Reconstruction and Prediction of Volterra Integral Equations Driven by Gaussian Noise
Zhihao Xu, Saisai Ding, Zhikun Zhang, Xiangjun Wang
TL;DR
The paper tackles parameter reconstruction for stochastic Volterra integral equations driven by Gaussian noise and introduces an improved multi-output physics-informed neural network (DNN) framework to learn the drift parameter $\theta$ while simultaneously tracking the integral term $v(t)$. By enforcing a physics-based output condition and using adaptive residual weighting, the approach achieves accurate parameter recovery and robust long-horizon prediction across linear and nonlinear kernels, even under substantial noise. Trajectory averaging over multiple simulations provides a smoother reference for learning, and a rigorous convergence discussion links training residuals to solution and parameter accuracy. The results demonstrate strong practical potential for robust data-driven reconstruction and forecasting in stochastic integral systems, with scalable architecture choices and explicit attention to noise effects. The method offers a principled pathway for extending data-driven identification to more complex perturbations and higher-dimensional stochastic integral models.
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.