Observational constraints using Bayesian statistics and Deep Learning in $f(Q)$ gravity
This study investigates the evolution of Friedmann-Robertson-Walker (FRW) cosmological models within the $f(Q)$ gravity framework, utilizing a specific $f(Q)$ formulation and a novel Hubble parameter $H(z)$ parameterization to probe the universe's accelerating expansion. A central aspect is the application of advanced machine learning techniques for cosmological parameter estimation, alongside comparisons with traditional Bayesian (MCMC) methods. We employ a hybrid Mixed Neural Network (MNN), which synergistically combines Artificial Neural Networks (ANNs) and Mixture Density Networks (MDNs), to enhance the accuracy and robustness of parameter constraints. This MNN architecture is integrated into the CoLFI (Cosmological Likelihood-Free Inference) framework. CoLFI facilitates likelihood-free inference, a significant methodological advancement that provides an efficient and robust alternative, particularly for complex models with computationally expensive or intractable likelihood functions. Training efficiency for the neural networks is optimized by generating data via hyperellipsoid sampling. The $f(Q)$ model, constrained using these diverse approaches, successfully describes a universe transitioning from an early decelerating phase to the current accelerated expansion, with a computed transition redshift of $z_t = 0.60$. The physical and kinematic properties of the model are discussed, underscoring the efficacy of the MNN-CoLFI methodology and its consistency with MCMC results, while highlighting its advantages for obtaining observational constraints in $f(Q)$ gravity.