Reconstruction and Prediction of Volterra Integral Equations Driven by Gaussian Noise

2025-06-01

AI Summary

This work addresses parameter identification (i.e., equation reconstruction) and out-of-interval trajectory prediction for stochastic Volterra integral equations corrupted by Gaussian noise. We propose the first deep neural network–based joint framework: a novel architecture explicitly models both the state variable and its integral term; Volterra-kernel–defined integral constraints are embedded into the loss function to enable simultaneous optimization of parameter estimation and extrapolatory prediction. Prediction reliability is quantified via 95% confidence intervals. Numerical experiments demonstrate that the method achieves high-accuracy parameter identification (mean error < 2.3%) and robust trajectory prediction across multiple noise levels—substantially outperforming conventional approaches—while exhibiting strong robustness and generalization capability.

Technology Category

Intelligent Robots: State EstimationReasoning under Uncertainty: Stochastic OptimizationMachine Learning: Calibration & Uncertainty Quantification

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Identify parameters in stochastic Volterra integral equations
• Predict system behavior beyond integration intervals
• Enhance accuracy using deep neural networks

Innovation

• Improved deep neural networks for parameter estimation
• Incorporates inter-output relationships in loss function
• Predicts system behavior beyond integration interval