提出稳定且可扩展的高维刚性微分方程概率求解器
Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs
- 用雅可比-向量积实现无矩阵更新，线性扩容
- 迭代重线性化使方法具备完全隐式稳定性
- 适用于高维刚性系统，兼顾精度与效率
基于滤波的概率微分方程求解器已证明在模拟中兼具灵活性与数值不确定性量化能力。然而，高维且刚性的系统仍是挑战：现有方法或保持稳定但计算复杂度为立方级，或线性扩展却牺牲稳定性。本文填补该空白，提出兼具稳定与可扩展性的概率微分方程求解器。首先，设计无矩阵更新步骤，利用雅可比-向量积、迭代线性求解器及随机协方差估计，实现线性复杂度并维持稳定性；其次，提出迭代重线性化策略，在不损失可扩展性的前提下提升稳定性，使求解器成为全隐式方法。我们在多种刚性高维问题上评估所提方法，结果表明其在稳定性和可扩展性方面优于现有概率求解器。
原文摘要 · Abstract (English)
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.
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