# Finite Element Method-enhanced Neural Network for Forward and Inverse Problems

arXiv: https://arxiv.org/abs/2205.08321 · HTML: https://mlsanity.com/paper/2205.08321/

## Key facts

- Cited by 13 papers in the map (data through 7 Oct 2026).
- Builds on 15 of its 64 references that are in the map.
- First posted on arXiv 17 May 2022.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [Label-Free Supervision of Neural Networks with Physics and Domain Knowledge](https://mlsanity.com/paper/1609.05566/) (2016)
- [Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures](https://mlsanity.com/paper/2002.10253/) (2020)

## Built on by (most cited first, 13 of 13)

- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Sobolev neural network with residual weighting as a surrogate in linear and non-linear mechanics](https://mlsanity.com/paper/2407.16466/) (2024)
- [Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems](https://mlsanity.com/paper/2409.04067/) (2024)
- [Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization](https://mlsanity.com/paper/2412.04596/) (2024)
- [ε-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics](https://mlsanity.com/paper/2412.05144/) (2024)
- [A discrete physics-informed training for projection-based reduced order models with neural networks](https://mlsanity.com/paper/2504.13875/) (2025)
- [Physics-Informed Graph-Mesh Networks for PDEs: A hybrid approach for complex problems](https://mlsanity.com/paper/2410.02819/) (2024)
- [A Hybrid Virtual Element Method and Deep Learning Approach for Solving One-Dimensional Euler-Bernoulli Beams](https://mlsanity.com/paper/2501.06925/) (2025)
- [The Finite Element Neural Network Method: One Dimensional Study](https://mlsanity.com/paper/2501.12508/) (2025)
- [Hybrid Quantum Classical Surrogate for Real Time Inverse Finite Element Modeling in Digital Twins](https://mlsanity.com/paper/2508.00029/) (2025)
- [Robin-Neumann Coupling of PINN and FEM Solvers: A Steklov-Poincar'e View, with Application to Fluid-Structure Interaction with Contact](https://mlsanity.com/paper/2606.14181/) (2026)

## Abstract (arXiv)

We introduce a novel hybrid methodology combining classical finite element methods (FEM) with neural networks to create a well-performing and generalizable surrogate model for forward and inverse problems. The residual from finite element methods and custom loss functions from neural networks are merged to form the algorithm. The Finite Element Method-enhanced Neural Network hybrid model (FEM-NN hybrid) is data-efficient and physics conforming. The proposed methodology can be used for surrogate models in real-time simulation, uncertainty quantification, and optimization in the case of forward problems. It can be used for updating the models in the case of inverse problems. The method is demonstrated with examples, and the accuracy of the results and performance is compared against the conventional way of network training and the classical finite element method. An application of the forward-solving algorithm is demonstrated for the uncertainty quantification of wind effects on a high-rise buildings. The inverse algorithm is demonstrated in the speed-dependent bearing coefficient identification of fluid bearings. The hybrid methodology of this kind will serve as a paradigm shift in the simulation methods currently used.