Dilated convolution neural operator for multiscale partial differential equations
<|reference_start|>Dilated convolution neural operator for multiscale partial differential 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.<|reference_end|>
@article{xu2024dilated,
    title={Dilated convolution neural operator for multiscale partial differential
  equations},
    author={Bo Xu, Xinliang Liu, Lei Zhang},
    journal={arXiv preprint arXiv:2408.00775},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.00775},
    primaryClass={cs.LG cs.NA math.NA}
}