Constructive Approximation of Random Process via Stochastic Interpolation Neural Network Operators

TL;DR

The paper develops stochastic interpolation neural network operators (SINNOs) that interpolate stochastic processes using random coefficients activated by a sigmoidal function. It establishes $L^2$-boundedness, interpolation at grid points, and both mean-square and path-wise convergence rates governed by the modulus of continuity $\mathcal{W}(X_t,h)$, including Hölder-type rates. The SINNO framework is validated numerically on the Ornstein–Uhlenbeck process and applied to real COVID-19 time-series data, demonstrating consistent MSE decay with increasing node count and robust hold-out performance across multiple countries. Overall, SINNOs offer a rigorous, operator-based approach to stochastic data approximation with practical relevance for time-series prediction and uncertainty quantification. Future work suggests connections to diffusion-type models and extensions to higher-dimensional stochastic processes.

Abstract

In this paper, we construct a class of stochastic interpolation neural network operators (SINNOs) with random coefficients activated by sigmoidal functions. We establish their boundedness, interpolation accuracy, and approximation capabilities in the mean square sense, in probability, as well as path-wise within the space of second-order stochastic (random) processes \( L^2(Ω, \mathcal{F},\mathbb{P}) \). Additionally, we provide quantitative error estimates using the modulus of continuity of the processes. These results highlight the effectiveness of SINNOs for approximating stochastic processes with potential applications in COVID-19 case prediction.

Figures (10)

• Figure 1: Sigmoidal ramp function $\eta_R$.
• Figure 2: Activation function $\varphi_R$ corresponding to the ramp function $\eta_R$.
• Figure 3: Structure of the above-defined SINNOs $\mathcal{S}_n(X_t,t)$.
• Figure 4: Approximation of the O-U process using SINNOs for $n = 5, 10, 20, 50$. The black curve represents the exact path, while colored curves denote the SINNOs interpolation.
• Figure 5: Uniform mean square error (MSE) vs. $n$ for a single realization of the O-U process.

Theorems & Definitions (28)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4: Discrete absolute moments of order $\alpha$
• Lemma 2.5
• Lemma 2.6: M5
• proof
• Definition 2.7: Mean Square Convergence
• Definition 3.1
• Theorem 3.2: Mean-square boundedness of SINNOs