# Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN) arXiv: https://arxiv.org/abs/2108.13137 ยท HTML: https://mlsanity.com/paper/2108.13137/ ## Key facts - Cited by 18 papers in the map (data through 14 Oct 2026). - Builds on 10 of its 69 references that are in the map. - First posted on arXiv 30 Aug 2021. ## Builds on (most cited first, 10 of 10) - [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015) - [Deep learning of thermodynamics-aware reduced-order models from data](https://mlsanity.com/paper/2007.03758/) (2020) - [A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches](https://mlsanity.com/paper/2010.06050/) (2020) - [Extracting Interpretable Physical Parameters from Spatiotemporal Systems using Unsupervised Learning](https://mlsanity.com/paper/1907.06011/) (2019) - [Machine learning materials physics: Multi-resolution neural networks learn the free energy and nonlinear elastic response of evolving microstructures](https://mlsanity.com/paper/2001.01575/) (2019) - [Exploring the 3D architectures of deep material network in data-driven multiscale mechanics](https://mlsanity.com/paper/1901.04832/) (2019) - [GINNs: Graph-Informed Neural Networks for Multiscale Physics](https://mlsanity.com/paper/2006.14807/) (2020) - [Data-driven discovery of interpretable causal relations for deep learning material laws with uncertainty propagation](https://mlsanity.com/paper/2105.09980/) (2021) - [A robust solution of a statistical inverse problem in multiscale computational mechanics using an artificial neural network](https://mlsanity.com/paper/2011.11761/) (2020) - [Interface learning of multiphysics and multiscale systems](https://mlsanity.com/paper/2006.10112/) (2020) ## Built on by (most cited first, 18 of 18) - [A new family of Constitutive Artificial Neural Networks towards automated model discovery](https://mlsanity.com/paper/2210.02202/) (2022) - [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024) - [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023) - [tLaSDI: Thermodynamics-informed latent space dynamics identification](https://mlsanity.com/paper/2403.05848/) (2024) - [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023) - [Theory and implementation of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2311.06380/) (2023) - [Thermodynamics of learning physical phenomena](https://mlsanity.com/paper/2207.12749/) (2022) - [Spiking neural networks for nonlinear regression](https://mlsanity.com/paper/2210.03515/) (2022) - [Evolution TANN and the identification of internal variables and evolution equations in solid mechanics](https://mlsanity.com/paper/2209.13269/) (2022) - [EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomials](https://mlsanity.com/paper/2410.10897/) (2024) - [Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2407.19326/) (2024) - [Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous media](https://mlsanity.com/paper/2405.19019/) (2024) - [A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strains](https://mlsanity.com/paper/2510.04187/) (2025) - [Physically recurrent neural network for rate and path-dependent heterogeneous materials in a finite strain framework](https://mlsanity.com/paper/2404.17583/) (2024) - [Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity](https://mlsanity.com/paper/2407.11183/) (2024) - [A POD-TANN approach for the multiscale modeling of materials and macroelement derivation in geomechanics](https://mlsanity.com/paper/2408.07165/) (2024) - [Equivariant graph convolutional neural networks for the representation of homogenized anisotropic microstructural mechanical response](https://mlsanity.com/paper/2404.17584/) (2024) - [LaGSplat: Inferring Physics-Governed Interactive Simulation from Monocular Video Using Latent Lagrangian Gaussian Splatting](https://mlsanity.com/paper/2608.16324/) (2026) ## Abstract (arXiv) The mechanical behavior of inelastic materials with microstructure is very complex and hard to grasp with heuristic, empirical constitutive models. For this purpose, multiscale, homogenization approaches are often used for performing reliable, accurate predictions of the macroscopic mechanical behavior of solids and structures. Nevertheless, the calculation cost of such approaches is extremely high and prohibitive for real-scale applications involving inelastic materials. Here, we propose the so-called Thermodynamics-based Artificial Neural Networks (TANN) for the constitutive modeling of materials with inelastic and complex microstructure. Our approach integrates thermodynamics-aware dimensionality reduction techniques and thermodynamics-based deep neural networks to identify, in an autonomous way, the constitutive laws and discover the internal state variables of complex inelastic materials. The efficiency and accuracy of TANN in predicting the average and local stress-strain response, the free-energy and the dissipation rate is demonstrated for both regular and perturbed two- and three-dimensional lattice microstructures in inelasticity. TANN manage to identify the internal state variables that characterize the inelastic deformation of the complex microstructural fields. These internal state variables are then used to reconstruct the microdeformation fields of the microstructure at a given state. Finally, a double-scale homogenization scheme (FEMxTANN) is used to solve a large scale boundary value problem. The high performance of the homogenized model using TANN is illustrated through detailed comparisons with microstructural calculations at large scale. An excellent agreement is shown for a variety of monotonous and cyclic stress-strain paths.