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

Abstract

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.

Figures (25)

• Figure 1: Phase 1 and Phase 2 training in the RELift framework. In Phase 1, we learn a super-resolution operator that maps the PDE solution at the coarse grid to the fine grid. In Phase 2, we use this learned operator to compose an effective propagator $\hat{\mathcal{P}}_{f}(\Delta t) :=\mathcal{N}_\theta\circ\mathcal{P}_c(\Delta t)\circ P$ that approximates the true fine-grid propagator $\mathcal{P}_f(\Delta t)$.
• Figure 2: Phase 1 super-resolution results across different PDEs. For each PDE, we show results for an unseen test trajectory, with representative snapshots from the (Top) 2D heat equation, (Middle) 2D wave equation, and (Bottom) 2D Navier--Stokes equations (NSE). The first column shows the true fine-grid dynamics. The next two columns—labeled "Upsampled" and "FNO-AR"—correspond to the bicubic interpolation and autoregressive FNO baselines, respectively. All subsequent columns display predictions from different RELift models. The inset in each panel reports the relative $L_2$ reconstruction error.
• Figure 3: Phase 1 results across different resolutions and scaling factors.(a) Relative $L_2$ errors for all models and baselines evaluated at multiple fine-grid resolutions. Each curve is obtained by averaging over 20 unseen trajectories. Models that explicitly exploit spectral structure (FUnet, FNO) achieve the best accuracy, with FUnet exhibiting the slowest growth in error as the fine-grid resolution increases. (b) Relative $L_2$ error of FUnet at a fixed fine-grid resolution of $256\times256$ while varying the coarse-grid input size $N_c \in \{128,64,32\}$ (corresponding to upscale factors $\times 2, \times 4, \times 8$).
• Figure 4: Phase 2 future dynamics prediction results accross different PDEs. For each PDE, we show results for an unseen test trajectory, with representative snapshots from the (Top) 2D heat equation, (Middle) 2D wave equation, and (Bottom) 2D Navier--Stokes equations (NSE). The first column shows the true fine-grid dynamics. The next two columns—labeled "Upsampled" and "FNO-AR"—correspond to the bicubic interpolation and autoregressive FNO baselines, respectively. All subsequent columns display predictions from different RELift models. The inset in each panel reports the relative $L_2$ error.
• Figure 5: Phase 2 error accumulation for NSE.(Left) Relative $L_2$ error for a representative unseen test trajectory. At a fixed $4\times$ downsampling factor, the RELift Phase 2 predictions (solid, colored curves) are compared against the baselines (dashed, gray curves). Beyond timestep 400, the autoregressive FNO baseline rapidly diverges, whereas the RELift predictions exhibit substantially slower error growth. (Right) Phase 2 rollout error of RELift with the FUnet architecture, illustrating how prediction accuracy scales with the coarse input resolution $N_c \in \{32,64,128\}$ while keeping the fine grid fixed at $256\times256$.