# A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials

arXiv: https://arxiv.org/abs/1807.09829 · HTML: https://mlsanity.com/paper/1807.09829/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2018.

## Built on by (most cited first, 30 of 31)

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- [A Microstructure-based Graph Neural Network for Accelerating Multiscale Simulations](https://mlsanity.com/paper/2402.13101/) (2024)
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- [Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN](https://mlsanity.com/paper/2304.06044/) (2023)
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- [A single Long Short-Term Memory network for enhancing the prediction of path-dependent plasticity with material heterogeneity and anisotropy](https://mlsanity.com/paper/2204.01466/) (2022)
- [Heterogeneous Peridynamic Neural Operators: Discover Biotissue Constitutive Law and Microstructure From Digital Image Correlation Measurements](https://mlsanity.com/paper/2403.18597/) (2024)
- [Orientation-aware interaction-based deep material network in polycrystalline materials modeling](https://mlsanity.com/paper/2502.02457/) (2025)
- [Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions](https://mlsanity.com/paper/2505.01060/) (2025)
- [Cell division in deep material networks applied to multiscale strain localization modeling](https://mlsanity.com/paper/2101.07226/) (2021)
- [Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity](https://mlsanity.com/paper/2407.11183/) (2024)
- [Spectral Normalization and Voigt-Reuss net: A universal approach to microstructure-property forecasting with physical guarantees](https://mlsanity.com/paper/2504.00712/) (2025)
- [Microstructure-based Variational Neural Networks for Robust Uncertainty Quantification in Materials Digital Twins](https://mlsanity.com/paper/2512.18104/) (2025)
- [Crack-Net: Prediction of Crack Propagation in Composites](https://mlsanity.com/paper/2309.13626/) (2023)
- [Physics-informed neural networks (PINNs) for numerical model error approximation and superresolution](https://mlsanity.com/paper/2411.09728/) (2024)
- [Deep Micro Solvers for Rough-Wall Stokes Flow in a Heterogeneous Multiscale Method](https://mlsanity.com/paper/2507.13902/) (2025)
- [Systematic Performance Assessment of Deep Material Networks for Multiscale Material Modeling](https://mlsanity.com/paper/2602.07192/) (2026)
- [Deep material network for homogenization of piezoelectric composites](https://mlsanity.com/paper/2606.22566/) (2026)

## Abstract (arXiv)

In this paper, a new data-driven multiscale material modeling method, which we refer to as deep material network, is developed based on mechanistic homogenization theory of representative volume element (RVE) and advanced machine learning techniques. We propose to use a collection of connected mechanistic building blocks with analytical homogenization solutions which avoids the loss of essential physics in generic neural networks, and this concept is demonstrated for 2-dimensional RVE problems and network depth up to 7. Based on linear elastic RVE data from offline direct numerical simulations, the material network can be effectively trained using stochastic gradient descent with backpropagation algorithm, enhanced by model compression methods. Importantly, the trained network is valid for any local material laws without the need for additional calibration or micromechanics assumption. Its extrapolations to unknown material and loading spaces for a wide range of problems are validated through numerical experiments, including linear elasticity with high contrast of phase properties, nonlinear history-dependent plasticity and finite-strain hyperelasticity under large deformations. By discovering a proper topological representation of RVE with fewer degrees of freedom, this intelligent material model is believed to open new possibilities of high-fidelity efficient concurrent simulations for a large-scale heterogeneous structure. It also provides a mechanistic understanding of structure-property relations across material length scales and enables the development of parameterized microstructural database for material design and manufacturing.