# A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method

arXiv: https://arxiv.org/abs/2206.13103 · HTML: https://mlsanity.com/paper/2206.13103/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 61 references that are in the map.
- First posted on arXiv 27 Jun 2022.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications](https://mlsanity.com/paper/2010.02011/) (2020)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imaging](https://mlsanity.com/paper/2009.04525/) (2020)
- [The mixed deep energy method for resolving concentration features in finite strain hyperelasticity](https://mlsanity.com/paper/2104.09623/) (2021)
- [Stress field prediction in fiber-reinforced composite materials using a deep learning approach](https://mlsanity.com/paper/2111.05271/) (2021)
- [A deep learning energy method for hyperelasticity and viscoelasticity](https://mlsanity.com/paper/2201.08690/) (2022)
- [Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence](https://mlsanity.com/paper/2108.02837/) (2021)
- [Energy-based error bound of physics-informed neural network solutions in elasticity](https://mlsanity.com/paper/2010.09088/) (2020)
- [Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis](https://mlsanity.com/paper/2010.12060/) (2020)
- [Bayesian neural networks for weak solution of PDEs with uncertainty quantification](https://mlsanity.com/paper/2101.04879/) (2021)
- [Physics-informed neural network solution of thermo-hydro-mechanical (THM) processes in porous media](https://mlsanity.com/paper/2203.01514/) (2022)
- [Machine Learning-Accelerated Computational Solid Mechanics: Application to Linear Elasticity](https://mlsanity.com/paper/2112.08676/) (2021)

## Built on by (most cited first, 30 of 33)

- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems](https://mlsanity.com/paper/2410.01337/) (2024)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains](https://mlsanity.com/paper/2302.04954/) (2023)
- [Theory and implementation of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2311.06380/) (2023)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Physics-informed PointNet: On how many irregular geometries can it solve an inverse problem simultaneously? Application to linear elasticity](https://mlsanity.com/paper/2303.13634/) (2023)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2411.15111/) (2024)
- [Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning](https://mlsanity.com/paper/2511.06802/) (2025)
- [Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN](https://mlsanity.com/paper/2304.06044/) (2023)
- [A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2504.02459/) (2025)
- [Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method](https://mlsanity.com/paper/2209.02931/) (2022)
- [RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network](https://mlsanity.com/paper/2212.02861/) (2022)
- [Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations](https://mlsanity.com/paper/2401.02363/) (2024)
- [Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space](https://mlsanity.com/paper/2405.01975/) (2024)
- [Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEs](https://mlsanity.com/paper/2412.05719/) (2024)
- [Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometries](https://mlsanity.com/paper/2504.06327/) (2025)
- [EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations](https://mlsanity.com/paper/2504.07976/) (2025)
- [LNN-PINN: A Unified Physics-Only Training Framework with Liquid Residual Blocks](https://mlsanity.com/paper/2508.08935/) (2025)
- [Physics-aware deep learning framework for linear elasticity](https://mlsanity.com/paper/2302.09668/) (2023)
- [DPA-WNO: A gray box model for a class of stochastic mechanics problem](https://mlsanity.com/paper/2309.15128/) (2023)
- [Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity](https://mlsanity.com/paper/2407.11183/) (2024)
- [Heterogeneous Graph Representation of Stiffened Panels with Non-Uniform Boundary Conditions and Loads](https://mlsanity.com/paper/2510.02472/) (2025)
- [A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials](https://mlsanity.com/paper/2605.31231/) (2026)
- [Finite-PINN: A Physics-Informed Neural Network with Finite Geometric Encoding for Solid Mechanics](https://mlsanity.com/paper/2412.09453/) (2024)
- [Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations](https://mlsanity.com/paper/2512.14980/) (2025)
- [AI enhanced finite element multiscale modelling and structural uncertainty analysis of a functionally graded porous beam](https://mlsanity.com/paper/2211.01970/) (2022)
- [Detecting hidden structures from a static loading experiment: topology optimization meets physics-informed neural networks](https://mlsanity.com/paper/2303.09280/) (2023)
- [Physics-informed radial basis network (PIRBN): A local approximating neural network for solving nonlinear PDEs](https://mlsanity.com/paper/2304.06234/) (2023)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering problems. The main contribution of the current work is to promote using the spatial gradient of the primary variable as an output from separated neural networks. Later on, the strong form which has a higher order of derivatives is applied to the spatial gradients of the primary variable as the physical constraint. In addition, the so-called energy form of the problem is applied to the primary variable as an additional constraint for training. The proposed approach only required up to first-order derivatives to construct the physical loss functions. We discuss why this point is beneficial through various comparisons between different models. The mixed formulation-based PINNs and FE methods share some similarities. While the former minimizes the PDE and its energy form at given collocation points utilizing a complex nonlinear interpolation through a neural network, the latter does the same at element nodes with the help of shape functions. We focus on heterogeneous solids to show the capability of deep learning for predicting the solution in a complex environment under different boundary conditions. The performance of the proposed PINN model is checked against the solution from FEM on two prototype problems: elasticity and the Poisson equation (steady-state diffusion problem). We concluded that by properly designing the network architecture in PINN, the deep learning model has the potential to solve the unknowns in a heterogeneous domain without any available initial data from other sources. Finally, discussions are provided on the combination of PINN and FEM for a fast and accurate design of composite materials in future developments.