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.

翻译：基于滤波的常微分方程概率数值求解器已被确立为一种灵活高效的仿真框架，并具备内置的数值不确定性量化能力。然而，同时具有刚性和高维特性的问题仍构成挑战，因为现有方法要么稳定但计算代价与常微分方程维度呈三次方关系，要么线性可扩展但牺牲了稳定性。本文弥合了这一差距，开发了兼具稳定性和可扩展性的概率常微分方程求解器。我们提出两种互补策略：首先，设计了一种无矩阵更新步骤，通过使用雅可比向量积、迭代线性求解器和随机协方差估计实现线性扩展，同时保持稳定性；其次，提出迭代线性化方法来进一步改善稳定性而不牺牲可扩展性，将概率常微分方程求解器转化为完全隐式方法。我们在多种刚性和高维问题上评估了所提出的方法，结果表明相较于现有概率求解器在稳定性和可扩展性上均有显著提升。