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

2024-10-03

arXiv.org

AI Summary

Solving partial differential equations (PDEs) efficiently requires accurate continuous dynamical modeling and effective capture of long-range dependencies—challenges where standard architectures like Transformers suffer from high computational cost and poor dynamical fidelity. Method: We propose the Mamba Neural Operator (MNO), the first framework to theoretically unify structured state space models (SSMs) with neural operators. MNO embeds SSMs into the neural operator paradigm, enabling explicit continuous-time dynamics modeling and linear-complexity global interaction representation. Contribution/Results: Theoretically, MNO enjoys universal approximation capability for PDE solution operators. Empirically, it significantly outperforms Transformer-based baselines across diverse PDEs—including Navier–Stokes and Burgers equations—in both accuracy and inference speed. By bridging SSMs and neural operators, MNO establishes a new data-driven paradigm for efficient, physics-informed PDE solving.

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• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Diffusion Models for Vision

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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.

Problem

• Solving partial differential equations efficiently with neural networks
• Overcoming Transformers' limitations in continuous dynamics representation
• Enhancing long-range dependency capture in PDE solution methods

Innovation

• Mamba Neural Operator combines state-space models with neural operators
• It captures long-range dependencies and continuous dynamics effectively
• MNO enhances expressive power and accuracy for PDE solutions