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

We perform a late-time cosmological study; we compare the performance of two Dirac-Born-Infeld (DBI)-type k-essence scalar field extensions of the $\Lambda$CDM model to the standard framework and a wCDM scenario using the Chevallier-Polarski-Linder (CPL) equation of state parametrization. We solve background dynamics numerically as functions of redshift and incorporate them into a Bayesian inference pipeline accelerated by machine learning. We use a Flax-based surrogate emulator to replace repeated direct integrations of the ODE system, reducing computational cost. A hybrid scheme that combines Stochastic Variational Inference (SVI) with No-U-Turn Hamiltonian Monte Carlo constrains cosmological parameters using the Pantheon$+$SH0ES Type Ia supernova sample, DESI BAO (DR2) data, and cosmic chronometer $H(z)$ measurements without CMB-based priors. In both DBI k-essence formulations, present-day dark energy equations of state are consistent with cosmic acceleration, indicating a $\Lambda$CDM-like regime with a modest redshift dependence. The $w$CDM model is marginally favored by conventional model selection measures such as $\chi^2$, AIC, BIC, and DIC, which are based on goodness of fit and penalized. However, Bayesian predictive measures like WAIC and PSIS-LOO show no significant differences between $\Lambda$CDM, $w$CDM, and DBI k-essence scenarios. All have similar model weights and out-of-sample predictive performance for the datasets. Thus, DBI k-essence models mimic the success of the classic $\Lambda$CDM paradigm while allowing controlled, redshift-dependent deviations from a strict cosmological constant that are consistent with present late-time observations.