# Physics informed neural networks for continuum micromechanics

arXiv: https://arxiv.org/abs/2110.07374 · HTML: https://mlsanity.com/paper/2110.07374/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 55 references that are in the map.
- First posted on arXiv 14 Oct 2021.

## Builds on (most cited first, 20 of 20)

- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [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)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [Deep Learning in Bioinformatics](https://mlsanity.com/paper/1603.06430/) (2016)
- [Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture](https://mlsanity.com/paper/2011.13511/) (2020)
- [Machine learning for metal additive manufacturing: Predicting temperature and melt pool fluid dynamics using physics-informed neural networks](https://mlsanity.com/paper/2008.13547/) (2020)
- [Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imaging](https://mlsanity.com/paper/2009.04525/) (2020)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling](https://mlsanity.com/paper/2109.11028/) (2021)
- [Energy-based error bound of physics-informed neural network solutions in elasticity](https://mlsanity.com/paper/2010.09088/) (2020)
- [A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity](https://mlsanity.com/paper/2108.07243/) (2021)

## Built on by (most cited first, 30 of 46)

- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations](https://mlsanity.com/paper/2211.00713/) (2022)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Recent Advances and Applications of Machine Learning in Experimental Solid Mechanics: A Review](https://mlsanity.com/paper/2303.07647/) (2023)
- [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)
- [Three-dimensional microstructure generation using generative adversarial neural networks in the context of continuum micromechanics](https://mlsanity.com/paper/2206.01693/) (2022)
- [Spiking neural networks for nonlinear regression](https://mlsanity.com/paper/2210.03515/) (2022)
- [Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)](https://mlsanity.com/paper/2406.03172/) (2024)
- [Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problem](https://mlsanity.com/paper/2405.12380/) (2024)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [Physics-informed Discretization-independent Deep Compositional Operator Network](https://mlsanity.com/paper/2404.13646/) (2024)
- [NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elements](https://mlsanity.com/paper/2407.21217/) (2024)
- [Deep learning for full-field ultrasonic characterization](https://mlsanity.com/paper/2301.02378/) (2023)
- [Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance](https://mlsanity.com/paper/2303.03918/) (2023)
- [Stochastic stiffness identification and response estimation of Timoshenko beams via physics-informed Gaussian processes](https://mlsanity.com/paper/2309.11875/) (2023)
- [Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations](https://mlsanity.com/paper/2407.19707/) (2024)
- [Physics-Informed Neural Networks for Material Model Calibration from Full-Field Displacement Data](https://mlsanity.com/paper/2212.07723/) (2022)
- [Learning from Integral Losses in Physics Informed Neural Networks](https://mlsanity.com/paper/2305.17387/) (2023)
- [Physics-guided training of GAN to improve accuracy in airfoil design synthesis](https://mlsanity.com/paper/2308.10038/) (2023)
- [Physics-informed UNets for Discovering Hidden Elasticity in Heterogeneous Materials](https://mlsanity.com/paper/2306.01204/) (2023)
- [Identifying Constitutive Parameters for Complex Hyperelastic Materials using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.15640/) (2023)
- [Probabilistic Physics-integrated Neural Differentiable Modeling for Isothermal Chemical Vapor Infiltration Process](https://mlsanity.com/paper/2311.07798/) (2023)
- [Input convex neural networks: universal approximation theorem and implementation for isotropic polyconvex hyperelastic energies](https://mlsanity.com/paper/2502.08534/) (2025)
- [EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations](https://mlsanity.com/paper/2504.07976/) (2025)
- [Generalized invariants meet constitutive neural networks: A novel framework for hyperelastic materials](https://mlsanity.com/paper/2508.12063/) (2025)
- [Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks](https://mlsanity.com/paper/2405.18311/) (2024)
- [Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networks](https://mlsanity.com/paper/2509.07579/) (2025)
- [A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials](https://mlsanity.com/paper/2605.31231/) (2026)

## Abstract (arXiv)

Recently, physics informed neural networks have successfully been applied to a broad variety of problems in applied mathematics and engineering. The principle idea is to use a neural network as a global ansatz function to partial differential equations. Due to the global approximation, physics informed neural networks have difficulties in displaying localized effects and strong non-linear solutions by optimization. In this work we consider material non-linearities invoked by material inhomogeneities with sharp phase interfaces. This constitutes a challenging problem for a method relying on a global ansatz. To overcome convergence issues, adaptive training strategies and domain decomposition are studied. It is shown, that the domain decomposition approach is able to accurately resolve nonlinear stress, displacement and energy fields in heterogeneous microstructures obtained from real-world μCT-scans.