Geological carbon sequestration in saline aquifers is a promising route to mitigate CO₂ emissions by permanently storing CO₂ in subsurface formations. While many studies emphasise optimising trapping mechanisms to maximise storage efficiency, the transient migration of CO₂ plumes—especially the onset and growth of interfacial instabilities (fingering) driven by viscosity and density contrasts between CO₂ and brine—remains challenging to characterize. In practice, the development of model-based operational/control strategies is further constrained by the high computational cost and limited accessibility of high-fidelity multiphase simulations. Hence, affordable yet physics-consistent surrogate solvers are needed to enable rapid analysis and decision support. In this paper, we develop a physics-informed neural network (PINN) to reconstruct unstable fronts in immiscible CO₂–brine displacement. The network takes (x,y,t) as input and predicts nondimensional pressure p̃(x,y,t) and CO₂ saturation S_CO₂(x,y,t). To better represent the coupled but distinct behaviours of pressure diffusion and saturation transport, we adopt a TwoNetPINN “divide-and-conquer” architecture with separate subnetworks for p̃ and S_CO₂, and employ Fourier feature mapping to enhance the approximation of sharp, multiscale fingering structures. The loss function combines (i) PDE residuals derived from two-phase mass conservation and extended Darcy flux relations, evaluated via automatic differentiation, (ii) explicit initial/boundary constraints, and (iii) supervision from multi-time observations. CO₂ compressibility is incorporated through a differentiable density surrogate (a Chebyshev-polynomial proxy ρ_CO₂(p)) embedded consistently in the residuals. Moreover, a residual–gradient adaptive refinement strategy concentrates collocation points near large residuals and strong |∇S_CO₂|, enabling efficient “front-tracking” training and improved resolution of finger growth and branching. We validate the proposed approach against high-fidelity results from the ICFERST control volume finite element method (CVFEM) framework, demonstrating close agreement in pressure–saturation evolution and instability dynamics.