# Meta-learning Pseudo-differential Operators with Deep Neural Networks

## TL;DR

This paper presents a meta-learning method using deep neural networks to efficiently approximate parameterized pseudo-differential operators, enabling accurate solutions for complex PDEs with limited computations.

## Contribution

It introduces a novel meta-learning framework that combines wavelet transforms and neural networks to approximate pseudo-differential operators from minimal data.

## Key findings

- Efficient approximation of Green's functions for elliptic PDEs.
- Accurate modeling of radiative transfer equations.
- Reduced computational cost for operator evaluation.

## 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.