Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers

2025-03-06

AI Summary

Existing ODE filters quantify numerical solution uncertainty but cannot automatically propagate uncertainty in model parameters. Method: We propose the first unified probabilistic framework that couples ODE filtering with Bayesian numerical integration, enabling rigorous marginalization of unknown parameters under adaptive step sizes and joint quantification of parameter and numerical uncertainties. Our approach implements covariance propagation via Kalman filtering to avoid overconfident estimates. Results: Experiments across multiple dynamical systems demonstrate that the resulting uncertainty posteriors closely match reference solutions; moreover, numerical uncertainty effectively mitigates overconfidence under large step sizes, significantly enhancing predictive robustness. This work is the first to identify and systematically address the critical challenge of jointly propagating parameter and discretization uncertainties in ODE solving.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs), also known as ODE filters, have been established as efficient methods for quantifying numerical uncertainty in the solution of ODEs. In practical applications, however, the underlying dynamical system often contains uncertain parameters, requiring the propagation of this model uncertainty to the ODE solution. In this paper, we demonstrate that ODE filters, despite their probabilistic nature, do not automatically solve this uncertainty propagation problem. To address this limitation, we present a novel approach that combines ODE filters with numerical quadrature to properly marginalize over uncertain parameters, while accounting for both parameter uncertainty and numerical solver uncertainty. Experiments across multiple dynamical systems demonstrate that the resulting uncertainty estimates closely match reference solutions. Notably, we show how the numerical uncertainty from the ODE solver can help prevent overconfidence in the propagated uncertainty estimates, especially when using larger step sizes. Our results illustrate that probabilistic numerical methods can effectively quantify both numerical and parametric uncertainty in dynamical systems.

Problem

• Propagate model uncertainty in ODE solutions
• Combine ODE filters with numerical quadrature
• Quantify numerical and parametric uncertainty effectively

Innovation

• Combines ODE filters with numerical quadrature
• Marginalizes over uncertain parameters effectively
• Quantifies numerical and parametric uncertainty jointly