Cosmology-informed Neural Networks to infer dark energy equation-of-state

We present a framework that combines physics-informed neural networks (PINNs) with Markov Chain Monte Carlo (MCMC) inference to constrain dynamical dark energy models using the Pantheon+ Type Ia supernova compilation. First, we train a physics-informed neural network to learn the solution of the Friedmann equation and accurately reproduce the matter density term x_m(z) = Omega_m,0 (1+z)^3 across a range of Omega_m,0. For each of five two-parameter equation-of-state (EoS) forms: Chevallier-Polarski-Linder (CPL), Barboza-Alcaniz (BA), Jassal-Bagla-Padmanabhan (JBP), Linear-z, and Logarithmic-z, we derive the analytic dark energy factor x_de(z), embed the trained surrogate within a GPU-accelerated likelihood pipeline, and sample the posterior of (h0, Omega_m,0, w0, wa, M0) using the emcee ensemble sampler with the full Pantheon+ covariance. All parameterizations remain consistent with a cosmological constant (w0 = -1, wa = 0) at the 95% credible level, with the tightest bounds from the CPL form. While the surrogate does not reduce computation time for a single run in simple models, it becomes advantageous for repeated analyses of the same EoS or for models with expensive likelihood evaluations, and can be shared as a reusable tool with different datasets within the training range of SNe redshifts. This flexibility makes the approach a scalable tool for future cosmological inference, especially in regimes where conventional ODE-based methods are computationally prohibitive.