A Structured Matrix Method for Nonequispaced Neural Operators

Levi E. Lingsch, Mike Yan Michelis, Emmanuel de Bezenac, Sirani M. Perera, Robert K. Katzschmann, Siddhartha Mishra

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Keywords: Nonequispaced Neural Operators, Operator Learning, Fourier Neural Operator, Deep Learning for Scientific Computing

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TL;DR: We demonstrate efficient Fourier and spherical harmonic transformations via structured rectangular matrices for neural operator learning on nonequispaced data.

Abstract: The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. However, as FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. We address this issue by proposing a novel method that leverages batch matrix multiplications to efficiently construct Vandermonde-structured matrices and compute forward and inverse transforms, on arbitrarily distributed points. An efficient implementation of such *structured matrix methods* is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows one to extend neural operators to very general point distributions with significant gains in training speed over baselines, while retaining or improving accuracy.

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Submission Number: 5379