🤖 AI Summary
This work addresses the significant degradation in solution trajectories and spatial fidelity caused by insufficient resolution in coarse-grid partial differential equation (PDE) solvers. To overcome this limitation, the authors propose RECAST, a novel framework that uniquely integrates recurrent error correction with super-resolution reconstruction into the PDE solving pipeline. Specifically, a recurrent neural network models and corrects temporal errors on the coarse grid, and high-resolution solutions are subsequently reconstructed from the corrected historical states. Evaluated across six classes of one-dimensional PDE systems, RECAST achieves high-fidelity long-horizon rollouts, demonstrating strong generalization to unseen initial conditions and parameters. Compared to uncorrected coarse-grid solvers, it reduces time-averaged relative errors by 50%–92% and outperforms existing methods over 5,000-step predictions.
📝 Abstract
Coarse-grid numerical solvers can substantially reduce the computational cost of time-dependent PDE simulation, but under-resolution often degrades both the trajectory and the spatial fidelity of the solution. We introduce RECAST (Recurrent Error Correction And Super-resolution of coarse-grid Trajectories), a machine-learning framework designed to restore this lost accuracy while retaining coarse-grid evolution. RECAST combines learned correction within the numerical time-stepping loop with reconstruction of the corresponding fine-grid state from the corrected coarse history. We evaluate the framework on six one-dimensional PDE systems spanning transport, diffusion, dispersion, reaction, and wave dynamics, using spatial grids coarsened by factors of 8-16 and 1000-step closed-loop rollouts from unseen initial conditions. Across the test cases, RECAST remains closely aligned with the fine-grid reference solutions and reduces time-averaged relative error by approximately 50-92% compared with the corresponding uncorrected coarse-grid solvers. Additional tests show generalization to unseen PDE parameter values, while comparison with a contemporary coarse-correction architecture shows that RECAST achieves lower error and better long-horizon agreement with the fine-grid reference over 5000-step rollouts. These results demonstrate that the learned correction and reconstruction capabilities of RECAST can enable substantially coarser PDE evolution without the corresponding loss of solution fidelity, providing a proof-of-concept route toward machine-learning acceleration of higher-dimensional numerical simulations across science and engineering.