蒲公英：一种用于行星动力学神经模拟的球形 Flower 架构

中文摘要 AI 辅助

许多动力学过程在球面上展开，但默认的科学机器学习架构是欧几里得的。将这些架构应用于规则的经纬度网格会引发问题：笛卡尔卷积在高纬度发生变形；傅里叶神经算子中的二维快速傅里叶变换错误地假设存在双周期性；视觉Transformer（ViTs）中的笛卡尔位置编码会扭曲球面测地距离。近期研究转向原生球形基元，包括球形卷积（如DeepSphere或DISCO）、球形傅里叶神经算子（SFNOs）以及测地注意力。本文提出Dandelion，一种基于变形（warp）的神经偏微分方程（PDE）求解器Flower的球形版本。Dandelion的各层预测切平面位移并沿大圆传输特征。我们通过完全在球谐域中实现分层池化，得到类似U-Net的结构。因此不存在卷积：空间混合仅通过球坐标变化或变形实现。为对比Dandelion与现有球形架构，我们发布了一个包含挑战性原生球形PDE数据集的演化基准套件，包括改进的Galewsky急流、异常链式湍流、Cahn-Hilliard分解、球形黎曼激波、Held-Suarez干大气传输及全球海洋动力学。该新基准填补了现有球形数据集的缺口，现有数据集要么规模过小且过于程式化，要么（如ERA5）规模过大不适合模型迭代。Dandelion在所有数据集上均为最优或次优，且与非变形基线的差距随分辨率增大而扩大：在256×512分辨率下，Dandelion与Flower2D在单步预测和滚动预测中均占据前两位。

英文摘要

Many dynamical processes unfold on the sphere but the default scientific machine learning architectures are Euclidean. Applying these architectures on a regular lat-lon grid causes problems: Cartesian convolutions become distorted at high latitude; 2D FFTs in Fourier neural operators incorrectly assume double periodicity; Cartesian positional encodings in ViTs distort spherical geodesic distances. Recent work moves towards natively spherical primitives, including spherical convolutions (e.g., DeepSphere or DISCO), Spherical Fourier Neural Operators (SFNOs), and geodesic attention. Here we propose Dandelion, a spherical version of Flower, a warp-based neural PDE solver. Layers of Dandelion predict a tangent-plane displacement and transport features along great circles. We obtain a U-Net-like structure by implementing hierarchical pooling entirely in the spherical-harmonic domain. There are thus no convolutions: spatial mixing is achieved only through spherical coordinate changes, or warps. To compare Dandelion with existing spherical architectures, we release an evolving benchmark suite of challenging, natively-spherical PDE datasets including a modified Galewsky jet, anomalous chained turbulence, Cahn-Hilliard decomposition, spherical Riemann shocks, Held-Suarez dry atmospheric transport and global ocean dynamics. This new benchmark fills the gap in existing spherical datasets which are either too small and stylized, or much too large (ERA5) for model iteration. Dandelion is best or second-best on every dataset, and the gap to non-warp baselines widens with resolution: at $256\times 512$, Dandelion and Flower2D occupy the top two slots in both single-step prediction and rollout.

发表机构

• University of Basel（巴塞尔大学）

机构由 AI 辅助整理，请以论文原文为准。