Mamba神经算子：谁胜出？Transformer与状态空间模型在PDEs中的比较
Mamba Neural Operator: Who Wins? Transformers vs. State-Space Models for PDEs
AI总结：
Mamba神经算子通过结合状态空间模型的优势，提升了求解PDEs的表达能力和准确性，成为优于Transformer的高效解决方案。
摘要：
偏微分方程（PDEs）被广泛用于建模复杂的物理系统，但高效求解仍然是一项重大挑战。最近，Transformer因其能够捕捉复杂依赖关系而成为PDEs的首选架构。然而，它们在表示连续动态和长程相互作用方面存在困难。为克服这些限制，我们引入了Mamba神经算子（MNO），一种新的框架，它增强了基于神经算子的技术以求解PDEs。MNO建立了结构化状态空间模型（SSMs）与神经算子之间的正式理论联系，提供了一种统一的结构，可以适应多种架构，包括基于Transformer的模型。通过利用SSMs的结构设计，MNO比传统Transformer更有效地捕捉长程依赖和连续动态。通过广泛分析，我们证明MNO显著提升了神经算子的表达能力和准确性，使其不仅成为补充，而是PDE相关任务的更优框架，弥合了高效表示和准确解近似之间的差距。我们的代码可在https://github.com/Math-ML-X/Mamba-Neural-Operator上获得。
英文摘要：
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
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
