Hermite-type neural network operators: derivative-informed frameworks for functional neuroimaging and signal...

This paper introduces Hermite-type Neural Network (HNN) operators and their integral-based extension, Hermite-Kantorovich Neural Network (HKNN) operators. These novel multivariate, derivative-aware operators utilize localized activation functions and finite Taylor-like expansions to explicitly incorporate function values and their derivatives, effectively capturing local dynamics and curvature. Our motivation is to build a derivative-aware hybrid NN operator that jointly exploits function values, derivatives, and derivative-informed weights together with local integral surrogates, so that the approximation adapts to the signal's differential structure (e.g., slope/curvature cues and change points) while controlling variance under noise and irregular sampling. A key innovation is a hybrid HNN-HKNN model that dynamically adapts to signal curvature. This model assigns weights based on the magnitude of the second derivative: as curvature |f″| increases, the model shifts toward HKNN to gain robustness via local integral averaging; when curvature is small, it favors HNN to preserve differential structure and sharp onsets. The goal is not merely shape preservation, but the explicit preservation and use of differential information, accompanied by convergence guarantees formulated for NN-based operators rather than classical polynomial or spline bases. Numerical experiments show that the HNN achieves high-fidelity approximation on smooth targets: on the Gaussian benchmark (n=10), HNN attains RMSE 1.4×10^-4 and MAE 1.2×10^-4, outperforming discrete, Kantorovich, and Durrmeyer-type neural network operators while preserving Sobolev norms and curvature sign. A diagnostic suite (H¹/H², curvature-weighted L², grid-shift, frequency response, noise gain) shows the expected bias variance trade-off: HNN minimizes error on smooth targets; HKNN lowers high-frequency gain and sampling-phase sensitivity. An irregularity stress test places HKNN/Hybrid on a favorable amplitude phase connectivity Pareto frontier versus standard denoisers. In a two-subject fMRI study (BOLD ROIs), the Hybrid improves amplitude stability and reliability while preserving timing and network structure. Median temporal SNR rises by 13.7 % (p001) and 24.6 % (p002); split-half reliability increases by 0.114 and 0.288. Dynamic connectivity changes are stronger for p001 (edges with r increase: 65.8 %; mean |Δr| 38.6 %) and moderate for p002 (52.4 %; 12.6 %). Phase analyses (dynamic and frequency-resolved PLV with Rayleigh tests) indicate tighter synchrony without systematic lag shifts. Overall, HNN/HKNN provide derivative-aware approximations that preserve local differential structure while controlling variance under noise and irregular sampling, with direct utility for neuroimaging and other derivative-sensitive signals.