🤖 AI Summary
Existing probabilistic ODE solvers struggle to simultaneously achieve numerical stability and scalability when applied to stiff, high-dimensional problems. This work proposes two complementary strategies to address this challenge: first, a matrix-free update mechanism leveraging Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to attain linear computational complexity while ensuring numerical stability; second, an iterative re-linearization scheme that reformulates the solver into a fully implicit form, further enhancing stability without compromising scalability. The resulting method constitutes the first probabilistic ODE solver that combines high stability with linear scalability, demonstrating substantial improvements over state-of-the-art approaches across multiple benchmark stiff, high-dimensional ODE systems.
📝 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.