Revolutionizing Fractional Calculus with Neural Networks: Voronovskaya-Damasclin Theory for Next-Generation AI Systems
It provides foundational mathematical insights and deployable engineering solutions for complex system modeling and signal processing, but is incremental as it extends classical approximation theory to fractional calculus.
This work tackles the problem of establishing rigorous convergence rates for neural network operators in fractional calculus, demonstrating that Kantorovich operators achieve o(n^{-β(N-ε)}) convergence rates and basic operators show O(n^{-βN}) error decay, with deep networks achieving O(L^{-β(N-ε)}) approximation bounds.
This work introduces rigorous convergence rates for neural network operators activated by symmetrized and perturbed hyperbolic tangent functions, utilizing novel Voronovskaya-Damasclin asymptotic expansions. We analyze basic, Kantorovich, and quadrature-type operators over infinite domains, extending classical approximation theory to fractional calculus via Caputo derivatives. Key innovations include parameterized activation functions with asymmetry control, symmetrized density operators, and fractional Taylor expansions for error analysis. The main theorem demonstrates that Kantorovich operators achieve \(o(n^{-β(N-\varepsilon)})\) convergence rates, while basic operators exhibit \(\mathcal{O}(n^{-βN})\) error decay. For deep networks, we prove \(\mathcal{O}(L^{-β(N-\varepsilon)})\) approximation bounds. Stability results under parameter perturbations highlight operator robustness. By integrating neural approximation theory with fractional calculus, this work provides foundational mathematical insights and deployable engineering solutions, with potential applications in complex system modeling and signal processing.