We introduce CLINE (Computational Learning and Identification of Nullclines), a neural network-based method that uncovers the hidden structure of nullclines from oscillatory time series data. Unlike traditional approaches aiming at direct prediction of system dynamics, CLINE identifies static geometric features of the phase space that encode the (non)linear relationships between state variables. It overcomes challenges such as multiple time scales and strong nonlinearities while producing interpretable results convertible into symbolic differential equations. We validate CLINE on various oscillatory systems, showcasing its effectiveness.

翻译：我们提出了CLINE（零斜线计算学习与识别方法），这是一种基于神经网络的方法，能够从振荡时间序列数据中揭示零斜线的隐藏结构。与旨在直接预测系统动力学的传统方法不同，CLINE识别相空间中编码状态变量之间（非）线性关系的静态几何特征。它克服了诸如多时间尺度和强非线性等挑战，同时产生可解释的结果，并可转化为符号微分方程。我们在多种振荡系统上验证了CLINE，展示了其有效性。