GeoMaNO: Geometric Mamba Neural Operator for Partial Differential Equations

TL;DR

GeoMaNO introduces a geometric neural operator with linear complexity and improved accuracy for solving PDEs, outperforming Transformer-based methods on various benchmarks.

Contribution

It presents GeoMaNO, a novel neural operator framework combining geometric rigor and linear complexity, addressing limitations of existing Transformer-based NOs.

Findings

Achieves up to 58.9% improvement over baselines.

Demonstrates effectiveness on Darcy flow and Navier-Stokes problems.

Provides a scalable, geometrically rigorous approach for PDE solutions.

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%.