# Reconstruction and Prediction of Volterra Integral Equations Driven by Gaussian Noise

arXiv: https://arxiv.org/abs/2506.00933 · HTML: https://mlsanity.com/paper/2506.00933/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 34 references that are in the map.
- First posted on arXiv 1 Jun 2025.

## Builds on (most cited first, 8 of 8)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [Separable nonlinear least-squares parameter estimation for complex dynamic systems](https://mlsanity.com/paper/1908.03717/) (2019)

## Abstract (arXiv)

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.