Multiwavelet-based operator learning for 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. Described herein is a multiwavelet-based neural operator learning scheme that compresses the associated operator's kernel using fine-grained wavelets. The system embeds the inverse multiwavelet filters to 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. These techniques exploit the fundamental properties of the operator's kernel, which enables numerically efficient representation. These techniques show significantly higher accuracy in a large range of datasets. By learning the mappings between function spaces, these techniques can be used to find the solution of a high-resolution input after learning from lower-resolution data.

Dependencies are inferred from the claim text.