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

Shiyuan Li, Hossein Salahshoor

TL;DR

FEENet addresses the challenge of solving PDEs on complex geometries by embedding the solution space in a geometry-aware spectral basis obtained from the Finite Element Method. A Branch Net learns spectral coefficients for each input, enabling a fast, resolution-independent reconstruction of the solution via $u(x)=\sum_{k=1}^M c_k \psi_k(x)$ (with time-dependent problems incorporating $e^{-\lambda_k t}$ when appropriate). Across Poisson and heat benchmarks on 2D and 3D geometries, FEENet demonstrates superior accuracy and efficiency compared to DeepONet, while offering interpretability and natural generalization to nonlocal operators through functional calculus on the spectrum. The framework decouples geometric encoding from operator learning, and the one-time eigenbasis computation can be amortized across multiple PDE problems on the same domain, enabling scalable many-query solving with physics-consistent dynamics.

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.

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

TL;DR

FEENet addresses the challenge of solving PDEs on complex geometries by embedding the solution space in a geometry-aware spectral basis obtained from the Finite Element Method. A Branch Net learns spectral coefficients for each input, enabling a fast, resolution-independent reconstruction of the solution via (with time-dependent problems incorporating when appropriate). Across Poisson and heat benchmarks on 2D and 3D geometries, FEENet demonstrates superior accuracy and efficiency compared to DeepONet, while offering interpretability and natural generalization to nonlocal operators through functional calculus on the spectrum. The framework decouples geometric encoding from operator learning, and the one-time eigenbasis computation can be amortized across multiple PDE problems on the same domain, enabling scalable many-query solving with physics-consistent dynamics.

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.
Paper Structure (26 sections, 1 theorem, 33 equations, 15 figures, 4 tables)

This paper contains 26 sections, 1 theorem, 33 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Finite Element Eigenfunction Network
  3. Preliminaries: Spectral representation of elliptic operators
  4. Illustration of the FEENet idea
  5. Remark on the connection of FEENet with existing operator learning frameworks
  6. FEENet Architecture
  7. Training Objective and Loss Function
  8. Numerical Experiments
  9. Training and Evaluation Protocols
  10. Eigenvalue Problem Solving
  11. Poisson Problem
  12. Homogeneous Heat Problem
  13. Inhomogeneous Heat Problem
  14. Discussion
  15. Resolution Independence Study
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\mathcal{L}$ be a self-adjoint linear elliptic differential operator of order $2m$ on a bounded domain $\Omega \subset \mathbb{R}^d$ with homogeneous Dirichlet boundary conditions. Then the (real) eigenfunctions of $\mathcal{L}$ form a complete system in $H^m(\Omega)$. In particular, for any $u

Figures (15)

  • Figure 1: Finite Element Eigenfunction Network (FEENet). This framework learns the solution operator of a differential equation governed by operator $\mathcal{L}$ by decoupling the problem into two components: (Top) An offline FEM solver pre-computes the eigenvalue--eigenfunction pairs $\{\lambda_k, \psi_k\}_{k=1}^M$ by solving the eigenvalue problem $\mathcal{L}\psi_k = \lambda_k \psi_k$ in $\Omega$ with homogeneous boundary conditions. These eigenfunctions serve as fixed, geometry-aware basis functions. (Bottom) For each training sample, a branch network maps the input function $f$ to spectral coordinates $\{c_k(\boldsymbol{\theta})\}_{k=1}^M$, and the predicted solution is reconstructed via a linear combination of basis functions and corresponding learned coordinates.
  • Figure 2: Poisson problem on the Unit Square. Randomly selected representative result for (top row) input function and corresponding reference solution; (middle row) FEENet prediction and the corresponding absolute error. (bottom row) DeepONet prediction and the corresponding absolute error.
  • Figure 3: Poisson problem on the Fins. Randomly selected representative result for (top row) input function and corresponding reference solution; (middle row) FEENet prediction and the corresponding absolute error; (bottom row) DeepONet prediction and the corresponding absolute error.
  • Figure 4: Poisson problem on the Bunny. Randomly selected representative result for (top row) input function and corresponding reference solution; (middle row) FEENet prediction and the corresponding absolute error; (bottom row) DeepONet prediction and the corresponding absolute error.
  • Figure 5: Homogeneous heat problem on the Unit Square. From top to bottom: Reference solutions at selected time steps; FEENet predictions; DeepONet predictions; absolute error of FEENet; absolute error of DeepONet.
  • ...and 10 more figures

Theorems & Definitions (1)

  • Theorem 1: Completeness of elliptic eigenfunctions Browder1952Browder1953