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

Abstract

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 ($Λ$CDM, CPL, GEDE, $Λ_s$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 more fundamental constraint than cosmological model priors, with DDR violations of 30--52 mmag under physical constraints versus 85--2330 mmag without. 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 and finding no significant $M_B$ evolution in $z \in [0.3, 1.5]$. The heteroscedastic method identifies a persistent $2-3σ$ residual at $z \sim 0.4-0.5$ that is consistent across all four models and both fiducials; the Fisher method finds no significant pointwise deviation in $z\in[0.3,1.5]$ but shows a systematic separation of redshift-binned $M_B$ distributions consistent with 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.

Figures (6)

• Figure 1: Goodness-of-fit summary for all models, configurations, and fiducials on the Pantheon+ dataset. Left and centre: $\chi^2/\mathrm{dof}$ for BAO and SN Ia (diagonal errors), respectively, on a shared logarithmic scale. The dotted line marks $\chi^2/\mathrm{dof} = 1$. Right: total variation $\sum|\Delta M_B(z)|$ of the reconstructed $M_B(z)$ curve, a measure of high-frequency oscillation. Filled circles indicate DESI+CMB fiducial; open triangles indicate DESI+PP. Colours denote the constraint configuration: H-TT (blue), H-TF (green), H-FT (red), H-FF (purple). See text for a discussion on the outliers
• Figure 2: Summary of reconstructed mean $M_B$ for all models, configurations, and fiducials on the Pantheon+ dataset. Left panel shows DESI+CMB, right: DESI+PP. Colours denote the cosmological model; marker shapes denote the constraint configuration. Thick error bars show the scatter-based standard error on the mean $\sigma_\mathrm{err}$; thin error bars show the epistemic uncertainty from the network $\sigma_\mathrm{ep}$. The dashed line and the shaded band marks $M_{B,\mathrm{fid}} = -19.3 \pm 0.5$ mag.
• Figure 3: Significance of $M_B(z)$ deviation from its mean value $|M_B(z) - \langle M_B\rangle|/\sigma_{M_B}$, for H-TT (solid) and H-TF (dashed) configurations. Left panel: DESI+CMB fiducial; right panel: DESI+PP fiducial. Colors indicate cosmological model. Gray shading marks the $< 1\sigma$ region. Horizontal dotted lines indicate $1\sigma$, $3\sigma$, and $5\sigma$ thresholds. The uncertainty $\sigma_{M_B}$ is the observationally-floored standard error of the mean per redshift bin (Eq. \ref{['eq:sigma_floor']}). The plot is restricted to $0.15 < z < 2$ where SN Ia coverage is reliable.
• Figure 4: $M_B$ distributions reconstructed by the Fisher two-network method, colour-coded by redshift bin ($z \in [0.0, 0.3]$, $[0.3, 0.8]$, $[0.8, 1.5]$, $[1.5, 2.5]$). Top row: DESI+CMB fiducial; bottom row: Pantheon+ (DESI+PP) fiducial. GEDE and CPL show broader distributions reflecting additional dark energy freedom.
• Figure 5: $\Delta M_B(z) = M_B(z) - M_B^\mathrm{fid}$ reconstructed by the Fisher method. Left: DESI+CMB fiducial; right: DESI+PP fiducial. Shaded bands show $1\sigma$ uncertainties. All models are consistent with zero deviation in $z \in [0.3, 2]$. The oscillatory feature at $z \lesssim 0.2$ is data-driven and common to all models. At $z > 1.5$ models diverge as SN coverage becomes sparse and reconstruction relies on BAO.