Flowers: 神经PDE求解器的曲速引擎
Flowers: A Warp Drive for Neural PDE Solvers
AI总结：
提出Flowers架构，通过多头扭曲场实现线性代价的自适应全局交互，在2D/3D时变PDE基准上超越傅里叶、卷积和注意力基线。
摘要：
我们引入了Flowers，一种完全由多头扭曲构建的神经架构，用于学习PDE解算子。除了逐点通道混合和多尺度支架外，Flowers不使用傅里叶乘子、点积注意力或卷积混合。每个头预测一个位移场并扭曲混合后的输入特征。受物理和计算效率的启发，位移是逐点预测的，没有任何空间聚合，非局域性仅通过每个头在源坐标处的稀疏采样引入。在多尺度残差块中堆叠扭曲得到Flowers，它以线性代价实现自适应的全局交互。我们通过三个互补视角从理论上论证了这一设计：守恒律的流图、非均匀介质中的波以及动力学理论的连续极限。Flowers在一系列2D和3D时变PDE基准上取得了优异性能，特别是流和波。一个紧凑的17M参数模型持续优于相似规模的傅里叶、卷积和注意力基线，而一个150M参数变体在参数、数据和训练计算量多得多的情况下，超越了近期基于transformer的基础模型。
英文摘要：
We introduce Flowers, a neural architecture for learning PDE solution operators built entirely from multihead warps. Aside from pointwise channel mixing and a multiscale scaffold, Flowers use no Fourier multipliers, no dot-product attention, and no convolutional mixing. Each head predicts a displacement field and warps the mixed input features. Motivated by physics and computational efficiency, displacements are predicted pointwise, without any spatial aggregation, and nonlocality enters only through sparse sampling at source coordinates, one per head. Stacking warps in multiscale residual blocks yields Flowers, which implement adaptive, global interactions at linear cost. We theoretically motivate this design through three complementary lenses: flow maps for conservation laws, waves in inhomogeneous media, and a kinetic-theoretic continuum limit. Flowers achieve excellent performance on a broad suite of 2D and 3D time-dependent PDE benchmarks, particularly flows and waves. A compact 17M-parameter model consistently outperforms Fourier, convolution, and attention-based baselines of similar size, while a 150M-parameter variant improves over recent transformer-based foundation models with much more parameters, data, and training compute.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
