From Noisy Neural Black Boxes to Concise Physical Formulas: Structure-Constrained Exact Formula Recovery

Neural networks can fit noisy physical observations accurately but rarely expose the compact laws governing the underlying system. We present a structure-constrained neural-to-symbolic framework for recovering verifiable formulas from neural black-box surrogates. The method combines bounded search over physically motivated expression families, exactification of continuous coefficients into sparse symbolic constants, and noisy-data cross-fitting with structural and stability checks. This separates denoised structure proposal from discrete identification and validation on the original observations. We evaluate 13 development formulas and 20 held-out formulas over five seeds and Gaussian noise levels from 0% to 50%. On held-out formulas, exact-equivalence recovery decreases from 57.9% without noise to 19.2% at 50% noise; the development rate decreases from 69.2% to 61.5%. Removing exactification reduces recovery to 0% at both 20% and 50% noise, while removing raw-data refit cross-fitting reduces the 50%-noise development rate from 66.7% to 28.3%. Compared with PySR, TPSR, uDSR, Operon, and DSR, our method achieves the highest recovery rate at every nonzero noise level in the evaluated comparison. These results show that numerical agreement alone is insufficient for scientific formula recovery: exact constants, structural validity, and noise-aware validation are essential.