Mamba Neural Operator: Who Wins? Transformers vs. State-Space Models for PDEs

TL;DR

The paper introduces Mamba Neural Operator (MNO), a new framework that unifies state-space models and neural operators to improve PDE solving by capturing long-range dependencies and continuous dynamics more effectively than Transformers.

Contribution

MNO provides a formal connection between structured state-space models and neural operators, enhancing PDE solution accuracy and efficiency beyond traditional Transformer approaches.

Findings

MNO outperforms Transformers in capturing long-range dependencies.

MNO significantly improves the accuracy of neural operators for PDEs.

The framework unifies diverse architectures under a common structure.

Abstract

Partial differential equations (PDEs) are widely used to model complex physical systems, but solving them efficiently remains a significant challenge. Recently, Transformers have emerged as the preferred architecture for PDEs due to their ability to capture intricate dependencies. However, they struggle with representing continuous dynamics and long-range interactions. To overcome these limitations, we introduce the Mamba Neural Operator (MNO), a novel framework that enhances neural operator-based techniques for solving PDEs. MNO establishes a formal theoretical connection between structured state-space models (SSMs) and neural operators, offering a unified structure that can adapt to diverse architectures, including Transformer-based models. By leveraging the structured design of SSMs, MNO captures long-range dependencies and continuous dynamics more effectively than traditional Transformers. Through extensive analysis, we show that MNO significantly boosts the expressive power and accuracy of neural operators, making it not just a complement but a superior framework for PDE-related tasks, bridging the gap between efficient representation and accurate solution approximation. Our code is available on https://github.com/Math-ML-X/Mamba-Neural-Operator