RELift: Learned Coarse-to-Fine Propagators for Time-Dependent PDEs with Applications to Electron Dynamics
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914392046567424 |
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| author | Bassi, Hardeep Zhu, Yuanran Ye, Erika Ren, Pu Dektor, Alec Mahoney, Michael W. Bhat, Harish S. Yang, Chao |
| author_facet | Bassi, Hardeep Zhu, Yuanran Ye, Erika Ren, Pu Dektor, Alec Mahoney, Michael W. Bhat, Harish S. Yang, Chao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | RELift: Learned Coarse-to-Fine Propagators for Time-Dependent PDEs with Applications to Electron Dynamics Bassi, Hardeep Zhu, Yuanran Ye, Erika Ren, Pu Dektor, Alec Mahoney, Michael W. Bhat, Harish S. Yang, Chao Numerical Analysis Analysis of PDEs 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. |
| title | RELift: Learned Coarse-to-Fine Propagators for Time-Dependent PDEs with Applications to Electron Dynamics |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2509.12220 |