Reconstruction and Prediction of Volterra Integral Equations Driven by Gaussian Noise

TL;DR

This paper introduces a deep neural network framework for accurately identifying parameters and predicting the behavior of stochastic Volterra integral equations driven by Gaussian noise, with robust performance demonstrated through numerical experiments.

Contribution

It proposes an improved neural network approach that enhances parameter estimation and prediction accuracy for stochastic Volterra integral equations, addressing a limited area of prior research.

Findings

Effective parameter estimation under varying noise levels

Accurate system behavior prediction outside the data interval

Robust performance demonstrated through numerical experiments

Abstract

Integral equations are widely used in fields such as applied modeling, medical imaging, and system identification, providing a powerful framework for solving deterministic problems. While parameter identification for differential equations has been extensively studied, the focus on integral equations, particularly stochastic Volterra integral equations, remains limited. This research addresses the parameter identification problem, also known as the equation reconstruction problem, in Volterra integral equations driven by Gaussian noise. We propose an improved deep neural networks framework for estimating unknown parameters in the drift term of these equations. The network represents the primary variables and their integrals, enhancing parameter estimation accuracy by incorporating inter-output relationships into the loss function. Additionally, the framework extends beyond parameter identification to predict the system's behavior outside the integration interval. Prediction accuracy is validated by comparing predicted and true trajectories using a 95% confidence interval. Numerical experiments demonstrate the effectiveness of the proposed deep neural networks framework in both parameter identification and prediction tasks, showing robust performance under varying noise levels and providing accurate solutions for modeling stochastic systems.