1. Inferring the functional shape of ecological and evolutionary processes from time-series data can be challenging because processes are often not describable with simple equations. The dynamical coupling between variables in time series further complicates the identification of equations through model selection as the inference of a given process is contingent on the accurate depiction of all other processes.
2. We present a novel method, neural ordinary differential equations (NODEs), for learning ecological and evolutionary processes from time-series data by modelling dynamical systems as ordinary differential equations and dynamical functions with artificial neural networks (ANNs). Upon successful training, the ANNs converge to functional shapes that best describe the biological processes underlying the dynamics observed, in a way that is robust to mathematical misspecifications of the dynamical model.
3. We demonstrate NODEs in a population dynamic context and show how they can be used to infer ecological interactions, dynamical causation and equilibrium points. We tested NODEs by analysing well-understood hare and lynx time-series data, which revealed that prey–predator oscillations were mainly driven by the interspecific interaction, as well as intraspecific densitydependence, and characterised by a single equilibrium point at the centre of the oscillation.
4. Our approach is applicable to any system that can be modelled with differential equations, and particularly suitable for linking ecological, evolutionary and environmental dynamics where parametric approaches are too challenging to implement, opening new avenues for theoretical and empirical investigations.

中文翻译：

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用于生态和进化时间序列分析的神经常微分方程

1. 从时间序列数据推断生态和进化过程的功能形状可能具有挑战性，因为过程通常无法用简单的方程来描述。时间序列中变量之间的动态耦合进一步使通过模型选择确定方程变得复杂，因为给定过程的推断取决于所有其他过程的准确描述。
2. 我们提出了一种新方法，神经常微分方程 (NODE)，通过将动态系统建模为常微分方程和人工神经网络 (ANN) 的动态函数，从时间序列数据中学习生态和进化过程。成功训练后，人工神经网络会收敛到最能描述所观察到的动力学背后的生物过程的功能形状，其方式对动力学模型的数学错误规范具有鲁棒性。
3. 我们在种群动态环境中展示了节点，并展示了如何使用它们来推断生态相互作用、动态因果关系和平衡点。我们通过分析众所周知的野兔和猞猁时间序列数据测试了 NODE，结果表明，猎物-捕食者的振荡主要由种间相互作用和种内密度依赖性驱动，并以振荡中心的单个平衡点为特征.
4. 我们的方法适用于任何可以用微分方程建模的系统，特别适用于将参数化方法难以实施的生态、进化和环境动力学联系起来，为理论和实证研究开辟了新的途径。