RELift: Learned Coarse-to-Fine Propagators for Time-Dependent PDEs with Applications to Electron Dynamics

We present RELift (Restrict, Evolve, Lift), a two-phase learning framework that couples coarse-grid numerical solvers with neural operators to super-resolve and forecast fine-grid dynamics for time-dependent partial differential equations (PDEs). In Phase 1, RELift learns a super-resolution operator that maps the solution on a coarse grid to a fine grid. In Phase 2, this learned operator is composed with a coarse-grid numerical integrator to construct an effective fine-grid propagator for the governing equation. We benchmark RELift on three canonical two-dimensional PDEs of increasing dynamical complexity -- the heat equation, the wave equation, and the incompressible Navier--Stokes equations -- and we further demonstrate its performance on a kinetic electron dynamics case study via the 1D1V Vlasov--Poisson system. Across all examples, RELift delivers high-fidelity super-resolution (Phase 1) and accurate long-horizon rollouts (Phase 2), outperforming standard super-resolution and neural operator baselines in both field-level error metrics and physics-relevant diagnostics. Finally, we provide error analysis of the effective fine-grid propagator, characterizing how approximation errors accumulate over time and explaining the observed numerical stability of the RELift framework.