Meta-learning Pseudo-differential Operators with Deep Neural Networks

Abstract

This paper introduces a meta-learning approach for parameterized pseudo-differential operators with deep neural networks. With the help of the nonstandard wavelet form, the pseudo-differential operators can be approximated in a compressed form with a collection of vectors. The nonlinear map from the parameter to this collection of vectors and the wavelet transform are learned together from a small number of matrix-vector multiplications of the pseudo-differential operator. Numerical results for Green's functions of elliptic partial differential equations and the radiative transfer equations demonstrate the efficiency and accuracy of the proposed approach.

arXiv comment21 pages, 9 figures

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• A Mathematical Guide to Operator Learning
• Operator Learning Using Random Features: A Tool for Scientific Computing
• Solving Optical Tomography with Deep Learning
• Learning the structure of wind: A data-driven nonlocal turbulence model for the atmospheric boundary layer
• Meta-MgNet: Meta Multigrid Networks for Solving Parameterized Partial Differential Equations
• A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks
• Principled interpolation of Green's functions learned from data
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