# Finite Element Operator Network for Solving Elliptic-type parametric PDEs

arXiv: https://arxiv.org/abs/2308.04690 · HTML: https://mlsanity.com/paper/2308.04690/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 23 references that are in the map.
- First posted on arXiv 9 Aug 2023.

## Builds on (most cited first, 7 of 7)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)

## Built on by (most cited first, 14 of 14)

- [Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients](https://mlsanity.com/paper/2507.10241/) (2025)
- [A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2504.02459/) (2025)
- [ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines](https://mlsanity.com/paper/2501.09395/) (2025)
- [FourierSpecNet: Neural Collision Operator Approximation Inspired by the Fourier Spectral Method for Solving the Boltzmann Equation](https://mlsanity.com/paper/2504.20408/) (2025)
- [Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs](https://mlsanity.com/paper/2510.01039/) (2025)
- [Efficient High-Accuracy PDEs Solver with the Linear Attention Neural Operator](https://mlsanity.com/paper/2510.16816/) (2025)
- [Reduced-Basis Deep Operator Learning for Parametric PDEs with Independently Varying Boundary and Source Data](https://mlsanity.com/paper/2511.18260/) (2025)
- [Discontinuous Galerkin finite element operator network for solving non-smooth PDEs](https://mlsanity.com/paper/2601.03668/) (2026)
- [Overlapping Schwarz Attention: Hierarchical Attention via Domain Decomposition](https://mlsanity.com/paper/2606.18525/) (2026)
- [Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers](https://mlsanity.com/paper/2606.21828/) (2026)
- [Learning from Imperfect Data: Robust Inference of Dynamic Systems using Simulation-based Generative Model](https://mlsanity.com/paper/2507.10884/) (2025)
- [Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEs](https://mlsanity.com/paper/2601.00672/) (2026)
- [Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation](https://mlsanity.com/paper/2605.07323/) (2026)
- [GA-VINO: A Geometry-Aware Variational Physics-informed Neural Operator for Mindlin-Reissner Plates](https://mlsanity.com/paper/2606.16624/) (2026)

## Abstract (arXiv)

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.