  Neural operators improve conventional neural networks by expanding their
capabilities of functional mappings between different function spaces to solve
partial differential equations (PDEs). One of the most notable methods is the
Fourier Neural Operator (FNO), which draws inspiration from Green's function
method and directly approximates operator kernels in the frequency domain.
However, after empirical observation followed by theoretical validation, we
demonstrate that the FNO approximates kernels primarily in a relatively
low-frequency domain. This suggests a limited capability in solving complex
PDEs, particularly those characterized by rapid coefficient changes and
oscillations in the solution space. Such cases are crucial in specific
scenarios, like atmospheric convection and ocean circulation. To address this
challenge, inspired by the translation equivariant of the convolution kernel,
we propose a novel hierarchical Fourier neural operator along with
convolution-residual layers and attention mechanisms to make them complementary
in the frequency domain to solve complex PDEs. We perform experiments on
forward and reverse problems of multiscale elliptic equations, Navier-Stokes
equations, and other physical scenarios, and find that the proposed method
achieves superior performance in these PDE benchmarks, especially for equations
characterized by rapid coefficient variations.
