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.

通过改进的傅里叶层和注意机制，我们提出了一种新颖的分层神经算子，旨在捕获所有细节并在不同尺度上处理它们，以解决多尺度偏微分方程问题。在多个物理场景中进行实验，并在现有偏微分方程基准测试中取得卓越性能，尤其是具有快速系数变化特征的方程。