用深度神经网络同时求解随机积分方程的参数与未来轨迹
Reconstruction and Prediction of Volterra Integral Equations Driven by Gaussian Noise
- 构建含积分关系的神经网络，通过输出间约束提升参数估计精度
- 在不同噪声水平下均实现高精度参数识别与轨迹预测
- 适用于需要建模随机动态系统的科研与工程场景
积分方程广泛应用于应用建模、医学成像和系统辨识等领域，为求解确定性问题提供强大框架。尽管微分方程的参数识别已得到广泛研究，但对积分方程，特别是由高斯噪声驱动的随机伏尔泰拉积分方程的研究仍有限。本文针对此类方程的参数识别问题（也称方程重构问题）提出改进的深度神经网络框架，用于估计漂移项中的未知参数。该网络同时表示主变量及其积分，并通过将输出间关系引入损失函数来提升参数估计精度。此外，该框架还拓展至积分区间外的系统行为预测，通过95%置信区间对比预测轨迹与真实轨迹验证预测准确性。数值实验表明，所提深度神经网络框架在参数识别与预测任务中均表现有效，且在不同噪声水平下具备鲁棒性能，可为随机系统建模提供准确解。
原文摘要 · Abstract (English)
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.
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