Discovering Symbolic Differential Equations with Symmetry Invariants

2025-05-17

AI Summary

This work addresses the challenge of balancing physical consistency and search efficiency in data-driven symbolic discovery of differential equations. We propose a physics-constrained embedding method grounded in differential invariants derived from Lie group symmetries. Our key innovation is the first use of such differential invariants—as opposed to conventional monomial bases—as fundamental building blocks for symbolic regression, ensuring that discovered equations inherently satisfy conservation laws and physical symmetries. Integrating sparse regression, genetic programming, and Lie group theory, we construct an efficient and interpretable equation discovery framework. Evaluated on fluid dynamics and reaction–diffusion systems, our approach successfully recovers concise, accurate, and physically self-consistent governing equations. It significantly improves discovery efficiency, generalizability across unseen conditions, and consistency with known physical principles.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Search
• Machine Learning: Neuro-Symbolic 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
• Security and Privacy: Data transparency and provenance

Abstract

Discovering symbolic differential equations from data uncovers fundamental dynamical laws underlying complex systems. However, existing methods often struggle with the vast search space of equations and may produce equations that violate known physical laws. In this work, we address these problems by introducing the concept of extit{symmetry invariants} in equation discovery. We leverage the fact that differential equations admitting a symmetry group can be expressed in terms of differential invariants of symmetry transformations. Thus, we propose to use these invariants as atomic entities in equation discovery, ensuring the discovered equations satisfy the specified symmetry. Our approach integrates seamlessly with existing equation discovery methods such as sparse regression and genetic programming, improving their accuracy and efficiency. We validate the proposed method through applications to various physical systems, such as fluid and reaction-diffusion, demonstrating its ability to recover parsimonious and interpretable equations that respect the laws of physics.

Problem

• Discover symbolic differential equations from complex data
• Reduce vast search space in equation discovery
• Ensure discovered equations obey physical symmetry laws

Innovation

• Uses symmetry invariants in equation discovery
• Integrates with sparse regression and genetic programming
• Ensures discovered equations respect physical laws