Modelling geometric and dynamical observables with machine learning

We present a physically coupled inverse-emulation framework for the joint reconstruction of cosmic expansion and linear growth dynamics using observational H(z) and fσ8(z) data. The methodology combines a unified forward model linking expansion history, comoving geometry, Alcock–Paczynski distortions, and perturbation growth with machine-learning inverse emulators trained on physically generated realizations. Reference Bayesian inference is performed through Markov Chain Monte Carlo sampling, while robustness is examined using bootstrap resampling, K-fold cross-validation, observational jitter propagation, and prior-volume analyses. We show that the inverse-emulation models successfully recover both posterior parameter constraints and the dominant geometry–growth coupling structure of the Bayesian posterior. In particular, the reconstructed expansion and growth trajectories remain remarkably stable across independent machine-learning architectures, indicating that the learned inverse-emulation space preserves the underlying physical coupling between geometry and structure growth. The most stable reconstructions emerge in the Alcock–Paczynski geometric sector, whereas the largest deviations appear along higher-order kinematic degeneracy directions. These results demonstrate that machine-learning cosmological inference can move beyond phenomenological regression toward physically interpretable reconstruction of coupled geometry–growth dynamics in observable space.