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

TL;DR

The paper develops Kantorovich-type Stochastic Neural Network Operators (K-SNNOs) to approximate second-order stochastic processes in the mean-square sense. It embeds stochastic neurons via Wiener-integral kernels $X_t(\omega)= \int_{\mathcal{T}} \zeta(t,s) dW_s$ and a sigmoidal density $\mathcal{L}^\sigma$ to build operators $\mathcal{X}_n(X_t,\omega)$. It proves $\mathbb{E}| \mathcal{X}_n(X_t,\omega) - X_t(\omega)|^2 \to 0$ with a rate $\mathcal{O}(n^{-\mu})$ where $\mu=\min\{\theta,\beta,2u\}$ and depends on the modulus of continuity $\mathcal{W}_2(\zeta,\delta)$. Numerical validation on a kernel-driven Gaussian process demonstrates accurate path reconstruction and a decreasing MSE with $n$, confirming both convergence and practical utility. The framework connects deterministic operator theory, stochastic analysis, and neural approximation, offering a scalable approach to learning and simulating stochastic dynamics.

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.

Figures (7)

• Figure 1: Structural diagram of the K-SNNOs defined in (\ref{['SNNO_by_S.neuron_definition']})
• Figure 2: 10 realizations of the process $X_t(\cdot)$ as defined in (\ref{['OriganX_teq.']})
• Figure 3: One realization of the approximation $(\ref{['approximation4.2']})$ corresponding to a fixed realization of $dW_s$ for $n=20.$
• Figure 4: Comparison between the 5 realizations of original stochastic process $X_t(\omega)$ (\ref{['OriganX_teq.']}) (solid lines) and its corresponding approximations via K-SNNO \ref{['approximation4.2']} (dashed lines) for fixed $n=20$.
• Figure 5: Comparison between the 5 realizations of original stochastic process $X_t(\omega)$ (\ref{['OriganX_teq.']}) (solid lines) and its corresponding approximations via K-SNNO \ref{['approximation4.2']} (dashed lines) for $n=5,10,15,....,100$.

Theorems & Definitions (14)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 2.6
• Definition 3.1
• Lemma 3.2
• proof
• Definition 3.3