GeoMaNO: Geometric Mamba Neural Operator for Partial Differential Equations
This work addresses the computational and geometric limitations in PDE-solving neural operators for researchers and practitioners in scientific computing, though it appears incremental as it builds on existing Mamba and neural operator frameworks.
The paper tackled the problem of inefficient and geometrically limited Transformer-based neural operators for solving partial differential equations (PDEs), resulting in the proposed GeoMaNO framework that improves solution operator approximation by up to 58.9% on benchmarks like Darcy flow and Navier-Stokes problems.
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%.