Kantorovich-Type Stochastic Neural Network Operators for the Mean-Square Approximation of Certain Second-Order Stochastic Processes
This work addresses the under-explored extension of neural network operators to random dynamics, providing a method for modeling stochastic signals, though it appears incremental as it builds on existing deterministic ANNOs with a stochastic adaptation.
The authors tackled the problem of approximating second-order stochastic processes by constructing Kantorovich-type Stochastic Neural Network Operators (K-SNNOs) that incorporate randomness through stochastic neurons, achieving mean-square convergence with quantitative error estimates and demonstrating accurate sample path reconstruction and rapid MSE decay in simulations.
Artificial neural network operators (ANNOs) have been widely used for approximating deterministic input-output functions; however, their extension to random dynamics remains comparatively unexplored. In this paper, we construct a new class of \textbf{Kantorovich-type Stochastic Neural Network Operators (K-SNNOs)} in which randomness is incorporated not at the coefficient level, but through \textbf{stochastic neurons} driven by stochastic integrators. This framework enables the operator to inherit the probabilistic structure of the underlying process, making it suitable for modeling and approximating stochastic signals. We establish mean-square convergence of K-SNNOs to the target stochastic process and derive quantitative error estimates expressing the rate of approximation in terms of the modulus of continuity. Numerical simulations further validate the theoretical results by demonstrating accurate reconstruction of sample paths and rapid decay of the mean square error (MSE). Graphical results, including sample-wise approximations and empirical MSE behaviour, illustrate the robustness and effectiveness of the proposed stochastic-neuron-based operator.