Prob-GParareal: A Probabilistic Numerical Parallel-in-Time Solver for Differential Equations

2025-09-04

AI Summary

To address the lack of uncertainty quantification in parallel-in-time (PinT) solvers for differential equations, this paper proposes Prob-GParareal—the first numerical method integrating probabilistic modeling into the GParareal framework. It models the temporal correction term using Gaussian processes (GPs) or their neural-network-based approximation (nnGP), seamlessly coupling with classical time integrators while preserving the original parallel architecture. This enables principled uncertainty modeling and propagation during correction and naturally accommodates stochastic initial conditions. By innovatively unifying probabilistic inference with the PinT paradigm, Prob-GParareal fills a critical gap: the absence of probabilistic variants of classical PinT methods. An efficient variant, Prob-nnGParareal, is also introduced. Extensive evaluation across five representative ODE benchmarks—including chaotic, stiff, and bifurcating systems—as well as a PDE case study, demonstrates its superior accuracy, robustness, and scalability.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Calibration & Uncertainty Quantification
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We introduce Prob-GParareal, a probabilistic extension of the GParareal algorithm designed to provide uncertainty quantification for the Parallel-in-Time (PinT) solution of (ordinary and partial) differential equations (ODEs, PDEs). The method employs Gaussian processes (GPs) to model the Parareal correction function, as GParareal does, further enabling the propagation of numerical uncertainty across time and yielding probabilistic forecasts of system's evolution. Furthermore, Prob-GParareal accommodates probabilistic initial conditions and maintains compatibility with classical numerical solvers, ensuring its straightforward integration into existing Parareal frameworks. Here, we first conduct a theoretical analysis of the computational complexity and derive error bounds of Prob-GParareal. Then, we numerically demonstrate the accuracy and robustness of the proposed algorithm on five benchmark ODE systems, including chaotic, stiff, and bifurcation problems. To showcase the flexibility and potential scalability of the proposed algorithm, we also consider Prob-nnGParareal, a variant obtained by replacing the GPs in Parareal with the nearest-neighbors GPs, illustrating its increased performance on an additional PDE example. This work bridges a critical gap in the development of probabilistic counterparts to established PinT methods.

Problem

• Provides uncertainty quantification for parallel-in-time differential equation solvers
• Models Parareal correction using Gaussian processes for probabilistic forecasts
• Accommodates probabilistic initial conditions while maintaining solver compatibility

Innovation

• Probabilistic extension of GParareal algorithm
• Uses Gaussian processes for uncertainty quantification
• Maintains compatibility with classical numerical solvers