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

1. Problem Statement

Solving time-dependent Partial Differential Equations (PDEs) at high resolution is computationally prohibitive due to the scale of dynamics evolution. While machine learning (ML) has emerged as an accelerator for PDE solvers, existing approaches face significant trade-offs:

• Reduced-Order Models: Project dynamics to lower dimensions but often require learning complex closure terms for missing subgrid physics.
• Fully Data-Driven Operators: Learn the complete propagator directly from data. While flexible, these models often lack explicit physical constraints, leading to rapid error accumulation and instability during long-horizon extrapolation (rollouts).
• Standard Super-Resolution (SR): Upsampling coarse solutions often fails to capture the correct temporal evolution or fine-scale physics when used for forecasting.

The core challenge is to achieve accurate, stable, long-time forecasting of fine-grid PDE dynamics using only coarse-grid data, without incurring the computational cost of full fine-grid simulations or the instability of purely data-driven autoregressive models.

2. Methodology: The RELift Framework

The authors propose RELift (Restrict, Evolve, Lift), a two-phase learning framework that couples a coarse-grid numerical solver with a learned neural operator. The framework constructs an effective fine-grid propagator by composing these components.

Phase 1: Learning the Super-Resolution Operator

• Goal: Learn a neural operator $N_\theta$ that maps a coarse-grid solution $u_c$ to its fine-grid counterpart $u_f$.
• Training Data: Pairs of coarse and fine snapshots $\{(u_c^{(i)}, u_f^{(i)})\}$.
• Loss Function: Minimizes the $L_1$ norm difference between the true fine solution and the neural prediction:
  $$J_{P1}(\theta) = \frac{1}{K} \sum \| u_f^{(i)} - N_\theta[u_c^{(i)}] \|_{\ell_1}$$
• Architectures Tested: U-Net, EDSR, FNO-SR (Fourier Neural Operator adapted for SR), and a novel FUnet (Fourier U-Net), which integrates spectral convolutional layers into a U-Net bottleneck to better capture frequency content.

Phase 2: Constructing the Effective Propagator

• Goal: Create a stable operator for long-time forecasting by composing the learned SR operator with a fixed coarse-grid integrator.
• The Effective Propagator: Defined as $\hat{P}_f(\Delta t; \theta) = N_\theta \circ P_c(\Delta t) \circ P$, where:
  • $P$: Projection operator (coarsening/downsampling).
  • $P_c(\Delta t)$: Fixed coarse-grid numerical integrator (e.g., finite difference or spectral method).
  • $N_\theta$: The learned super-resolution operator from Phase 1.
• Training Strategy: The network is fine-tuned using fine-grid trajectory pairs. The loss minimizes the discrepancy between the true fine-grid evolution and the composed operator's output:
  $$J_{P2}(\theta) = \frac{1}{K} \sum \| P_f(\Delta t) u_f^{(i)} - \hat{P}_f(\Delta t; \theta) [u_f^{(i)}] \|_{\ell_1}$$
• Mechanism: This creates a predictor-corrector loop. The coarse solver advances the state (preserving macro-scale physics), and the neural operator corrects the spatial details (lifting to fine grid). This avoids the need for backpropagation through the PDE solver, significantly reducing training overhead.

3. Key Contributions

1. Novel Framework: Introduction of RELift, which occupies a "middle ground" between strictly equation-constrained solvers and fully equation-free neural operators. It leverages the stability of coarse solvers while using data-driven learning for spatial refinement.
2. Theoretical Error Analysis: The authors derive local and global approximation error bounds for the effective propagator.
   • They prove that the global error growth is governed by the Lipschitz constant of the composed operator ($L_{N} L_{c} C_P$).
   • Crucially, they show that this composite Lipschitz constant is significantly smaller than that of purely autoregressive models (like FNO-AR), explaining the observed numerical stability in long-horizon predictions.
   • They demonstrate that the error is dominated by the super-resolution approximation rather than the coarse integration error.
3. Comprehensive Benchmarking: Evaluation across three canonical 2D PDEs (Heat, Wave, Navier-Stokes) and a kinetic electron dynamics case (1D1V Vlasov-Poisson system).
4. Efficiency Gains: Demonstration of significant storage and runtime reductions. By storing only coarse trajectories and the neural weights, the framework achieves up to 125x storage compression and 1.7x speedup in wall-clock time compared to full fine-grid solvers.

4. Experimental Results

A. Standard PDEs (Heat, Wave, Navier-Stokes)

• Phase 1 (Reconstruction): RELift models (particularly FUnet and FNO-SR) outperformed naive bicubic upsampling by 1–3 orders of magnitude in relative $L_2$ error.
• Phase 2 (Long-Horizon Forecasting):
  • Stability: Purely autoregressive models (FNO-AR) and upsampling baselines diverged rapidly for nonlinear systems (Navier-Stokes). RELift maintained stability for hundreds of time steps.
  • Accuracy: For the Navier-Stokes equations, RELift reduced the relative $L_2$ error from $\approx 1.58 \times 10^1$ (FNO-AR) to $\approx 9.36 \times 10^{-2}$ (FUnet) at $t=7$.
  • Metrics: RELift excelled in Structural Similarity (SSIM) and Spectral MSE, preserving fine-scale vorticity structures that baselines smoothed out or distorted.

B. Electron Dynamics (Vlasov-Poisson System)

• Application: Applied to the 1D1V Vlasov-Poisson system, a high-dimensional kinetic equation critical for plasma physics.
• Physics-Based Diagnostics: Evaluated using negative mass penalty, entropy error, and electric field error.
• Results:
  • RELift (FUnet) reduced field-level $L_2$ error by two orders of magnitude compared to bicubic upsampling.
  • It significantly improved the accuracy of derived physical observables (charge density $\rho$, current density $J$, and kinetic energy $M$).
  • The method maintained physical coherence (e.g., non-negative distribution functions) and stability over extrapolation horizons far beyond the training window (up to $T=64$).

5. Significance and Impact

• Bridging the Gap: RELift successfully bridges the gap between the physical fidelity of numerical solvers and the flexibility of machine learning. It allows researchers to use existing, robust coarse-grid solvers while recovering fine-scale details via learning.
• Scalability: The framework offers a practical path for high-resolution forecasting in scenarios where fine-grid data is too expensive to generate or store (e.g., climate modeling, fusion plasma simulation).
• Theoretical Insight: The error analysis provides a rigorous justification for why "coarse-solver + neural-lift" is more stable than "neural-only" propagation, offering a blueprint for designing future physics-informed ML architectures.
• Generalizability: The framework is not tied to a specific data generation paradigm (sensor-style downsampling or simulation-to-simulation) and is applicable to both linear and highly nonlinear systems.

In conclusion, RELift represents a significant advancement in scientific machine learning, offering a stable, accurate, and computationally efficient method for predicting complex time-dependent PDE dynamics at fine scales.