Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching

2024-09-02

arXiv.org

AI Summary

Symbolic discovery of ordinary differential equations (ODEs) suffers from low accuracy and poor robustness under limited trajectory data. Method: We propose a phase-portrait-guided active learning framework that replaces conventional single-point initial-value sampling with batched, adaptive trajectory acquisition—guided by geometric analysis of the phase space to identify information-rich regions. This approach significantly reduces data storage overhead while enhancing modeling stability. Our method integrates phase-space structural priors, sparse regression, and evolutionary symbolic regression, augmented with dynamical-systems stability criteria to ensure physical consistency. Results: Experiments across multiple benchmark systems demonstrate that our method achieves higher-accuracy symbolic ODE models using fewer trajectories than state-of-the-art symbolic regression approaches. It exhibits superior generalizability and interpretability, offering both computational efficiency and physically grounded model discovery.

Technology Category

• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Active Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Differential Equation Discovery
• Sensitivity Analysis
• Data-driven Learning

Innovation

• APPS method
• data selection
• differential equation discovery