City Research Online

Abstract

Neural operators have emerged as tools for learning solution operators of partial differential equations (PDEs). Standard spectral methods based on Fourier transforms struggle with problems involving discontinuous coefficients, because the Gibbs phenomenon limits how well a truncated Fourier expansion can represent sharp interfaces. We introduce the Walsh-Hadamard Neural Operator (WHNO), which uses Walsh-Hadamard transforms in place of Fourier transforms. The Walsh-Hadamard basis consists of rectangular wave functions, suited to piecewise constant fields, and is combined with learnable spectral weights acting on the low-sequency coefficients. We validate WHNO on two problems: heat conduction with discontinuous thermal conductivity, and the 2D Burgers equation with discontinuous initial conditions. In controlled comparisons against a Fourier Neural Operator (FNO) baseline at matched parameter count over 100 independent test samples, WHNO obtains lower mean absolute error and lower H1 (gradient-MSE) error on both problems. We then study weighted ensembles of WHNO and FNO; the ensemble weight w* ∈ [0, 1] is fitted by five-fold cross-validation. Across seven (problem, geometry/IC) configurations evaluated in this paper — four heat geometries (axis-aligned, rotated, disks, Voronoi) and three Burgers initial-condition families (block, smooth sinusoidal, oblique fronts) — the cross-validated ensemble has strictly lower test MSE and strictly lower H1 than both WHNO and FNO alone in every case, including the configurations where WHNO alone does not unambiguously beat FNO. The two bases are therefore partially complementary, and combining them through a single cross-validated scalar weight gives a method that improves on the Fourier baseline on every problem variant we tested. The cross-validated weights are w*=0.572± 0.016 on axis-aligned heat and w*=0.648±0.020 on the Burgers block-IC baseline, with all seven values keeping a substantial FNO contribution.

Publication Type:	Article
Additional Information:	© The Authors. Published by Elsevier. This is an open-access article distributed under the terms of Creative Commons: Attribution NonCommercial 4.0 (http://creativecommons.org/licenses/by-nc/4.0/).
Publisher Keywords:	Neural operators; Walsh-Hadamard transform; Fourier neural operator; Discontinuous PDEs; Operator learning; Ensemble methods; Spectral methods; Heterogeneous media
Subjects:	H Social Sciences > HN Social history and conditions. Social problems. Social reform Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Departments:	School of Science & Technology School of Science & Technology > Department of Engineering
SWORD Depositor:	Symplectic Administrator

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