Neural operators extend the capabilities of traditional neural networks by allowing them to handle mappings between function spaces for the purpose of solving partial differential equations (PDEs). One of the most notable methods is the Fourier Neural Operator (FNO), which is inspired by Green's function method and approximate operator kernel directly in the frequency domain. In this work, we focus on predicting multiscale dynamic spaces, which is equivalent to solving multiscale PDEs. Multiscale PDEs are characterized by rapid coefficient changes and solution space oscillations, which are crucial for modeling atmospheric convection and ocean circulation. To solve this problem, models should have the ability to capture rapid changes and process them at various scales. However, the FNO only approximates kernels in the low-frequency domain, which is insufficient when solving multiscale PDEs. To address this challenge, we propose a novel hierarchical neural operator that integrates improved Fourier layers with attention mechanisms, aiming to capture all details and handle them at various scales. These mechanisms complement each other in the frequency domain and encourage the model to solve multiscale problems. We perform experiments on dynamic spaces governed by forward and reverse problems of multiscale elliptic equations, Navier-Stokes equations and some other physical scenarios, and reach superior performance in existing PDE benchmarks, especially equations characterized by rapid coefficient variations.

翻译：神经算子通过允许传统神经网络处理函数空间之间的映射，扩展了其能力，旨在求解偏微分方程（PDEs）。其中最显著的方法之一是傅立叶神经算子（FNO），该方法受格林函数法启发，直接在频域中近似算子核。本文聚焦于预测多尺度动态空间，这等价于求解多尺度偏微分方程。多尺度偏微分方程的特点是系数快速变化和解空间振荡，这对于模拟大气对流和海洋环流至关重要。为解决此问题，模型应具备捕捉快速变化并在不同尺度上处理这些变化的能力。然而，FNO仅在低频域中近似核，这在求解多尺度偏微分方程时是不够的。为此，我们提出了一种新颖的层级神经算子，该算子将改进的傅立叶层与注意力机制相结合，旨在捕捉所有细节并在不同尺度上处理它们。这些机制在频域中相互补充，并促使模型求解多尺度问题。我们在由多尺度椭圆方程的正向和反向问题、纳维-斯托克斯方程及其他物理场景所控制的动态空间上进行实验，并在现有PDE基准测试中达到了优越性能，特别是针对具有快速系数变化特征的方程。