U-WNO: U-Net Enhanced Wavelet Neural Operator for Solving Parametric Partial Differential Equations

TL;DR

U-WNO enhances wavelet neural operators with a U-Net architecture and adaptive activation to better capture high-frequency features in solving parametric PDEs, significantly improving accuracy over previous methods.

Contribution

The paper introduces U-WNO, a novel neural operator architecture that combines U-Net and wavelet layers with adaptive activation to improve high-frequency feature learning in PDE solutions.

Findings

Achieves 45-83% error reduction compared to baseline WNO.

Demonstrates effectiveness across seven PDE families.

Provides a framework for multiresolution analysis in operator learning.

Abstract

High-frequency features are critical in multiscale phenomena such as turbulent flows and phase transitions, since they encode essential physical information. The recently proposed Wavelet Neural Operator (WNO) utilizes wavelets' time-frequency localization to capture spatial manifolds effectively. While its factorization strategy improves noise robustness, it suffers from high-frequency information loss caused by finite-scale wavelet decomposition. In this study, a new U-WNO network architecture is proposed. It incorporates the U-Net path and residual shortcut into the wavelet layer to enhance the extraction of high-frequency features and improve the learning of spatial manifolds. Furthermore, we introduce an adaptive activation mechanism to mitigate spectral bias through trainable slope parameters. Extensive benchmarks across seven PDE families (Burgers, Darcy flow, Navier-Stokes, etc.) show that U-WNO achieves 45--83% error reduction compared to baseline WNO, with mean $L_2$ relative errors ranging from 0.043% to 1.56%. This architecture establishes a framework combining multiresolution analysis with deep feature learning, addressing the spectral-spatial tradeoff in operator learning. Code and data used are available on https://github.com/WeiminLei/U-WNO.git.