Joint reconstruction of $H(z)$ and $f\sigma_8(z)$ with physics informed neural networks

Abstract

We present a model-independent joint reconstruction of the Hubble parameter $H(z)$ and the growth rate $f\sigma_8(z)$ using a dual-head physics-informed neural network trained on four complementary late-universe datasets: Cosmic Chronometers, Redshift-Space Distortions, the DESI DR2 BAO mean vector and full covariance, and the Pantheon$+$SH0ES supernova compilation. The two output heads share a backbone and are coupled through the linear growth equation of general relativity, penalizing the ODE residual at 1000 collocation points per training step via automatic differentiation. Uncertainty is quantified by an ensemble of 100 networks, each trained on an independent parametric-bootstrap resample of the data and its own draw of the fiducial cosmological parameters from Planck 2018 priors, so that the ensemble spread captures data-noise, initialization, and fiducial-cosmology systematics simultaneously. The physics coupling weight $\lambda$ is selected via an L-curve analysis over six values; the curve is nearly flat in total data $\chi^2$, indicating that the joint dataset is intrinsically consistent with the growth ODE. Without any $H_0$ prior, the free reconstruction yields $H_0 = 69.0 \pm 4.7$\,km\,s$^{-1}$\,Mpc$^{-1}$, consistent with the Planck 2018 CMB value and with the DESI DR2 inverse distance-ladder determination, and approximately $0.9\sigma$ below the SH0ES local measurement. The reconstructed $H(z)$ lies systematically below the flat $\Lambda$CDM prediction at $z \sim 0.7$-$0.8$, consistent with the dark energy evolution suggested by DESI DR2. As conditional analyses, we also anchor $H_0$ to the SH0ES value $73.04 \pm 1.04$\,km\,s$^{-1}$\,Mpc$^{-1}$ and the Local Distance Network consensus $73.50 \pm 0.81$\,km\,s$^{-1}$\,Mpc$^{-1}$; both anchored reconstructions yield a suppressed $f\sigma_8$, illustrating the propagation of the $H_0$--$\sigma_8$ link through the ODE coupling.