Meta-learning Pseudo-differential Operators with Deep Neural Networks
This work addresses computational challenges in solving partial differential equations for researchers in scientific computing, though it appears incremental as it builds on existing wavelet and neural network methods.
The paper tackles the problem of approximating parameterized pseudo-differential operators using a meta-learning approach with deep neural networks, achieving efficiency and accuracy as demonstrated in numerical results for Green's functions of elliptic PDEs and radiative transfer equations.
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.