AI 中文总结
研究针对ODE模型数值解的离散化误差，提出基于集合卡尔曼滤波的贝叶斯推断框架，引入相关马尔可夫先验并通过两类数值实验验证其有效性。
AI 中文摘要
我们提出一种贝叶斯框架，用于基于观测数据量化常微分方程（ODE）模型数值解中的离散化误差。将离散化误差建模为随机变量，其均值（称为离散化误差均值）由观测数据推断得到。通过对离散化误差均值的时间演化引入马尔可夫先验，我们将该问题构建为具有线性高斯观测过程的状态空间模型，可通过集合卡尔曼滤波（Ensemble Kalman Filter）实现高效推断。我们还提出了一种由经典离散化误差分析启发的特定形式马尔可夫先验，其中全局误差由局部误差累积而成；该先验依赖于数值求解器的步长，且我们证明了当步长趋于零时，该先验在概率意义下的收敛速率。针对单摆系统和FitzHugh-Nagumo模型的数值实验验证了所提方法的有效性。
英文摘要
We propose a Bayesian framework to quantify discretization errors in numerical solutions of ODE models based on observational data. The discretization error is modeled as a random variable, and its mean-referred to as the discretization error mean-is inferred from the observations. By introducing a Markov prior on the temporal evolution of the discretization error mean, we formulate the problem as a state-space model with a linear Gaussian observation process, which enables efficient inference via the Ensemble Kalman Filter. We also propose a specific form of a Markov prior motivated by classical discretization error analysis, in which global errors accumulate from local errors. The proposed prior depends on the step size of a numerical solver, and we establish its convergence rate in probability as the step size tends to zero. Numerical experiments on the pendulum system and the FitzHugh-Nagumo model demonstrate the effectiveness of the proposed approach.
CommentsTo be published in the proceedings of the Second International Conference on Probabilistic Numerics (ProbNum 2026), Finland, 2026
