MSR-PP: Modular Symbolic Regression with Physics Priors for PDE Discovery

Abstract

Discovering governing Partial Differential Equations (PDEs) from observational data is a fundamental challenge in AI for Science. While Symbolic Regression (SR) dominates this task, existing token-level methods trigger a combinatorial explosion of search spaces, frequently yielding mathematically valid yet physically inconsistent candidates. To overcome this, we propose Modular Symbolic Regression with Physics Priors (MSR-PP), a knowledge-guided framework that conceptualizes one-dimensional PDEs as structured compositions of semantic modules (e.g., convection, diffusion) rather than random symbol sequences. MSR-PP employs an RL-optimized LSTM agent to sequentially construct equation skeletons based on syntactic and semantic dependencies, utilizing a modular search strategy to significantly prune the search space. Addressing the critical scarcity of standardized, high-fidelity benchmarks for 1D PDEs, we constructed an extended dataset for rigorous evaluation. Extensive experiments demonstrate that MSR-PP outperforms state-of-the-art baselines. Real-world validation on the NGSIM US101 traffic dataset shows MSR-PP successfully identifies a convection-diffusion model featuring a critical second-order term. Notably, the negative coefficient of this term (-u_xx) aligns with theoretical models, accurately capturing the string instability and wave-amplifying dynamics characteristic of stop-and-go traffic. Moreover, the discovered equation exhibits robust out-of-distribution (OOD) generalization on the unseen NGSIM I-80 dataset, underscoring MSR-PP's capability to uncover physically meaningful laws. Datasets: https://github.com/jinyangdu5/MSR-PP.