发表机构
Indian Institute of Technology Delhi; Yardi School of Artificial Intelligence (ScAI); Department of Applied Mechanics（印度德里理工学院; 亚迪人工智能学院; 应用力学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对参数化偏微分方程，提出超图自适应小波算子HALO，在多类基准测试中精度优异且滚动稳定，可扩展至工业空气动力学几何，性能优于或媲美相关基线模型。
AI 中文摘要
物理系统通常由解算子建模，这些算子将输入场、参数、几何形状或过去状态映射到稳态或未来物理状态。学习这些映射很困难，尤其是对于必须同化历史并在自回归滚动中保持稳定的时变系统。许多神经算子在规则结构化网格上表现最佳，而实际模拟通常需要非结构化网格或点云来解析复杂几何；在这种情况下，以网格为中心的表示会损失精度。图神经算子通过消息传递或谱图滤波处理这些域，但成对边无法直接捕捉网格单元、局部邻域或守恒体积之间的组耦合。我们引入超图自适应小波算子（HALO），它将域提升到超图并在其谱小波域中学习。HALO通过切比雪夫多项式小波滤波器避免显式超图拉普拉斯特征分解，以线性稀疏矩阵成本提供局部化谱核。其可训练的二进小波尺度被正则化为紧框架覆盖，使频率响应适应每个偏微分方程，同时鼓励稳定的多尺度谱覆盖。在结构化和非结构化离散化的2D和3D基准测试中，HALO在基于频率、Transformer、DeepONet、状态空间和图的基线中实现了最佳或接近最佳的精度，并维持稳定的多步滚动。同一模型可扩展到工业空气动力学几何：在数十万个点的网格上，其性能与最强的固定离散化Transformer相当或更优，同时保持分辨率等变性。
英文摘要
Physical systems are often modeled by solution operators that map input fields, parameters, geometries, or past states to steady or future physical states. Learning these maps is difficult, especially for time-dependent systems that must assimilate history and remain stable under autoregressive rollout. Many neural operators work best on regular, structured grids, while realistic simulations often require unstructured meshes or point clouds to resolve complex geometries; in such settings, grid-centric representations can lose accuracy. Graph neural operators handle these domains through message passing or spectral graph filtering, but pairwise edges do not directly capture group-wise couplings among mesh cells, local neighborhoods, or conservation volumes. We introduce the Hypergraph Adaptive waveLet Operator (HALO), which lifts the domain to a hypergraph and learns in its spectral wavelet domain. HALO avoids explicit hypergraph-Laplacian eigendecomposition through Chebyshev polynomial wavelet filters, giving localized spectral kernels at linear sparse-matrix cost. Its trainable dyadic wavelet scales are regularized toward tight-frame coverage, allowing the frequency response to adapt to each PDE while encouraging stable multi-scale spectral coverage. Across 2D and 3D benchmarks on structured and unstructured discretizations, HALO achieves best or near-best accuracy among frequency-, transformer-, DeepONet-, state-space-, and graph-based baselines and sustains stable multi-step rollouts. The same model scales to industrial aerodynamic geometries: on meshes of a few hundred thousand points it is on par with, or better than, the strongest fixed-discretization transformers, while remaining resolution-equivariant.
