mlsanityData to 14 Oct 2026

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

arXiv 2101.00962 ↗First posted 4 Jan 2021math.NA cs.LG cs.NAcited by 24

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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.

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

17 of 50 references.

References · 17

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  2. 2018Neural Ordinary Differential Equationscited by 3,779
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  4. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  5. 2020Universal Differential Equations for Scientific Machine Learningcited by 353
  6. 2017PDE-Net: Learning PDEs from Datacited by 311
  7. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 235
  8. 2018Deep Learning: A Critical Appraisalcited by 217
  9. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 210
  10. 2018Deep Neural Networks Motivated by Partial Differential Equationscited by 187
  11. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 148
  12. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 135
5 more
  1. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 116
  2. 2019Learning Dynamical Systems from Partial Observationscited by 36
  3. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  4. 2017Neural network augmented inverse problems for PDEscited by 11
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 2; 2022: 3; 2023: 7; 2024: 5; 2025: 4; 2026: 3.

24 papers.

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  1. 2021Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modelingcited by 59
  2. 2022Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamicscited by 44
  3. 2022Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networkscited by 34
  4. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  5. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  6. 2023Neural Spectral Methods: Self-supervised learning in the spectral domaincited by 14
  7. 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
  8. 2021NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEscited by 10
  9. 2025Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modelingcited by 8
  10. 2025A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometriescited by 5
  11. 2023Differentiable modeling to unify machine learning and physical models and advance Geosciencescited by 4
  12. 2023LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometrycited by 4
12 more
  1. 2024Sobolev neural network with residual weighting as a surrogate in linear and non-linear mechanicscited by 2
  2. 2024Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problemscited by 2
  3. 2025Bridging Data and Physics: A Graph Neural Network-Based Hybrid Twin Frameworkcited by 2
  4. 2023Physics-Informed Neural Networks for Discovering Localised Eigenstates in Disordered Mediacited by 1
  5. 2024Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimizationcited by 1
  6. 2024ε-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamicscited by 1
  7. 2023The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approachcited by 0
  8. 2023A conservative hybrid physics-informed neural network method for Maxwell-Amp`ere-Nernst-Planck equationscited by 0
  9. 2025Hybrid Quantum Classical Surrogate for Real Time Inverse Finite Element Modeling in Digital Twinscited by 0
  10. 2026Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problemscited by 0
  11. 2026Robin-Neumann Coupling of PINN and FEM Solvers: A Steklov-Poincar'e View, with Application to Fluid-Structure Interaction with Contactcited by 0
  12. 2026A Hybrid GNN-FEM Framework for Phase-Field Fracture Simulation. Physics-Preserving Hybridization for Generalizable Surrogate Modelingcited by 0