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Vandermonde Neural Operators

by L. Lingsch and M. Michelis and S. Perera and R. Katzschmann and S. Mishra

(Report number 2023-24)

Abstract
Fourier Neural Operators (FNOs) have emerged as very popular machine learning architectures for learning operators, particularly those arising in PDEs. However, as FNOs rely on the fast Fourier transform for computational efficiency, the architecture can be limited to input data on equispaced Cartesian grids. Here, we generalize FNOs to handle input data on non-equispaced point distributions. Our proposed model, termed as Vandermonde Neural Operator (VNO), utilizes Vandermonde-structured matrices to efficiently compute forward and inverse Fourier transforms, even on arbitrarily distributed points. We present numerical experiments to demonstrate that VNOs can be significantly faster than FNOs, while retaining comparable accuracy, and improve upon accuracy of comparable non-equispaced methods such as the Geo-FNO.

Keywords: Neural Operators, Vandermonde Matrices, Operator learning

@Techreport{LMPKM23_1061,
  author = {L. Lingsch and M. Michelis and S. Perera and R. Katzschmann and S. Mishra},
  title = {Vandermonde Neural Operators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-24.pdf },
  year = {2023}
}

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