Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching

TL;DR

This paper introduces APPS, an active learning approach that improves symbolic discovery of ODEs by focusing on informative phase space regions, leading to more accurate equations from trajectory data.

Contribution

We propose APPS, a novel active learning method that enhances ODE discovery by selecting informative phase space regions instead of individual initial conditions.

Findings

APPS outperforms baseline methods in discovering accurate ODEs.

APPS reduces the need for storing large amounts of trajectory data.

Experiments confirm APPS's effectiveness across various datasets.

Abstract

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.