Verifier-Guided AI Discovers Equations for Physical Systems

Accurate forecasting of nonlinear physical systems is fundamental to scientific discovery and engineering. While high-fidelity simulations are often too costly, and machine learning surrogates can lack transparency and embed assumptions, pretrained transformers that map synthetic trajectories to equations offer an interpretable alternative. However, reliably transferring these symbolic transformers to complex, high-dimensional physical data has remained a significant challenge. This paper presents a verifier-guided (VG) workflow built around ODEFormer, a symbolic transformer backbone. The VG approach uses dynamic and physical-admissibility criteria to select the most appropriate equations from a pool of candidates generated from multiple trajectories, thereby enabling robust transfer. On canonical Van der Pol oscillators, VG demonstrated superior performance over the original ODEFormer workflow, particularly for unseen initial conditions. The methodology was further applied to vortex shedding, a critical phenomenon in atmospheric and plasma systems. Through coordinate reduction and symbolic discovery, VG successfully identified fixed-parameter reduced-order equations that accurately capture the fundamental shedding oscillator and higher harmonics, without needing a predefined library or Navier-Stokes structure. Crucially, the cross-parameter model generalized well to unobserved regimes. The findings emphasize that reconstruction fidelity alone is insufficient for symbolic discoverability, highlighting the importance of aligning latent dynamics with the backbone's pretraining distribution. This work establishes a novel, interpretable, and physically auditable forecasting methodology for natural sciences.