Mechanism-preserved adaptive daubechies wavelet neural operator.

Abstract

Wavelets exhibit joint localization in spatial and frequency domains, making them effective for learning solutions of partial differential equations (PDEs) with complex geometries. However, existing adaptive Daubechies-based methods compromise between channel-wise adaptivity and orthogonality, while neglecting key wavelet mechanisms that often result in oscillatory solutions or blurred edges. To address these challenges, we propose a Mechanism-Preserved Adaptive Daubechies (MPAD) wavelet neural operator. MPAD constructs independent learnable wavelet bases for each feature channel and enforces orthogonality under the Quadrature Mirror Filter constraint, achieving non-redundant feature representations with channel- and coefficient-wise adaptivity. Furthermore, we incorporate explicit Direct Current Preservation and Vanishing Moment losses to enhance low-frequency energy stability and high-frequency detail resolution, respectively. This dual enhancement preserves the wavelet mechanisms and ensures stable solutions with sharp boundaries. Extensive experiments on parametric PDEs demonstrate that MPAD achieves significant performance improvements with negligible parameter overhead. For example, on the Burgers equation, MPAD-WNO reduces the relative L² error from 0.85 to 0.337 compared to the original WNO, while MPAD-UWNO further outperforms strong baselines across multiple PDE datasets. Moreover, MPAD exhibits robust cross-domain applicability. It achieves 32.01 dB and 34.92 dB PSNR on the BSD68 and CBSD68 datasets for image denoising, and yields a maximum error reduction of over 50% on the ERA5 Monthly weather forecasting task, demonstrating clear advantages in long-term temporal modeling.