Finite Element Eigenfunction Network (FEENet): A Hybrid Framework for Solving PDEs on Complex Geometries

Abstract: Neural operators aim to learn mappings between infinite-dimensional function spaces, but their performance often degrades on complex or irregular geometries due to the lack of geometry-aware representations. We propose the Finite Element Eigenfunction Network (FEENet), a hybrid spectral learning framework grounded in the eigenfunction theory of differential operators. For a given domain, FEENet leverages the Finite Element Method (FEM)toperformaone-timecomputationofaneigenfunctionbasisintrinsictothegeometry. PDE solutions are subsequently represented in this geometry-adapted basis, and learning is reduced to predicting the corresponding spectral coefficients. Numerical experiments conducted across a range of parameterized PDEs and complex two- and three-dimensional geometries, including benchmarks against the seminal DeepONet framework (1), demonstrate that FEENet consistently achieves superior accuracy and computational efficiency. We further highlight key advantages of the proposed approach, including resolution-independent inference, interpretability, and natural generalization to nonlocal operators defined as functions of differential operators. We envision that hybrid approaches of this form, which combine structure-preserving numerical methods with data-driven learning, offer a promising pathway toward solving real-world PDE problems on complex geometries.