Reconstruction and Prediction of Volterra Integral Equations Driven by Gaussian Noise
This work addresses the limited focus on stochastic integral equations for applied modeling and system identification, representing an incremental improvement with a novel method for a known bottleneck.
The research tackled the parameter identification problem in stochastic Volterra integral equations driven by Gaussian noise by proposing an improved deep neural networks framework, which demonstrated robust performance under varying noise levels and provided accurate solutions for modeling stochastic systems.
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.