Neural operators are effective tools for solving parametric partial differential equations (PDEs). They can predict solutions of PDEs with different initial and boundary conditions, as well as different input functions. The recently proposed Wavelet Neural Operator (WNO) utilizes the time-frequency localization of wavelets to capture spatial manifolds effectively. While WNO has shown promise as an operator learning method, it only parameterizes neural network weights under higher-order wavelet factorization. This approach avoids noise interference but may result in insufficient extraction of high-frequency features from the data. In this study, we propose a new network architecture called U-WNO. 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. Additionally, we introduce the Adaptive Activation Function into the wavelet layer to address the spectral bias of the neural network. The effectiveness of U-WNO is demonstrated through numerical experiments on various problems, including the Burgers equation, Darcy flow, Navier-Stokes equation, Allen-Cahn equation, Non-homogeneous Poisson equation, and Wave advection equation. This study also includes a comparative analysis of existing operator learning frameworks.

翻译：神经算子是求解参数化偏微分方程（PDE）的有效工具。它们能够预测具有不同初始条件、边界条件以及不同输入函数的PDE解。近期提出的小波神经算子（WNO）利用小波的时频局部化特性，有效捕捉空间流形。尽管WNO作为一种算子学习方法展现出潜力，但其仅在高阶小波分解下参数化神经网络权重。该方法虽避免了噪声干扰，但可能导致数据中高频特征提取不足。本研究提出一种名为U-WNO的新型网络架构。该架构将U-Net路径与残差捷径融入小波层，以增强高频特征提取并提升空间流形学习能力。此外，我们在小波层中引入自适应激活函数以缓解神经网络的谱偏差问题。通过对多种问题的数值实验（包括Burgers方程、达西流、Navier-Stokes方程、Allen-Cahn方程、非齐次泊松方程及波动平流方程），验证了U-WNO的有效性。本研究还包含对现有算子学习框架的比较分析。