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arXiv · 2505.12020

GeoMaNO: Geometric Mamba Neural Operator for Partial Differential Equations

Xi Han·Jingwei Zhang·Dimitris Samaras·Fei Hou·Hong Qin
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Abstract

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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Xi Han, Jingwei Zhang, Dimitris Samaras, Fei Hou, Hong Qin. 2025-05-17. GeoMaNO: Geometric Mamba Neural Operator for Partial Differential Equations. https://arxiv.org/abs/2505.12020

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