Finite Element Operator Network for Solving Elliptic-type parametric PDEs

TL;DR

FEONet presents an unsupervised operator-learning framework that fuses neural networks with the finite element method to solve parametric PDEs without requiring precomputed input-output data. By predicting FEM coefficients from problem inputs and reconstructing the solution via FEM basis functions, FEONet enforces the variational (weak) form and exact boundary conditions through a residual loss, with convergence theory linking its performance to FEM accuracy and network capacity. The authors demonstrate FEONet's versatility across 1D/2D domains, Dirichlet/Neumann boundaries, linear/nonlinear PDEs, and inputs such as variable coefficients and boundary data, and extend it to time-dependent problems and the Stokes system, including challenging singular perturbation scenarios via boundary-layer enrichment. They show competitive or superior generalization and accuracy with unsupervised training compared to supervised baselines like DeepONet, POD-DeepONet, and PIDeepONet, while highlighting practical aspects such as computational efficiency and the importance of mesh-based methods for complex geometries. The work offers a promising, theory-grounded alternative for rapid, real-time PDE solves in complex domains, with future directions addressing high-dimensional problems and further stability enhancements.

Abstract

Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates efficient numerical methods. In this paper, we propose a novel approach for solving parametric PDEs using a Finite Element Operator Network (FEONet). Our proposed method leverages the power of deep learning in conjunction with traditional numerical methods, specifically the finite element method, to solve parametric PDEs in the absence of any paired input-output training data. We performed various experiments on several benchmark problems and confirmed that our approach has demonstrated excellent performance across various settings and environments, proving its versatility in terms of accuracy, generalization, and computational flexibility. While our method is not meshless, the FEONet framework shows potential for application in various fields where PDEs play a crucial role in modeling complex domains with diverse boundary conditions and singular behavior. Furthermore, we provide theoretical convergence analysis to support our approach, utilizing finite element approximation in numerical analysis.

Figures (11)

• Figure 1: Domain triangulation for three different complex geometries.
• Figure 2: Schematic diagram of the Finite Element Operator Network (FEONet).
• Figure 3: Solution profiles for complex geometries.
• Figure 4: Solution profiles predicted by the trained FEONet for PDE \ref{['eq_varcoeff']} obtained from variable coefficients $c(x)$ as an input function.
• Figure 5: Solution profiles predicted by the trained FEONet for PDE \ref{['eq_bdry_input']} obtained from the boundary condition input. Note that the boundary condition $g(x,y)$ for the region with a circle hole inside $\Omega_{\text{cir}}$ is represented as $h(\theta)$ in polar coordinates. The average of relative $L^2$ errors across the entire test set when comparing the exact solution to the predictions is 0.0043.

Theorems & Definitions (15)

• Remark 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5
• Theorem 2.6
• Definition 2.7
• Theorem 2.8
• Theorem 2.9
• Remark 3.1