论文标题
PDE的非平衡傅立叶神经求解器
Non-equispaced Fourier Neural Solvers for PDEs
论文作者
论文摘要
求解部分微分方程很困难。最近提出的神经分辨率不变模型，尽管它们的有效性和效率通常需要稳定的数据点。但是，空间域中的采样有时不可避免地是在现实世界中不可平衡的，从而限制了它们的适用性。在本文中，我们提出了一个非平衡的傅立叶PDE求解器（\ textsc {nfs}），并在重新采样的arepaced点上具有自适应插值，以及作为其组件的傅立叶神经操作员的变体。复杂PDE的实验结果证明了其在准确性和效率方面的优势。与空间平衡的基准方法相比，它以$ 42.85 \％$的改进获得了卓越的性能，并且能够以微小的准确性损失处理非平衡数据。此外，据我们所知，\ textsc {nfs}是第一个基于ML的方法，具有网格不变的推理能力，可以在非平衡场景中成功建模湍流，并且在未见空间点上的错误略有偏差。
Solving partial differential equations is difficult. Recently proposed neural resolution-invariant models, despite their effectiveness and efficiency, usually require equispaced spatial points of data. However, sampling in spatial domain is sometimes inevitably non-equispaced in real-world systems, limiting their applicability. In this paper, we propose a Non-equispaced Fourier PDE Solver (\textsc{NFS}) with adaptive interpolation on resampled equispaced points and a variant of Fourier Neural Operators as its components. Experimental results on complex PDEs demonstrate its advantages in accuracy and efficiency. Compared with the spatially-equispaced benchmark methods, it achieves superior performance with $42.85\%$ improvements on MAE, and is able to handle non-equispaced data with a tiny loss of accuracy. Besides, to our best knowledge, \textsc{NFS} is the first ML-based method with mesh invariant inference ability to successfully model turbulent flows in non-equispaced scenarios, with a minor deviation of the error on unseen spatial points.
