Ordinary differential equations (ODEs) can provide mechanistic models of temporally local changes of processes, where parameters are often informed by external knowledge. While ODEs are popular in systems modeling, they are less established for statistical modeling of longitudinal cohort data, e.g., in a clinical setting. Yet, modeling of local changes could also be attractive for assessing the trajectory of an individual in a cohort in the immediate future given its current status, where ODE parameters could be informed by further characteristics of the individual. However, several hurdles so far limit such use of ODEs, as compared to regression-based function fitting approaches. The potentially higher level of noise in cohort data might be detrimental to ODEs, as the shape of the ODE solution heavily depends on the initial value. In addition, larger numbers of variables multiply such problems and might be difficult to handle for ODEs. To address this, we propose to use each observation in the course of time as the initial value to obtain multiple local ODE solutions and build a combined estimator of the underlying dynamics. Neural networks are used for obtaining a low-dimensional latent space for dynamic modeling from a potentially large number of variables, and for obtaining patient-specific ODE parameters from baseline variables. Simultaneous identification of dynamic models and of a latent space is enabled by recently developed differentiable programming techniques. We illustrate the proposed approach in an application with spinal muscular atrophy patients and a corresponding simulation study. In particular, modeling of local changes in health status at any point in time is contrasted to the interpretation of functions obtained from a global regression. This more generally highlights how different application settings might demand different modeling strategies.

翻译：常微分方程可为过程的局部时间变化提供机理模型，其参数通常由外部知识确定。尽管微分方程在系统建模中广受欢迎，但在纵向队列数据（如临床环境）的统计建模中尚未广泛应用。然而，鉴于个体当前状态，建模局部变化对于评估队列中个体近期轨迹同样具有吸引力，此时微分方程参数可由个体的进一步特征确定。但与基于回归的函数拟合方法相比，目前若干障碍限制了微分方程在此类场景的应用：队列数据中潜在的高噪声水平可能对微分方程产生不利影响，因为微分方程解的形状严重依赖初始值；此外，变量数量增加会加剧此类问题，微分方程可能难以处理。为此，我们提出将时间进程中的每个观测值作为初始值，获取多个局部微分方程解，并构建潜在动态的联合估计器。通过神经网络从潜在的大量变量中获取低维潜在空间以进行动态建模，并从基线变量中获取患者特异性微分方程参数。利用最新发展的可微编程技术，可同时识别动态模型与潜在空间。我们通过脊髓性肌萎缩症患者应用实例及相应模拟研究验证了所提方法。特别地，任何时间点健康状况局部变化的建模与全局回归所得函数的解释形成对比，这更广泛地揭示了不同应用场景可能需求不同建模策略。