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Brain Imaging

Brain Imaging

Brain imaging technologies provide critical insights into both the structure and function of the human brain, enabling medical professionals and researchers to diagnose, study, and treat neurological disorders or psychiatric disorders more effectively.
These technologies include computerized axial tomography (CAT or CT scans), positron-emission tomography (PET scans), magnetic resonance imaging (MRI), functional magnetic resonance imaging (fMRI), and Transcranial Magnetic Stimulation (TMS).

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

¹Faculty of Science and Arts, Department of Mathematics, Gazi University, Ankara, 06100, Turkey.

Neural Networks : the Official Journal of the International Neural Network Society

|December 4, 2025

Summary

Introducing Hermite-type Neural Network (HNN) and Hermite-Kantorovich Neural Network (HKNN) operators, this study presents derivative-aware approximations that adapt to signal curvature. These novel operators preserve differential structure and control variance for improved accuracy in various applications.

Keywords:

Derivative-informed neural operators Dynamic functional connectivity (dFC)Enhanced signal processing Hermite-type neural networks Neural network approximation

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Area of Science:

Machine Learning
Numerical Analysis
Signal Processing
Neuroimaging

Background:

Existing neural network operators often struggle to preserve differential structures like curvature and are sensitive to noise and irregular sampling.
There is a need for adaptive operators that can leverage both function values and their derivatives to improve approximation accuracy and robustness.

Purpose of the Study:

To introduce novel Hermite-type Neural Network (HNN) and Hermite-Kantorovich Neural Network (HKNN) operators that are derivative-aware.
To develop a hybrid HNN-HKNN model that dynamically adapts to signal curvature, balancing fidelity and robustness.
To provide convergence guarantees for these new NN-based operators.

Main Methods:

Development of HNN operators using localized activation functions and Taylor-like expansions to incorporate function values and derivatives.
Extension to HKNN operators for integral-based approximation, enhancing robustness.
Creation of a hybrid model that adaptively weights HNN and HKNN based on the magnitude of the second derivative (curvature).

Main Results:

HNN achieved high-fidelity approximation on smooth targets (e.g., Gaussian benchmark with RMSE 1.4×10^-4), outperforming existing methods.
The hybrid model demonstrated favorable performance in irregularity stress tests and improved amplitude stability and reliability in an fMRI study.
Numerical diagnostics confirmed the expected bias-variance trade-off, with HNN minimizing error on smooth signals and HKNN reducing high-frequency gain and sampling sensitivity.

Conclusions:

HNN and HKNN operators offer derivative-aware approximations that effectively preserve local differential structure and control variance under noise and irregular sampling.
The hybrid HNN-HKNN model provides adaptive robustness, making it suitable for complex signals with varying curvature.
These operators show significant utility in neuroimaging (fMRI) and other applications sensitive to differential information.