Estimating the parameters of ordinary differential equations (ODEs) is of fundamental importance in many scientific applications. While ODEs are typically approximated with deterministic algorithms, new research on probabilistic solvers indicates that they produce more reliable parameter estimates by better accounting for numerical errors. However, many ODE systems are highly sensitive to their parameter values. This produces deep local maxima in the likelihood function -- a problem which existing probabilistic solvers have yet to resolve. Here we present a novel probabilistic ODE likelihood approximation, DALTON, which can dramatically reduce parameter sensitivity by learning from noisy ODE measurements in a data-adaptive manner. Our approximation scales linearly in both ODE variables and time discretization points, and is applicable to ODEs with both partially-unobserved components and non-Gaussian measurement models. Several examples demonstrate that DALTON produces more accurate parameter estimates via numerical optimization than existing probabilistic ODE solvers, and even in some cases than the exact ODE likelihood itself.

翻译：常微分方程（ODEs）的参数估计在许多科学应用中具有基础重要性。虽然ODEs通常使用确定性算法进行近似，但关于概率求解器的新研究表明，通过更好地考虑数值误差，这些求解器能产生更可靠的参数估计。然而，许多ODE系统对其参数值高度敏感，这会导致似然函数中出现深局部最大值——现有概率求解器尚未解决这一问题。本文提出了一种新颖的概率ODE似然逼近方法DALTON，该方法通过以数据自适应方式从带噪ODE测量中学习，能够显著降低参数敏感性。我们的逼近在ODE变量和时间离散点两方面均呈线性扩展，并可应用于包含部分未观测分量和非高斯测量模型的ODEs。多个实例表明，与现有概率ODE求解器相比，DALTON通过数值优化产生的参数估计更精确，甚至在某些情况下优于精确ODE似然本身。