用可微物理方法自动简化非结构网格，大幅降计算量仍保精度。
Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation
- 基于k-means聚类与自动微分实现网格自适应粗化
- 网格点数最多减少10倍，关键区域动态保持一致
- 适用于任意演化型偏微分方程的仿真加速
由于现代数值模拟计算负载高，亟需在保持合理精度的前提下缩小离散问题规模。本文提出一种基于可微物理的新算法，通过k-means聚类、自动微分和随机优化实现非结构网格的粗化。我们在两个偏微分方程上验证：一个描述多孔介质中轻微可压缩流体流动的线性抛物方程，以及波动方程。结果表明，在所考虑场景下，网格点数最多可减少10倍，同时在关注点处仍能准确保留模拟变量的动力学特征。该方法可推广至任意由演化型偏微分方程描述的系统仿真。
原文摘要 · Abstract (English)
Due to the high computational load of modern numerical simulation, there is a demand for approaches that would reduce the size of discrete problems while keeping the accuracy reasonable. In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics. We achieve this by employing k-means clustering, autodifferentiation and stochastic minimization algorithms. We demonstrate performance of the designed algorithm on two PDEs: a linear parabolic equation which governs slightly compressible fluid flow in porous media and the wave equation. Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest. The proposed approach can be applied to the simulation of an arbitrary system described by evolutionary partial differential equations.
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