Walsh-Hadamard Neural Operators for Solving PDEs with Discontinuous Coefficients

TL;DR

The paper introduces Walsh-Hadamard Neural Operators (WHNO), a spectral method designed to better handle PDEs with discontinuous coefficients, outperforming Fourier-based methods in accuracy and interface preservation.

Contribution

We develop WHNO, leveraging Walsh-Hadamard transforms for improved representation of discontinuous PDE solutions, and demonstrate its superiority over Fourier Neural Operators through extensive experiments.

Findings

WHNO outperforms FNO in accuracy for discontinuous PDEs.

Weighted ensembles of WHNO and FNO significantly reduce errors.

WHNO better preserves sharp interfaces in solutions.

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.