Symmetry- and Semantic-Aware Symbolic Discovery of Ordinary Differential Equations

Recovering symbolic ordinary differential equations (ODEs) from trajectories requires solutions that are both expressive and consistent with known physics. Library-based sparse regression methods provide structural guarantees only within predefined invariant/equivariant bases, while free-form symbolic regression explores many symmetry-violating expressions and obtains behavioral feedback mainly from completed candidates. From this perspective, we develop Symmetry- and Semantic-Aware Graph Symbolic Regression (S${}^2$GSR) by searching over canonical, symmetry-typed expression graphs equipped with sub-expression-level semantics. The local typing rules of expression graphs guarantee the prescribed symmetry of the discovered ODEs by construction, and path-conditioned semantic features compare closed sub-expressions with position-specific behavioral targets. This framework makes invariant-basis methods a special case of S${}^2$GSR while supporting *free-form, non-polynomial* dynamics, and further integrates path-conditioned semantic guidance into a neural-guided Monte Carlo Tree Search pipeline. Experiments on polynomial and non-polynomial symmetry-constrained systems and on ODEBench demonstrate improved symbolic recovery and trajectory reconstruction of S${}^2$GSR while maintaining physical consistency.