Multiwavelet-based Operator Learning for Differential Equations

Gaurav Gupta, Xiongye Xiao, Paul Bogdan

Keywords: Neural operators, multiwavelet transform, partial differential equations

Abstract: The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a $\textit{multiwavelet-based neural operator learning scheme}$ that compresses the associated operator's kernel using fine-grained wavelets. By explicitly embedding the inverse multiwavelet filters, we learn the projection of the kernel onto fixed multiwavelet polynomial bases. The projected kernel is trained at multiple scales derived from using repeated computation of multiwavelet transform. This allows learning the complex dependencies at various scales and results in a resolution-independent scheme. Compare to the prior works, we exploit the fundamental properties of the operator's kernel which enable numerically efficient representation. We perform experiments on the Korteweg-de Vries (KdV) equation, Burgers' equation, Darcy Flow, and Navier-Stokes equation. Compared with the existing neural operator approaches, our model shows significantly higher accuracy and achieves state-of-the-art in a range of datasets. For the time-varying equations, the proposed method exhibits a ($2X-10X$) improvement ($0.0018$ ($0.0033$) relative $L2$ error for Burgers' (KdV) equation). By learning the mappings between function spaces, the proposed method has the ability to find the solution of a high-resolution input after learning from lower-resolution data.

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TL;DR: We propose a novel multiwavelet-based Neural Operator scheme for learning partial differential equations, where the fundamental property of kernel smoothness enables efficient learning by multiwavelets.

Code: https://github.com/gaurav71531/mwt-operator

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2109.13459/code)