# Hybrid FEM-NN models: Combining artificial neural networks with the finite element method

arXiv: https://arxiv.org/abs/2101.00962 · HTML: https://mlsanity.com/paper/2101.00962/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 4 Jan 2021.

## Builds on (most cited first, 17 of 17)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Deep Learning: A Critical Appraisal](https://mlsanity.com/paper/1801.00631/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Learning Dynamical Systems from Partial Observations](https://mlsanity.com/paper/1902.11136/) (2019)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [Neural network augmented inverse problems for PDEs](https://mlsanity.com/paper/1712.09685/) (2017)
- [Revisiting Calderon's Problem](https://mlsanity.com/paper/1912.02970/) (2019)

## Built on by (most cited first, 24 of 24)

- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)
- [Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks](https://mlsanity.com/paper/2203.08852/) (2022)
- [Partial Differential Equations Meet Deep Neural Networks: A Survey](https://mlsanity.com/paper/2211.05567/) (2022)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Neural Spectral Methods: Self-supervised learning in the spectral domain](https://mlsanity.com/paper/2312.05225/) (2023)
- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs](https://mlsanity.com/paper/2110.01601/) (2021)
- [Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modeling](https://mlsanity.com/paper/2504.11383/) (2025)
- [A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2504.02459/) (2025)
- [Differentiable modeling to unify machine learning and physical models and advance Geosciences](https://mlsanity.com/paper/2301.04027/) (2023)
- [LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometry](https://mlsanity.com/paper/2302.01518/) (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)
- [Bridging Data and Physics: A Graph Neural Network-Based Hybrid Twin Framework](https://mlsanity.com/paper/2512.15767/) (2025)
- [Physics-Informed Neural Networks for Discovering Localised Eigenstates in Disordered Media](https://mlsanity.com/paper/2305.06802/) (2023)
- [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)
- [The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach](https://mlsanity.com/paper/2302.08309/) (2023)
- [A conservative hybrid physics-informed neural network method for Maxwell-Amp`ere-Nernst-Planck equations](https://mlsanity.com/paper/2312.05891/) (2023)
- [Hybrid Quantum Classical Surrogate for Real Time Inverse Finite Element Modeling in Digital Twins](https://mlsanity.com/paper/2508.00029/) (2025)
- [Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems](https://mlsanity.com/paper/2603.03259/) (2026)
- [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)
- [A Hybrid GNN-FEM Framework for Phase-Field Fracture Simulation. Physics-Preserving Hybridization for Generalizable Surrogate Modeling](https://mlsanity.com/paper/2606.19378/) (2026)

## Abstract (arXiv)

We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in the optimisation as apposed to making them part of the loss function. The resulting models are discretised in space by the finite element method (FEM). The method applies to both stationary and transient as well as linear/nonlinear PDEs. We describe implementation of the approach as an extension of the existing FEM framework FEniCS and its algorithmic differentiation tool dolfin-adjoint. Through series of examples we demonstrate capabilities of the approach to recover coefficients and missing PDE operators from observations. Further, the proposed method is compared with alternative methodologies, namely, physics informed neural networks and standard PDE-constrained optimisation. Finally, we demonstrate the method on a complex cardiac cell model problem using deep neural networks.