Parameter inference for ordinary differential equations (ODEs) is of fundamental importance in many scientific applications. While ODE solutions are typically approximated by 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 minima in the likelihood function -- a problem which existing probabilistic solvers have yet to resolve. Here, we show that a Bayesian filtering paradigm for probabilistic ODE solution can dramatically reduce sensitivity to parameters by learning from the noisy ODE observations in a data-adaptive manner. Our method is applicable to ODEs with partially unobserved components and with arbitrary non-Gaussian noise. Several examples demonstrate that it is more accurate than existing probabilistic ODE solvers, and even in some cases than the exact ODE likelihood.

翻译：常微分方程的参数推断在许多科学应用中具有根本重要性。虽然常微分方程的解通常由确定性算法近似，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，许多常微分方程系统对其参数值高度敏感，这导致似然函数中出现深度局部极小值——现有概率求解器尚未解决这一问题。在此，我们证明，用于概率常微分方程解的贝叶斯滤波范式能够通过以数据自适应方式从含噪常微分方程观测中学习，显著降低对参数的敏感性。我们的方法适用于具有部分未观测分量和任意非高斯噪声的常微分方程。多个示例表明，该方法比现有概率常微分方程求解器更精确，甚至在某些情况下优于精确常微分方程似然。