Our recent work lays out a general framework for inferring information about the parameters and associated dynamics of a differential equation model from a discrete set of data points collected from the system being modeled. Rigorous mathematical results have justified this approach and have identified some common features that arise for certain classes of integrable models. In this work we present a thorough numerical investigation that shows that several of these core features extend to a paradigmatic linear-in-parameters model, the Lotka-Volterra (LV) system, which we consider in the conservative case as well as under the addition of terms that perturb the system away from this regime. A central construct for this analysis is a concise representation of parameter features in the data space that we call the $P_n$-diagram, which is particularly useful for visualization of results for low-dimensional (small $n$) systems. Our work also exposes some new properties related to non-uniqueness that arise for these LV systems, with non-uniqueness manifesting as a multi-layered structure in the associated $P_2$-diagrams.
翻译:我们最近的工作为从正在建模的系统所收集的一组离散数据点中推断关于差异方程模型参数和相关动态的信息提供了一个总体框架。 严格的数学结果证明这种做法是合理的,并确定了某些类别不可变模型产生的一些共同特征。 在这项工作中,我们提出一个彻底的数字调查,表明这些核心特征中的若干特征延伸到一个范式线性参数模型,即Lotka-Volterra(LV)系统,我们在保守的案例中以及在使系统脱离该系统的附加条件下,认为它属于Lotka-Volterra(LV)系统。这一分析的中心结构简洁地展示了我们称之为$P_n-diagram的数据空间中的参数特征,这对低度(小美元)系统结果的可视化特别有用。 我们的工作还暴露了一些与这些LV系统产生的非独有性有关的新特性,在相关的 $P_2美元-diagram中以多层结构的形式展示了非内在性。