Local Convolution Enhanced Global Fourier Neural Operator For Multiscale
  Dynamic Spaces Prediction
Xuanle Zhao Yue Sun Tielin Zhang Bo Xu
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.