WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations

2026-05-23

AI Summary

This work addresses the limitation of existing Laplacian Neural Operators (LNOs) in explicitly capturing spatial multiscale features inherent in solutions to partial differential equations. To overcome this, the authors propose the first integration of Haar wavelet-based multiscale decomposition with LNOs, employing parallel wavelet branches to extract LL, LH, HL, and HH subband features. A learnable sigmoid gating mechanism is introduced to adaptively fuse multiscale spatial information with temporal dynamics. The architecture leverages discrete wavelet transforms, 1×1 convolutions, an LNO backbone, and inverse wavelet reconstruction. Evaluated across five benchmark problems—including diffusion, Burgers’, reaction–diffusion, Darcy flow, and Navier–Stokes equations—the method consistently outperforms the original LNO, demonstrating particularly significant improvements in complex scenarios involving shocks or vortex structures.

Abstract

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 $1\times1$ convolutions to each subband before reconstruction via the inverse DWT. The two branches are fused through a learnable sigmoid-gated weight $α_\mathrm{wav}$, 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.

Problem

• partial differential equations
• neural operator
• multi-scale features
• spatial localization
• Laplace-domain

Innovation

• Wavelet-Laplace Neural Operator
• Haar wavelet decomposition
• Laplace Neural Operator
• multi-scale spatial features
• neural operators for PDEs