<p>In the context of CO₂ storage in deep saline aquifers, when CO₂ flows through the aquifer, three major regions can be distinguished: i) dry-out region (single-phase CO₂), ii) two-phase region (brine and CO₂), and iii) brine region (single-phase brine). Assuming fully immiscible fluids and neglecting local capillary pressure, the transition zone between CO₂ and brine can be considered a sharp interface. Additionally, due to the density difference between the displacing CO₂ and displaced (brine) fluid (CO₂ is less dense than the resident brine), gravity plays a significant role. The CO₂ displacement is governed by a nonlinear ordinary differential equation (ODE) that presents variations of the CO₂-brine interface with respect to time and (radial) space. To numerically solve this highly unstable ODE, this study presents and compares two distinct solutions: finite difference method (FDM), and physics-informed neural networks (PINNs). Although some researchers addressed the governing ODE in question previously, they neither specified the solution methods used nor provided solutions for the entire CO₂-brine interface, motivating a deeper exploration of solution approaches. Our findings indicate that the tested finite difference schemes (forward, backward, central) fail to provide accurate and stable solutions within analytically benchmarked cases. In contrast, the proposed PINN framework accurately captures the physics of the governing ODE and closely matches the analytical solution. After validation, the PINN model was optimized through a grid search and a loss-weighting sensitivity analysis to identify the most robust framework. Using this optimized model, transfer learning was performed beyond the analytical regime, revealing that the pretrained PINN model generalizes to moderately stiffer conditions, but fails in strongly stiff regimes.</p>