类谱无网格离散化学习

中文摘要 AI 辅助

带有核修正的光滑粒子流体动力学（SPH）、径向基函数生成的有限差分（RBF-FD）、局部各向异性基函数方法（LABFM）等无网格方法，通过在局部模板上施加多项式一致性来构造离散微分算子。对于包含节点数多于一致性约束数的模板，所得线性系统是欠定的，剩余自由度通过核的选择、基预处理或最小范数条件隐式确定。多项式一致性仅在低波数极限下约束算子，构造过程中没有任何部分针对细尺度内容所在的波数选择精度。我们提出类谱神经离散化（SpeND），将这些自由度的选择转化为一个学习问题：模板权重由以局部节点几何为条件的神经网络参数化，训练目标是在可分辨频带上近似谱算子的模态响应。一层硬约束投影将网络输出映射到一致权重的仿射子空间，因此多项式一致性是构造性地精确成立，而非作为惩罚项。训练是自监督且与物理无关的，无需参考解；目标函数最小化规定带限函数空间上的色散和耗散误差。对无序二维节点分布的模态分析表明，在相同模板尺寸下，学习到的四阶算子在宽得多的频带上遵循精确响应，优于显式LABFM，也优于结构化网格上的四阶有限差分，同时在加密时恢复了预期的四阶收敛速率。

英文摘要

Numerical methods for partial differential equations (PDEs) discretise differential operators, ideally reproducing the action of the continuous operators across all wavenumbers permitted by a given discretisation. Spectral-type methods approach this ideal, but rely on structured grids or high-order meshes that are hard to generate for complex geometries. In contrast, mesh-free methods are geometrically flexible, yet how faithfully they reproduce the operator across resolved scales is strongly influenced by a heuristically chosen kernel, which selects one of many weight sets satisfying the same consistency conditions without regard to the resulting spectral response. To address this, we introduce Spectrally optimised Neural Discretisations (SpeND), a framework that learns the map from local stencil geometry to discretisation weights on unstructured point clouds. A projection layer constrains every predicted stencil to the affine set defined by the discrete moment conditions, so polynomial consistency, and hence formal order of accuracy, holds exactly. Since consistency is enforced by the architecture, the weights can additionally be optimised for accuracy on a prescribed function space. Herein, we use Fourier modes and target the exact differentiation response over a chosen wavenumber band, so training is unsupervised. The loss function is a design interface: changing how it weights wavenumbers yields operators with distinct accuracy profiles. The trained discrete differential operators are PDE-agnostic and are applied without retraining to the Poisson, Burgers and Navier--Stokes equations, to near-boundary stencils, and across resolutions. At the same order and resolution, SpeND matches or improves on established mesh-free discretisations, and has been shown to reduce the wall-clock time between $3-20\times$ to achieve equivalent error as the baselines.

发表机构

• School of Engineering, University of Manchester（曼彻斯特大学工程学院）
• School of Engineering, Cardiff University（卡迪夫大学工程学院）

机构由 AI 辅助整理，请以论文原文为准。