN by preserving phase relationships; (ii) Derivative-based temporal characterization : pseudo-derivative features θ 1 (slope) and θ 2 (curvature) extracted directly from the fitted model encode local geometric structure of the BOLD signal without requiring stimulus onset information or explicit HRF modeling; (iii) Event detection : automated narrative boundary identification via slope-magnitude threshold exceedance; (iv) Multi-level connectivity framework : amplitude, slope, and curvature correlations expose qualitatively distinct coupling patterns across n = 8 subjects, with 50% of ROI pairs exhibiting sign reversals in group-mean slope connectivity relative to sliding window, and 67% exhibiting negative group-mean curvature correlation ( p < 0.001, phase-randomized permutation, FDR-corrected), consistent with complementary acceleration dynamics; mechanistic interpretation requires causal validation beyond the scope of the present work. Critically, AHKAN’s adaptive order mechanism allocates complexity based on intrinsic geometric structure rather than signal amplitude: low-variance default mode network requires highest quadratic dominance (42.6% of timepoints) due to complex spontaneous dynamics, exceeding high-variance sensory cortex (39.5%), with complexity-variance correlation r = 0.250 confirming orthogonality of these signal properties. Systematic benchmarking across six 1D function classes establishes cubic polynomial order ( r = 3 ) as the robust default, with AHKAN outperforming its nearest competitor ReLU-KAN by 53% in mean MSE (1.112 vs 1.705) across heterogeneous scenarios; a preliminary 2D extension on KA-decomposable bivariate functions shows competitive performance, with AHKAN leading on heterogeneous-smoothness functions ( f sharp ,MSE = 0.015 vs 0.016 for BSpline-KAN). Robustness analyses confirm AHKAN reconstruction stability under temporal smoothing (FWHM 0–8 s, Δ r = − 0.008 ) and superior noise resilience relative to Savitzky-Golay curvature (SG stability collapses to r = 0.063 at σ = 2.0 vs AHKAN r = 0.446 ). This architecture resolves the fundamental tension between predictive accuracy and mechanistic interpretability: AHKAN’s pseudo-derivative structure prioritizes geometric fidelity over point-wise smoothing, with the + 25 % correlation advantage in medium-variance fMRI regions arising from derivative-informed temporal feature preservation rather than amplitude fitting. With applications in derivative-constrained learning, physics-informed neural networks for PDE solving, and neuroimaging time series analysis, AHKAN establishes a framework where universal approximation guarantees and geometric interpretability are mutually reinforcing rather than competing objectives.