用线性复杂度的Mamba提升偏微分方程求解的精度与几何一致性。
GeoMaNO: Geometric Mamba Neural Operator for Partial Differential Equations
- 采用Mamba结构替代Transformer，实现线性计算复杂度。
- 在多个标准PDE基准上，解算器精度最高提升58.9%。
- 适合需要高效高精度建模的物理模拟场景。
神经算子（NO）框架已成为求解偏微分方程（PDE）的强大工具。当前主流的NO基于Transformer架构，虽能捕捉长期依赖关系，但存在二次计算复杂度、缺乏几何严谨性，导致在规则网格上的表现欠佳。为此，我们提出几何Mamba神经算子（GeoMaNO），融合Mamba的建模能力、线性复杂度与几何严谨性，显著提升性能。我们在从达西流到纳维-斯托克斯问题的多个标准且广泛使用的PDE基准上评估了GeoMaNO，结果表明其在解算子逼近方面相比现有基线最高提升58.9%。
原文摘要 · Abstract (English)
The neural operator (NO) framework has emerged as a powerful tool for solving partial differential equations (PDEs). Recent NOs are dominated by the Transformer architecture, which offers NOs the capability to capture long-range dependencies in PDE dynamics. However, existing Transformer-based NOs suffer from quadratic complexity, lack geometric rigor, and thus suffer from sub-optimal performance on regular grids. As a remedy, we propose the Geometric Mamba Neural Operator (GeoMaNO) framework, which empowers NOs with Mamba's modeling capability, linear complexity, plus geometric rigor. We evaluate GeoMaNO's performance on multiple standard and popularly employed PDE benchmarks, spanning from Darcy flow problems to Navier-Stokes problems. GeoMaNO improves existing baselines in solution operator approximation by as much as 58.9%.
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