We develop a Bayesian approach to estimate the parameters of ordinary differential equations (ODE) from the observed noisy data. Our method does not need to solve ODE directly. We replace the ODE constraint with a probability expression and combine it with the nonparametric data fitting procedure into a joint likelihood framework. One advantage of the proposed method is that for some ODE systems, one can obtain closed form conditional posterior distributions for all variables which substantially reduce the computational cost and facilitate the convergence process. An efficient Riemann manifold based hybrid Monte Carlo scheme is implemented to generate samples for variables whose conditional posterior distribution cannot be written in terms of closed form. Our approach can be applied to situations where the state variables are only partially observed. The usefulness of the proposed method is demonstrated through applications to both simulated and real data.

中文翻译：

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估计常微分方程参数的贝叶斯方法

我们开发了一种贝叶斯方法，可从观察到的噪声数据估计常微分方程（ODE）的参数。我们的方法不需要直接求解ODE。我们将ODE约束替换为概率表达式，并将其与非参数数据拟合程序组合到联合似然框架中。所提出的方法的一个优点是，对于某些ODE系统，可以为所有变量获得封闭形式的条件后验分布，从而大大降低了计算成本并促进了收敛过程。实现了一种有效的基于黎曼流形的混合蒙特卡洛方案，以生成变量变量的样本，这些变量的条件后验分布不能用闭合形式表示。我们的方法可以应用于仅部分观察状态变量的情况。通过将其应用于模拟和真实数据，证明了该方法的有效性。