用傅里叶谱变换+Transformer预测非线性偏微分方程，少数据也能准且快。
The Fourier Spectral Transformer Networks For Efficient and Generalizable Nonlinear PDEs Prediction
- 将偏微分方程转为谱系数的常微分方程，用Transformer建模演化过程。
- 在二维纳维-斯托克斯和一维伯格斯方程上，长时预测误差低于传统方法。
- 适用于实时预测与控制复杂系统，尤其适合训练数据少的场景。
本文提出一种统一的傅里叶谱Transformer网络，融合经典谱方法与基于注意力的神经架构优势。通过将原始偏微分方程转换为谱空间的常微分方程，利用高精度数值求解器生成训练数据，并采用Transformer网络建模谱系数的演化过程。在二维不可压缩纳维-斯托克斯方程与一维伯格斯方程上验证了该方法的有效性。结果表明，即使训练数据有限，该谱Transformer仍能实现高精度的长期预测，优于传统数值方法和机器学习方法，在未来流场动态预测方面表现更优。所提框架对未见数据具有良好的泛化能力，为复杂动力系统的实时预测与控制提供了有前景的新范式。
原文摘要 · Abstract (English)
In this work we propose a unified Fourier Spectral Transformer network that integrates the strengths of classical spectral methods and attention based neural architectures. By transforming the original PDEs into spectral ordinary differential equations, we use high precision numerical solvers to generate training data and use a Transformer network to model the evolution of the spectral coefficients. We demonstrate the effectiveness of our approach on the two dimensional incompressible Navier-Stokes equations and the one dimensional Burgers' equation. The results show that our spectral Transformer can achieve highly accurate long term predictions even with limited training data, better than traditional numerical methods and machine learning methods in forecasting future flow dynamics. The proposed framework generalizes well to unseen data, bringing a promising paradigm for real time prediction and control of complex dynamical systems.
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