Observational Insights on DBI K-essence Models Using Machine Learning and Bayesian Analysis

Abstract: We present a comparative observational investigation of two Dirac-Born-Infeld (DBI) k-essence scalar field models that address late-time cosmic acceleration and dark sector dynamics. The first model involves a non-canonical scalar field representing dark energy alongside standard matter, which is regulated by the Lagrangian, $\mathcal{L} = -\left[f(\phi)\sqrt{1- \frac{2X}{f(\phi)}} - f(\phi) + V(\phi)\right] + \mathcal{L}m$. The second model combines dark matter and dark energy with $ \mathcal{L} = -V(\phi)\sqrt{1-2X}$. The background evolution is based on modified Friedmann and scalar field equations constrained by Pantheon+ Type Ia supernovae, Hubble parameter data, and BAO (including DESBAO). A hybrid inference methodology is chosen, which includes Bayesian parameter estimation using the NUTS sampler in {\it NumPyro} and a deep-learning emulator employing supervised neural networks to speed up likelihood computations. A nuisance parameter $\mu_0$ is added to reduce systematics and improve the estimate of $H_0$, $r_d$, and $\Omega{d0}$. The deceleration parameter $q(z)$ exhibits significant differences: Model I yields $q^{\text{Bayes}}_0 = -0.538$, $q^{\text{ML}}_0 = -0.509$, with $z^{\text{Bayes}}_{\text{trans}} = 0.650$, $z^{\text{ML}}_{\text{trans}} = 0.602$, consistent with $\Lambda$CDM, while Model II results in stronger present acceleration ($q^{\text{Bayes}}_0 = -0.831$, $q^{\text{ML}}_0 = -0.780$). Model I performs better statistically and dynamically than Model II, as shown by lower $\chi^2_\nu$, AIC, and BIC values. We additionally use symbolic regression ({\it PySR}) to rebuild analytic forms of $\omega_{\text{eff}}(z)$, which improves interpretability. Model I provides a feasible, statistically preferred alternative to $\Lambda$CDM that addresses the Hubble tension.