Dilated convolution neural operator for multiscale partial differential equations
This provides a promising solution for multiscale operator learning in scientific computing, though it appears incremental as it builds on existing neural operator methods with a hybrid approach.
This paper tackled the problem of learning operators for multiscale partial differential equations by introducing the Dilated Convolutional Neural Operator (DCNO), which preserves high-frequency information and achieves an optimal balance between accuracy and computational cost across various datasets like the multiscale elliptic and Navier-Stokes equations.
This paper introduces a data-driven operator learning method for multiscale partial differential equations, with a particular emphasis on preserving high-frequency information. Drawing inspiration from the representation of multiscale parameterized solutions as a combination of low-rank global bases (such as low-frequency Fourier modes) and localized bases over coarse patches (analogous to dilated convolution), we propose the Dilated Convolutional Neural Operator (DCNO). The DCNO architecture effectively captures both high-frequency and low-frequency features while maintaining a low computational cost through a combination of convolution and Fourier layers. We conduct experiments to evaluate the performance of DCNO on various datasets, including the multiscale elliptic equation, its inverse problem, Navier-Stokes equation, and Helmholtz equation. We show that DCNO strikes an optimal balance between accuracy and computational cost and offers a promising solution for multiscale operator learning.