Kantorovich-Type Stochastic Neural Network Operators for the Mean-Square Approximation of Certain Second-Order Stochastic Processes

TL;DR

This paper introduces a new class of stochastic neural network operators that incorporate randomness through stochastic neurons, enabling effective mean-square approximation of certain second-order stochastic processes with proven convergence and validated by numerical simulations.

Contribution

The paper develops Kantorovich-type stochastic neural network operators with stochastic neurons, providing the first framework for approximating stochastic processes with proven mean-square convergence.

Findings

Proven mean-square convergence of K-SNNOs to stochastic processes.

Quantitative error estimates based on the modulus of continuity.

Numerical simulations demonstrate accurate path reconstruction and rapid MSE decay.

Abstract

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.