RECAST：用于粗网格PDE求解器校正与超分辨率的机器学习框架
RECAST: A Machine-Learning Framework for Correction and Super-Resolution of Coarse-Grid PDE Solvers
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中文总结 AI 辅助
该研究提出机器学习框架RECAST，可在保留粗网格PDE演化的同时降低约50%-92%的时间平均相对误差，实现更粗网格的高精度PDE模拟，为高维数值模拟加速提供概念验证。
中文摘要 AI 辅助
粗网格数值求解器可大幅降低时偏微分方程（PDE）模拟的计算成本，但分辨率不足常导致解的轨迹和空间保真度下降。我们提出RECAST（粗网格轨迹的循环误差校正与超分辨率，Recurrent Error Correction And Super-resolution of coarse-grid Trajectories），这一机器学习框架旨在在保留粗网格演化的同时恢复丢失的精度。RECAST将数值时间步循环内的学习校正与从校正后的粗网格历史重建对应细网格状态相结合。我们在六个一维PDE系统（涵盖输运、扩散、色散、反应和波动动力学）上评估该框架，使用粗化因子为8-16的空间网格以及来自未见过的初始条件的1000步闭环滚动。在测试案例中，RECAST与细网格参考解保持高度一致，相比未校正的粗网格求解器，其时间平均相对误差降低约50%-92%。额外测试显示其能泛化到未见过的PDE参数值，与同期的粗校正架构对比表明，RECAST在5000步滚动中实现了更低的误差和与细网格参考更好的长程一致性。这些结果证明，RECAST的学习校正与重建能力可在不损失解保真度的情况下实现显著更粗的PDE演化，为科学与工程领域更高维数值模拟的机器学习加速提供了概念验证途径。
英文摘要
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.
