WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations
Not on the map. Why?
Abstract (arXiv)
This work introduces the Wavelet-Laplace Neural Operator (WLNO), a novel neural operator that fuses Haar wavelet multi-scale spatial decomposition with the Laplace-domain pole-residue formulation of the Laplace Neural Operator (LNO). While LNO captures transient and steady-state dynamics through learnable system poles and residues, it lacks an explicit mechanism for extracting spatially localized multi-scale features inherent in complex PDE solutions. WLNO addresses this by augmenting the LNO core with a parallel single-level Haar discrete wavelet transform (DWT) branch that decomposes the lifted feature map into four frequency subbands: approximation (LL), horizontal detail (LH), vertical detail (HL), and diagonal detail (HH) and applies independent learned 1×1 convolutions to each subband before reconstruction via the inverse DWT. The two branches are fused through a learnable sigmoid-gated weight αwav, initialized to give a small initial contribution to the wavelet branch, allowing the model to adaptively balance Laplace-domain dynamics against spatial multi-scale features throughout training. WLNO is evaluated against LNO on five benchmark PDE problems using identical hyperparameters, training data, and evaluation protocols: the diffusion equation, the Burgers equation, the reaction-diffusion system, Darcy flow, and the two-dimensional Navier-Stokes equation. WLNO consistently outperforms LNO on all five problems, with the most pronounced improvement on problems with strong spatial multi-scale structure, such as the Burgers equation with sharp shock fronts and the Navier-Stokes equation with coherent vortical structures, while remaining consistent across smoother and elliptic problems. These results demonstrate that wavelet-based multi-scale spatial decomposition is a principled and effective complement to Laplace-domain operator learning.
Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 34 of its 65 references that are in the map. First posted on arXiv 23 May 2026.
Builds on
34 of 65 references.
References · 34
- 2017Attention Is All You Needcited by 59,932
- 2015Deep Residual Learning for Image Recognitioncited by 51,809
- 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
- 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
- 2016Instance Normalization: The Missing Ingredient for Fast Stylizationcited by 1,187
- 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
- 2019DeepXDE: A deep learning library for solving differential equationscited by 617
- 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
- 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
- 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
22 more
- 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
- 2021Choose a Transformer: Fourier or Galerkincited by 333
- 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 293
- 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
- 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
- 2021Factorized Fourier Neural Operatorscited by 237
- 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
- 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
- 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 225
- 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
- 2024Poseidon: Efficient Foundation Models for PDEscited by 186
- 2022Learning Operators with Coupled Attentioncited by 88
- 2021Improved architectures and training algorithms for deep operator networkscited by 85
- 2019Multi-level Wavelet Convolutional Neural Networkscited by 59
- 2021Seismic wave propagation and inversion with Neural Operatorscited by 51
- 2023LNO: Laplace Neural Operator for Solving Differential Equationscited by 47
- 2024Generating synthetic data for neural operatorscited by 11
- 2025HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutionscited by 8
- 2025FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learningcited by 7
- 2025Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximationcited by 5
- 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
- 2026SIMR-NO: A Spectrally-Informed Multi-Resolution Neural Operator for Turbulent Flow Super-Resolutioncited by 3
Built on by
Citing papers per year: 2026: 1.
1 paper.