Most ordinary differential equation (ODE) models used to describe biological or physical systems must be solved approximately using numerical methods. Perniciously, even those solvers which seem sufficiently accurate for the forward problem, i.e., for obtaining an accurate simulation, may not be sufficiently accurate for the inverse problem, i.e., for inferring the model parameters from data. We show that for both fixed step and adaptive step ODE solvers, solving the forward problem with insufficient accuracy can distort likelihood surfaces, which may become jagged, causing inference algorithms to get stuck in local "phantom" optima. We demonstrate that biases in inference arising from numerical approximation of ODEs are potentially most severe in systems involving low noise and rapid nonlinear dynamics. We reanalyze an ODE changepoint model previously fit to the COVID-19 outbreak in Germany and show the effect of the step size on simulation and inference results. We then fit a more complicated rainfall-runoff model to hydrological data and illustrate the importance of tuning solver tolerances to avoid distorted likelihood surfaces. Our results indicate that when performing inference for ODE model parameters, adaptive step size solver tolerances must be set cautiously and likelihood surfaces should be inspected for characteristic signs of numerical issues.

翻译：大多数用于描述生物或物理系统的常微分方程（ODE）模型必须通过数值方法近似求解。值得注意的是，即使那些在前向问题（即获取精确模拟）中看似足够精确的求解器，在逆问题（即从数据中推断模型参数）中可能仍不够精确。我们证明，对于固定步长和自适应步长两种ODE求解器，若前向问题求解精度不足，会扭曲似然面，使其变得崎岖不平，导致推断算法陷入局部“伪”最优解。我们表明，由ODE数值近似引起的推断偏差在低噪声和快速非线性动力学系统中可能最为严重。我们重新分析了先前拟合德国COVID-19疫情的ODE变点模型，展示了步长对模拟和推断结果的影响。随后，我们将更复杂的降雨径流模型拟合至水文数据，阐明了调整求解器容差以避免似然面扭曲的重要性。我们的结果表明，在对ODE模型参数进行推断时，必须谨慎设置自适应步长求解器的容差，并应检查似然面是否存在数值问题的特征性迹象。