ODEFormer's verifier-guided workflow uncovers physical equations from data

A new paper from Farbod Faraji and Francesco Belardinelli introduces a verifier-guided (VG) workflow that enhances ODEFormer, a pretrained symbolic transformer, to discover interpretable equations for physical dynamical systems. Instead of relying on opaque machine-learning surrogates, ODEFormer maps synthetic ODE trajectories to candidate equations, but transferring it to high-dimensional real-world data has been challenging. The VG framework solves this by using dynamical and physical-admissibility criteria to select the best equations from a multi-trajectory candidate pool, ensuring the results are physically meaningful and generalizable.

The team validated the approach on canonical Van der Pol oscillators, where VG outperformed the original ODEFormer across held-out initial conditions. They then tackled vortex shedding—a phenomenon critical to atmospheric and plasma systems—using coordinate reduction and symbolic discovery. VG recovered the fundamental shedding oscillator and higher harmonics without any wake-specific candidate library or prescribed Navier-Stokes structure. Crucially, the cross-parameter model generalized to withheld Reynolds number regimes, showing that reconstruction fidelity alone doesn't guarantee symbolic discoverability. This work establishes a neural-to-symbolic methodology that yields transparent, physically auditable forecasting models, potentially replacing black-box surrogates in engineering and climate science.