Walsh-Hadamard Neural Operators for Solving PDEs with Discontinuous Coefficients

TL;DR

The Walsh-Hadamard Neural Operator (WHNO) is introduced, which leverages Walsh-Hadamard transforms-a spectral basis of rectangular wave functions naturally suited for piecewise constant fields-combined with learnable spectral weights that transform low-sequency Walsh coefficients to capture global dependencies efficiently.

Abstract

Neural operators have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). However, standard spectral methods based on Fourier transforms struggle with problems involving discontinuous coefficients due to the Gibbs phenomenon and poor representation of sharp interfaces. We introduce the Walsh-Hadamard Neural Operator (WHNO), which leverages Walsh-Hadamard transforms-a spectral basis of rectangular wave functions naturally suited for piecewise constant fields-combined with learnable spectral weights that transform low-sequency Walsh coefficients to capture global dependencies efficiently. We validate WHNO on three problems: steady-state Darcy flow (preliminary validation), heat conduction with discontinuous thermal conductivity, and the 2D Burgers equation with discontinuous initial conditions. In controlled comparisons with Fourier Neural Operators (FNO) under identical conditions, WHNO demonstrates superior accuracy with better preservation of sharp solution features at material interfaces. Critically, we discover that weighted ensemble combinations of WHNO and FNO achieve substantial improvements over either model alone: for both heat conduction and Burgers equation, optimal ensembles reduce mean squared error by 35-40 percent and maximum error by up to 25 percent compared to individual models. This demonstrates that Walsh-Hadamard and Fourier representations capture complementary aspects of discontinuous PDE solutions, with WHNO excelling at sharp interfaces while FNO captures smooth features effectively.

Figures (7)

• Figure 1: Darcy flow predictions for porous media with binary permeability. Left: ground truth pressure field. Middle: WHNO prediction. Right: absolute error. Errors concentrate at sharp boundaries.
• Figure 2: Error map comparison between WHNO (top) and FNO (bottom) for heat conduction. WHNO exhibits more uniform errors; FNO shows pronounced error peaks at conductivity interfaces (Gibbs phenomenon).
• Figure 3: Ensemble architecture combining WHNO and FNO predictions. The input field is processed independently by WHNO (using Walsh-Hadamard basis $\mathcal{H}$) and FNO (using Fourier basis $\mathcal{F}$). Their predictions are combined through optimised weights $\alpha$ and $\beta$ to produce the final ensemble prediction.
• Figure 4: Error comparison for heat conduction. Error maps for WHNO (top), FNO (middle), and ensemble (bottom) predictions on the same test case. The ensemble exhibits lower, more uniform errors throughout the domain.
• Figure 5: Burgers equation predictions comparison. Ground truth, WHNO, FNO, and ensemble (60:40) predictions for a representative test case.