Grammar-based Ordinary Differential Equation Discovery

2025-04-03

AI Summary

This work addresses the core challenge of interpretable and efficient discovery of ordinary differential equations (ODEs) from observational data to model complex physical system dynamics. We propose an end-to-end automated discovery framework integrating formal grammars, PCA-based dimensionality reduction, and stochastic search. Formal grammars constrain the symbolic expression space and embed domain knowledge, while dimensionality reduction enhances search efficiency; together, they enable structured, interpretable dynamical law identification. Compared to state-of-the-art approaches—including Transformer-based and genetic programming methods—our framework achieves significantly improved sample and parameter efficiency across multiple benchmark systems, notably structural dynamics. The resulting ODE models exhibit both higher predictive accuracy and greater structural simplicity. By unifying symbolic regression with physics-informed constraints and scalable optimization, our approach establishes a novel paradigm for physics-guided scientific discovery.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Structured Learning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The understanding and modeling of complex physical phenomena through dynamical systems has historically driven scientific progress, as it provides the tools for predicting the behavior of different systems under diverse conditions through time. The discovery of dynamical systems has been indispensable in engineering, as it allows for the analysis and prediction of complex behaviors for computational modeling, diagnostics, prognostics, and control of engineered systems. Joining recent efforts that harness the power of symbolic regression in this domain, we propose a novel framework for the end-to-end discovery of ordinary differential equations (ODEs), termed Grammar-based ODE Discovery Engine (GODE). The proposed methodology combines formal grammars with dimensionality reduction and stochastic search for efficiently navigating high-dimensional combinatorial spaces. Grammars allow us to seed domain knowledge and structure for both constraining, as well as, exploring the space of candidate expressions. GODE proves to be more sample- and parameter-efficient than state-of-the-art transformer-based models and to discover more accurate and parsimonious ODE expressions than both genetic programming- and other grammar-based methods for more complex inference tasks, such as the discovery of structural dynamics. Thus, we introduce a tool that could play a catalytic role in dynamics discovery tasks, including modeling, system identification, and monitoring tasks.

Problem

• Discovering ODEs for complex physical phenomena modeling
• Improving efficiency in dynamical systems discovery
• Enhancing accuracy in structural dynamics inference

Innovation

• Grammar-based ODE Discovery Engine (GODE)
• Combines formal grammars with dimensionality reduction
• More efficient than transformer-based models