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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.
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.
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.
