超网络参数化的空间自适应神经算子用于偏微分方程学习

中文摘要 AI 辅助

空间异质偏微分方程（PDE）表现出由几何和物理系数变化引起的位置相关动力学。现有的神经算子通过多尺度特征、注意力机制或区域分解来改进局部建模，但其更新规则通常仍然是空间共享的。基于超网络的方法在不同PDE实例间调整参数，但通常每个实例仅生成一个全局参数化。因此，共享算子可能对边界和高梯度区域拟合不足，这些局部误差在自回归滚动过程中会累积。我们提出了一种空间自适应神经算子（SANO），将这种空间共享的参数化替换为位置相关算子参数的空间连续场。SANO使用傅里叶编码坐标和坐标条件超网络在采样点生成空间算子条件编码。一种超神经单元（HNE）机制在局部子区域内插值这些编码，耦合相邻算子同时允许其更新规则在空间上变化，并且单位分解权重组装重叠的局部预测。在一维、二维和三维偏微分方程以及两个穿孔域椭圆基准上的实验表明，SANO始终优于具有竞争力的神经算子、基于超网络和物理信息驱动的基线方法。

英文摘要

Spatially heterogeneous partial differential equations (PDEs) exhibit location-dependent dynamics arising from variations in geometry and physical coefficients. Existing neural operators improve localized modeling through multiscale features, attention mechanisms, or domain decomposition, yet their update rules often remain spatially shared. Hypernetwork-based methods adapt parameters across PDE instances but typically generate only one global parameterization per instance. Consequently, shared operators may underfit boundaries and high-gradient regions, with these localized errors accumulating during autoregressive rollout. We propose a spatially adaptive neural operator (SANO), which replaces this spatially shared parameterization with a spatially continuous field of location-dependent operator parameters. SANO uses Fourier-encoded coordinates and a coordinate-conditioned hypernetwork to generate spatial operator-conditioning codes at sampling points. A Hyper-Neural Element (HNE) mechanism interpolates these codes within local subregions, coupling neighboring operators while allowing their update rules to vary across space, and partition-of-unity weights assemble the overlapping local predictions. Experiments on one-, two-, and three-dimensional PDEs and two perforated-domain elliptic benchmarks show that SANO consistently outperforms competitive neural-operator, hypernetwork-based, and physics-informed baselines.

发表机构

• UESTC（电子科技大学）
• KHU（庆熙大学）
• THU（清华大学）
• SJTU（上海交通大学）
• HIT（哈尔滨工业大学）

机构由 AI 辅助整理，请以论文原文为准。