Advanced observational techniques in f(Q) gravity using Bayesian methods and deep learning

In this study, we employ Bayesian statistical methods alongside deep learning techniques to construct Friedmann–Robertson–Walker (FRW) cosmological models within the framework of [Formula: see text] gravity. We specifically investigate the accelerating expansion of the universe by considering cosmological dynamics consistent with a cubic form of the [Formula: see text] gravity model. This is achieved through an innovative and analytically tractable parameterization of the Hubble parameter, expressed as [Formula: see text]. The free parameters [Formula: see text], [Formula: see text], and [Formula: see text] are constrained using Bayesian Markov Chain Monte Carlo (MCMC) methods. Our results demonstrate consistency with observational cosmology, affirming the viability of our model. Furthermore, we analyze the physical properties of the universe within this framework, examining key cosmological quantities such as energy density, pressure, and the equation of state parameter. We also explore kinematic diagnostics, including the Hubble parameter, deceleration parameter, and the Om(z) diagnostic, to gain deeper insights into cosmic evolution. To enhance parameter estimation accuracy, we introduce a novel approach leveraging a Mixed Neural Network (MNN), which integrates Artificial Neural Networks (ANNs) with Mixture Density Networks (MDN). By synergizing the strengths of these techniques, our method significantly improves the precision of cosmological parameter inference.