用神经微分方程从数据中学习并预测跨分岔的系统演化。
Neural Ordinary Differential Equations for Learning and Extrapolating System Dynamics Across Bifurcations
- 基于神经微分方程学习参数依赖的向量场，建模连续动态。
- 在训练数据外的参数区域仍能准确预测分岔行为。
- 适用于混沌、周期倍增和全局分岔等复杂动力系统。
预测系统在分岔附近及跨分岔时的行为对识别动力系统的潜在转变至关重要。尽管机器学习已用于从数据中学习临界转变和分岔结构，但多数研究局限于离散时间方法和局部分岔。为此，本文采用神经微分方程（Neural ODEs），构建了一个数据驱动的系统动力学学习框架。结果表明，神经微分方程可直接从时间序列数据中恢复底层分岔结构，通过学习参数依赖的向量场实现。值得注意的是，该方法可在训练数据覆盖的参数范围之外仍成功预测分岔。我们在三个测试案例上验证了该方法：洛伦兹系统从非混沌到混沌的转变，罗素系统从混沌到周期倍增的演化，以及一个展示全局分岔导致崩溃的捕食者-猎物模型。
原文摘要 · Abstract (English)
Forecasting system behaviour near and across bifurcations is crucial for identifying potential shifts in dynamical systems. While machine learning has recently been used to learn critical transitions and bifurcation structures from data, most studies remain limited as they exclusively focus on discrete-time methods and local bifurcations. To address these limitations, we use Neural Ordinary Differential Equations which provide a data-driven framework for learning system dynamics. Our results show that Neural Ordinary Differential Equations can recover underlying bifurcation structures directly from time-series data by learning parameter-dependent vector fields. Notably, we demonstrate that Neural Ordinary Differential Equations can forecast bifurcations even beyond the parameter regions represented in the training data. We demonstrate our approach on three test cases: the Lorenz system transitioning from non-chaotic to chaotic behaviour, the Rössler system moving from chaos to period doubling, and a predator-prey model exhibiting collapse via a global bifurcation.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。