Alias-Free Mamba Neural Operator

Benefiting from the booming deep learning techniques, neural operators (NO) are considered as an ideal alternative to break the traditions of solving Partial Differential Equations (PDE) with expensive cost.Yet with the remarkable progress, current solutions concern little on the holistic function features--both global and local information-- during the process of solving PDEs.Besides, a meticulously designed kernel integration to meet desirable performance often suffers from a severe computational burden, such as GNO with $O(N(N-1))$, FNO with $O(NlogN)$, and Transformer-based NO with $O(N^2)$.To counteract the dilemma, we propose a mamba neural operator with $O(N)$ computational complexity, namely MambaNO.Functionally, MambaNO achieves a clever balance between global integration, facilitated by state space model of Mamba that scans the entire function, and local integration, engaged with an alias-free architecture. We prove a property of continuous-discrete equivalence to show the capability ofMambaNO in approximating operators arising from universal PDEs to desired accuracy. MambaNOs are evaluated on a diverse set of benchmarks with possibly multi-scale solutions and set new state-of-the-art scores, yet with fewer parameters and better efficiency.