aiqbench analysis
This research resolves a critical trade-off in probabilistic ODE solvers by achieving both linear scalability and stability for stiff, high-dimensional problems. The authors introduce matrix-free update steps and iterative re-linearization techniques that transform these solvers into fully implicit methods, offering a robust alternative to existing approaches that sacrifice one property for the other. For practitioners in scientific computing and uncertainty quantification, this advancement enables more efficient and reliable simulations in complex dynamical systems.
Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs
arXiv:2606.08203v1 Announce Type: cross Abstract: Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs) have been established as a flexible and efficient simulation framework with built-in numerical uncertainty quantification. However, problems that are both stiff and high-dimensional remain a challenge, as current methods are either stable and have cubic cost in the ODE dimension, or scale linearly at the expense of stability. In this paper, we close this gap and develop probabilistic ODE solvers that are both stable and scalable. We propose two complementary strategies. First, we develop a matrix-free update step that uses Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to enable linear scaling, all while retaining stability. Second, we propose iterative re-linearization to further improve stability without sacrificing scalability, turning probabilistic ODE solvers into fully implicit methods. We evaluate the proposed approaches on a range of stiff and high-dimensional problems and demonstrate improved stability and scalability over established probabilistic solvers.
Abstract:Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs) have been established as a flexible and efficient simulation framework with built-in numerical uncertainty quantification. However, problems that are both stiff and high-dimensional remain a challenge, as current methods are either stable and have cubic cost in the ODE dimension, or scale linearly at the expense of stability. In this paper, we close this gap and develop probabilistic ODE solvers that are both stable and scalable. We propose two complementary strategies. First, we develop a matrix-free update step that uses Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to enable linear scaling, all while retaining stability. Second, we propose iterative re-linearization to further improve stability without sacrificing scalability, turning probabilistic ODE solvers into fully implicit methods. We evaluate the proposed approaches on a range of stiff and high-dimensional problems and demonstrate improved stability and scalability over established probabilistic solvers.
Numerical Analysis (math.NA); Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2606.08203 [math.NA]
(or arXiv:2606.08203v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2606.08203 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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