Multiwavelet-based Operator Learning for Differential Equations

TL;DR

This paper introduces a multiwavelet-based neural operator learning scheme that efficiently learns inverse operators for PDEs across multiple scales, achieving higher accuracy and resolution independence compared to prior methods.

Contribution

The paper proposes a novel multiwavelet-based neural operator that explicitly embeds inverse multiwavelet filters, enabling efficient, multi-scale learning of PDE solution operators with state-of-the-art accuracy.

Findings

Achieves significantly higher accuracy than existing neural operators.

Demonstrates 2x to 10x improvement in error for time-varying equations.

Capable of high-resolution solutions from low-resolution training data.

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.