Ordinary differential equation (ODE) is a mathematical model used in many application areas such as climatology, bioinformatics, and chemical engineering with its intuitive appeal to modeling. Despite ODE's wide usage in modeling, frequent absence of their analytic solutions makes it difficult to estimate ODE parameters from the data, especially when the model has lots of variables and parameters. This paper proposes a Bayesian ODE parameter estimating algorithm which is fast and accurate even for models with many parameters. The proposed method approximates an ODE model with a state-space model based on equations of a numeric solver. It allows fast estimation by avoiding computations of a whole numerical solution in the likelihood. The posterior is obtained by a variational Bayes method, more specifically, the approximate Riemannian conjugate gradient method (Honkela et al. 2010), which avoids samplings based on Markov chain Monte Carlo (MCMC). In simulation studies, we compared the speed and performance of the proposed method with existing methods. The proposed method showed the best performance in the reproduction of the true ODE curve with strong stability as well as the fastest computation, especially in a large model with more than 30 parameters. As a real-world data application, a SIR model with time-varying parameters was fitted to the COVID-19 data. Taking advantage of our proposed algorithm, 30 parameters were adequately fitted for each country.

翻译：普通普通方程式 (ODE) 是一个数学模型, 在许多应用领域使用, 如气候学、生物信息学和化学工程等, 以及其直观的建模。 尽管 ODE在建模中广泛使用, 经常缺乏分析解决方案, 这使得很难从数据中估计ODE参数, 特别是当模型有许多变量和参数时。 本文建议采用贝叶西亚的ODE参数估计算法, 即使在模型有许多参数的情况下也是快速和准确的。 拟议的方法与基于数字解算器方程式的州- 空间模型相近。 它通过避免计算整个数字解决方案的可能性而得以快速估算。 峰值通过变异的 Bayes 方法获得, 更具体地说, 近似于 里曼尼的 conjugate 梯度方法( Honkela 等人, 2010), 避免根据Markov 链 Monte Carlo (MC ) 进行抽样。 在模拟研究中, 我们比较了拟议方法的速度和性与现有方法相比, 拟议的方法的进度和性方法的性模型显示真正的 DS- 19 参数的复制最佳性模型, 的精确度比S- 30 的参数的精确度为S- 数据, 和最精确的精确的比 数据计算, 和最精确的精确的精确的精确的比 的比 的每个数据是 的 的 的 和最精确的比 的 的 的 的精确的计算。