用可学习的有限体积法加速流体模拟，兼顾精度与泛化能力
Learnable-Differentiable Finite Volume Solver for Accelerated Simulation of Flows
- 将有限体积法设计为可微架构，支持端到端训练
- 在粗网格上实现高精度模拟，误差修正提升稳定性
- 仅需少量数据训练，适合缺乏标注的物理仿真场景
流体模拟在气象、空气动力学和生物医学等领域至关重要。传统数值求解器需精细时空网格以保证稳定性、一致性和收敛性，导致计算成本高昂。尽管机器学习方法效率更高，但普遍存在可解释性差、泛化能力弱和依赖大量数据的问题。为此，我们提出一种可学习且可微的有限体积求解器（LDSolver），用于在时空粗网格上高效准确地模拟流体。LDSolver包含两个核心组件：(1) 可微有限体积求解器；(2) 一个可学习模块，用于近似通量（导数与插值）及在粗网格上的时间误差校正。即使训练数据有限（如仅少数轨迹），模型仍能显著加速模拟并保持高精度，且具备优异泛化能力。在多种流动系统（如Burgers方程、衰减流、强迫流和剪切流）上的实验表明，LDSolver达到当前最优性能，显著超越基线模型。
原文摘要 · Abstract (English)
Simulation of fluid flows is crucial for modeling physical phenomena like meteorology, aerodynamics, and biomedicine. Classical numerical solvers often require fine spatiotemporal grids to satisfy stability, consistency, and convergence conditions, leading to substantial computational costs. Although machine learning has demonstrated better efficiency, they typically suffer from issues of interpretability, generalizability, and data dependency. Hence, we propose a learnable and differentiable finite volume solver, called LDSolver, designed for efficient and accurate simulation of fluid flows on spatiotemporal coarse grids. LDSolver comprises two key components: (1) a differentiable finite volume solver, and (2) an learnable module providing equivalent approximation for fluxes (derivatives and interpolations), and temporal error correction on coarse grids. Even with limited training data (e.g., only a few trajectories), our model could accelerate the simulation while maintaining a high accuracy with superior generalizability. Experiments on different flow systems (e.g., Burgers, decaying, forced and shear flows) show that LDSolver achieves state-of-the-art performance, surpassing baseline models with notable margins.
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