Data-driven and self-supervised spectral operator learning methods for heat conduction equation with variable source functions

Abstract Operator learning methodologies have gained significant attention as a powerful machine learning paradigm for solving partial differential equations (PDEs), yet their reliance on large labeled datasets for training limits broad applicability. This work advances the spectral operator learning (SOL) framework to address this challenge. We specifically extend SOL to the steady-state heat conduction (Poisson) equation with variable source terms, a critical problem in applications such as integrated circuit design. The core methodological contribution is a novel self-supervised learning strategy for SOL, which eliminates dependence on labeled data by training directly from the PDE residuals. This strategy leverages the spectral basis of SOL to inherently and precisely satisfy boundary conditions as hard constraints, thereby removing the need for explicit boundary condition loss terms and substantially reducing optimization complexity. We provide theoretical grounding by proving the universal approximation theorem for the proposed self-supervised SOL models on solving the Poisson equation. Comprehensive numerical experiments demonstrate that our approach achieves significantly higher accuracy compared to benchmark operator learning models. Furthermore, the framework exhibits robust generalization across different spatial resolutions and maintains consistent accuracy at arbitrary query points, confirming its suitability for practical, high-fidelity simulation.