Reconstructing the Type Ia Supernova Absolute Magnitude with Two-Probe Physics-Informed Neural Networks

We apply two variants of Physics-Informed Neural Networks (PINNs) to reconstruct the Type~Ia supernova absolute magnitude $M_B(z)$ from joint BAO and supernova data under four cosmological models ($\Lambda$CDM, CPL, GEDE, $\Lambda_s$CDM) and two DESI~DR2 fiducial sets. A heteroscedastic single-network method tested across four constraint configurations establishes that the Etherington distance duality relation is a more fundamental constraint than cosmological model priors, reducing internal inconsistencies by up to an order of magnitude. Under full constraints all models recover $M_B \approx -19.3$~mag with biases below 0.05~mag. A Fisher information-weighted two-network variant trains independent networks on BAO and SN data, providing clean probe separation; it finds no significant pointwise $M_B$ evolution in $z \in [0.3, 1.5]$, but reveals a systematic separation of redshift-binned $M_B$ distributions. The heteroscedastic method identifies a persistent $2$--$3\sigma$ residual at $z \sim 0.4$--$0.5$ that is consistent across all four models and both fiducials, implying the same underlying tension. While the origin of this feature remains ambiguous, its model-independence and cross-method consistency warrant further investigation with forthcoming data.