通过绘制相图区域，高效发现微分方程，减少数据存储负担。
Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching
- 先定位相空间中信息丰富的区域，再批量采样初始条件。
- 在相同数据量下，发现的微分方程准确率显著高于传统方法。
- 适合需要高效探索动力系统规律的研究者使用。
从轨迹数据中进行常微分方程（ODE）的符号发现，在人工智能驱动的科学发现中具有关键作用。现有符号方法主要依赖固定、预先收集的训练数据集，常导致性能不佳，如图1所示案例研究。受主动学习启发，我们探究如何查询有信息量的轨迹数据以提升预测ODE的评估效果。然而，动力系统中的蝴蝶效应表明，初值的微小变化可引发截然不同的轨迹，传统主动学习需存储海量轨迹数据。为此，我们提出基于相图草图的主动符号发现方法（APPS）。APPS不直接选择单个初值，而是先识别相空间中信息丰富的区域，再从该区域批量采样初值。相比传统方法，APPS显著降低了大量数据存储的需求。大量实验表明，使用被动采集数据集时，APPS始终能发现比基线方法更准确的ODE表达式。
原文摘要 · Abstract (English)
The symbolic discovery of Ordinary Differential Equations (ODEs) from trajectory data plays a pivotal role in AI-driven scientific discovery. Existing symbolic methods predominantly rely on fixed, pre-collected training datasets, which often result in suboptimal performance, as demonstrated in our case study in Figure 1. Drawing inspiration from active learning, we investigate strategies to query informative trajectory data that can enhance the evaluation of predicted ODEs. However, the butterfly effect in dynamical systems reveals that small variations in initial conditions can lead to drastically different trajectories, necessitating the storage of vast quantities of trajectory data using conventional active learning. To address this, we introduce Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching (APPS). Instead of directly selecting individual initial conditions, our APPS first identifies an informative region within the phase space and then samples a batch of initial conditions from this region. Compared to traditional active learning methods, APPS mitigates the gap of maintaining a large amount of data. Extensive experiments demonstrate that APPS consistently discovers more accurate ODE expressions than baseline methods using passively collected datasets.
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