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.

翻译：积分方程在应用建模、医学成像和系统辨识等领域具有广泛应用，为求解确定性问题的提供了强大框架。尽管微分方程的参数辨识已得到广泛研究，但对积分方程，特别是随机Volterra积分方程的关注仍然有限。本研究针对高斯噪声驱动的Volterra积分方程中的参数辨识问题（亦称为方程重构问题）展开研究。我们提出了一种改进的深度神经网络框架，用于估计这些方程漂移项中的未知参数。该网络表示主要变量及其积分，通过将输出间关系纳入损失函数，提高了参数估计精度。此外，该框架不仅限于参数辨识，还能预测系统在积分区间外的行为。通过使用95%置信区间比较预测轨迹与真实轨迹，验证了预测准确性。数值实验证明了所提出的深度神经网络框架在参数辨识和预测任务中的有效性，在不同噪声水平下均表现出鲁棒性能，为随机系统建模提供了精确解决方案。