Null test of cosmic curvature using deep learning method

Determining the spatial curvature of the Universe, a fundamental parameter defining the global geometry of spacetime, remains crucial yet contentious due to existing observational tensions. Although Planck satellite measurements have provided precise constraints on spatial curvature, discrepancies persist regarding whether the Universe is flat or closed. Here, we introduce a model-independent approach leveraging deep learning techniques, specifically residual neural networks (ResNet), to reconstruct the dimensionless Hubble parameter $E(z)$ and the normalized comoving distance $D(z)$ from $H(z)$ data and multiple SNe Ia compilations. Our dual-block ResNet architecture, which integrates a model-driven block informed by ΛCDM and a purely data-driven block, yields smooth and robust reconstructions and enables the derivation $D'(z)$. By combining these reconstructed quantities, we assess the curvature diagnostic function Oₖ(z). Analyses of the Pantheon+ sample support spatial flatness at the 1σ level over $0<z<2.5$, with a mild tendency toward negative curvature at high redshift. Reconstructions based on Union3 and DESY5, however, show stronger departures toward negative curvature at intermediate and high redshifts. These results highlight the need for expanded and refined observational datasets to conclusively resolve these tensions and comprehensively investigate cosmic geometry.