A Bayesian Collocation Integral Method for Parameter Estimation in Ordinary Differential Equations

Abstract–<b>Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on basis functions or Gaussian processes to approximate the ODE solution and its derivatives. Due to the sensitivity of the ODE solution to its derivatives, these methods can be hindered by estimation error, especially when only sparse time-course observations are available. We present a Bayesian collocation framework that operates on the integrated form of the ODEs and also avoids the expensive use of numerical solvers. Our methodology has the capability to handle general nonlinear ODE systems. We demonstrate the accuracy of the proposed method through simulation studies, where the estimated parameters and recovered system trajectories are compared with other recent methods. A real data example is also provided.</b>

摘要：从带噪观测数据中推断常微分方程（ordinary differential equations，ODEs）的参数，是诸多科学领域的核心研究问题之一。目前，大多数无需进行数值积分的参数估计方法，往往依托基函数或高斯过程来近似ODE的解及其导数。由于ODE的解对其导数具有高度敏感性，这类方法极易受估计误差的干扰，尤其在仅能获取稀疏时序观测数据的场景下，这一问题更为突出。本文提出一种基于ODE积分形式的贝叶斯配点框架，该框架同时规避了对数值求解器的高成本调用。所提方法可适用于通用非线性ODE系统。本文通过仿真实验验证了该方法的准确性，将估计得到的参数与恢复的系统轨迹与其他近年提出的同类方法开展对比分析。此外还提供了一个真实数据应用案例。