用动态模态分解构建神经算子，高效预测时空演化过程。
A Neural Operator based on Dynamic Mode Decomposition
- 结合动态模态分解与深度学习，从数据中自动提取关键模式。
- 在热方程等三类方程上实现高精度重建，计算成本低于传统方法。
- 适合需要快速、轻量级建模的科学计算场景，如物理模拟与仿真。
科学计算方法与人工智能技术的融合仍是热门研究方向。如何在模型轻量化与计算准确性之间取得平衡，是该领域的基础挑战。本文提出一种基于动态模态分解（DMD）的神经算子，用于映射函数空间，将DMD与深度学习（DL）结合，实现对时空过程的高效建模。求解各类初值与边界条件下的偏微分方程（PDE）通常需大量计算资源。所提方法可自动提取关键模态与系统动力学特征，并据此构建预测，显著降低计算开销。通过与最接近的对比方法（DeepONet和FNO）在热方程、拉普拉斯方程和伯格斯方程近似求解中的性能比较，验证了其高效性，实现了高精度重构。
原文摘要 · Abstract (English)
The scientific computation methods development in conjunction with artificial intelligence technologies remains a hot research topic. Finding a balance between lightweight and accurate computations is a solid foundation for this direction. The study presents a neural operator based on the dynamic mode decomposition algorithm (DMD), mapping functional spaces, which combines DMD and deep learning (DL) for spatiotemporal processes efficient modeling. Solving PDEs for various initial and boundary conditions requires significant computational resources. The method suggested automatically extracts key modes and system dynamics using them to construct predictions, reducing computational costs compared to traditional numerical methods. The approach has demonstrated its efficiency through comparative analysis of performance with closest analogues DeepONet and FNO in the heat equation, Laplaces equation, and Burgers equation solutions approximation, where it achieves high reconstruction accuracy.
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