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. This thesis proposes the Finite Element Eigenfunction Network (FEENet), a novel hybrid spectral learning framework grounded in the eigenfunction theory of differential operators. For a given domain, FEENet leverages the Finite Element Method (FEM) to perform a one-time computation of an eigenfunction basis intrinsic to the geometry. Partial Differential Equation (PDE) solutions are subsequently represented in this geometry-adapted basis, and the operator learning process is elegantly reduced to predicting the corresponding spectral coefficients.Furthermore, this thesis extends the proposed FEENet framework to address the significant computational challenges associated with nonlocal operators, specifically focusing on the spectral fractional Laplacian. Unlike local PDEs, nonlocal operators imply that the state at a given point depends on the state of the entire domain. Traditional numerical methods typically yield dense stiffness matrices when discretizing such fractional operators, leading to prohibitive computational costs. By embedding the spectral decomposition of the standard Laplacian directly into the network architecture, FEENet provides a geometry-adapted basis that naturally respects boundary conditions and seamlessly handles the global nature of fractional operators without the need for dense matrix inversions. Numerical experiments conducted across a range of parameterized PDEs and complex two- and three-dimensional geometries demonstrate that FEENet consistently achieves superior accuracy and computational efficiency. Benchmarks against seminal frameworks, including DeepONet and its multi-input variant MIONet, validate the model's robustness. 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. Ultimately, this thesis envisions that hybrid approaches of this form, which combine structure-preserving numerical methods with data-driven learning, offer a promising and scalable pathway toward solving real-world PDE problems on complex geometries.