Verifier-Guided Model Discovery for Physical Dynamical Systems with Pretrained Symbolic Transformers

2026-08-01

AI Summary

This study addresses the challenge of reliably transferring pretrained symbolic Transformers to high-dimensional physical systems for interpretable and generalizable dynamical modeling. The authors propose a verifier-guided workflow that integrates dynamical consistency and physical admissibility criteria to select valid low-dimensional governing equations from a pool of candidates generated across multiple trajectories, without requiring system-specific priors. Built upon ODEFormer, coordinate reduction, and multi-trajectory filtering, the method successfully discovers equations capturing fundamental frequencies and higher-order harmonics in benchmark systems such as the Van der Pol oscillator and vortex shedding. The resulting models generalize across unseen parameter regimes, demonstrating that reconstruction fidelity alone does not dictate symbolic discoverability and significantly enhancing both the reliability and physical interpretability of symbolic regression.

Abstract

Reliable forecasting of nonlinear physical systems underpins scientific discovery and engineering decision-making. Yet high-fidelity simulations are prohibitively costly, and machine-learning surrogates can be opaque and encode assumptions about system dynamics, limiting generalizability. Pretrained transformers mapping synthetic ODE trajectories to equations offer interpretable alternatives, promising transfer without system-specific equation knowledge. Transferring them reliably to high-dimensional physical data, however, remains an open challenge. We develop a verifier-guided (VG) workflow around ODEFormer as a symbolic backbone, using dynamical and physical-admissibility criteria to select from a multi-trajectory candidate equation pool, enabling transfer. On canonical Van der Pol oscillators, VG outperforms the original ODEFormer workflow across held-out initial conditions. We then address vortex shedding, a phenomenon occurring in atmospheric and plasma systems of societal relevance, through coordinate reduction and symbolic discovery at fixed and varying Reynolds numbers. VG discovers fixed-parameter reduced-order equations that recover the fundamental shedding oscillator and higher harmonics without a wake-specific candidate library or prescribed Navier-Stokes structure, while the cross-parameter model generalizes to withheld regimes. Reconstruction fidelity alone did not determine symbolic discoverability, highlighting the importance of compatibility between latent dynamics and the backbone's pretraining distribution. This work establishes a verifier-guided neural-to-symbolic methodology for interpretable and physically auditable forecasting in the natural sciences.

Problem

• symbolic discovery
• physical dynamical systems
• model transfer
• interpretable forecasting
• high-dimensional data

Innovation

• verifier-guided discovery
• symbolic transformers
• physical dynamical systems
• ODEFormer
• neural-to-symbolic