# Explainable artificial intelligence for mechanics: physics-informing neural networks for constitutive models

arXiv: https://arxiv.org/abs/2104.10683 · HTML: https://mlsanity.com/paper/2104.10683/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 72 references that are in the map.
- First posted on arXiv 20 Apr 2021.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Hyperband: A Novel Bandit-Based Approach to Hyperparameter Optimization](https://mlsanity.com/paper/1603.06560/) (2016)
- [Unmasking Clever Hans Predictors and Assessing What Machines Really Learn](https://mlsanity.com/paper/1902.10178/) (2019)
- [Methods for Interpreting and Understanding Deep Neural Networks](https://mlsanity.com/paper/1706.07979/) (2017)
- [LSTM: A Search Space Odyssey](https://mlsanity.com/paper/1503.04069/) (2015)
- [Non-stochastic Best Arm Identification and Hyperparameter Optimization](https://mlsanity.com/paper/1502.07943/) (2015)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Explaining Recurrent Neural Network Predictions in Sentiment Analysis](https://mlsanity.com/paper/1706.07206/) (2017)
- [iNNvestigate neural networks!](https://mlsanity.com/paper/1808.04260/) (2018)
- [Unsupervised discovery of interpretable hyperelastic constitutive laws](https://mlsanity.com/paper/2010.13496/) (2020)
- [A machine learning based plasticity model using proper orthogonal decomposition](https://mlsanity.com/paper/2001.03438/) (2020)
- [FEA-Net: A Physics-guided Data-driven Model for Efficient Mechanical Response Prediction](https://mlsanity.com/paper/2002.01893/) (2020)

## Built on by (most cited first, 1 of 1)

- [Physically recurrent neural network for rate and path-dependent heterogeneous materials in a finite strain framework](https://mlsanity.com/paper/2404.17583/) (2024)

## Abstract (arXiv)

(Artificial) neural networks have become increasingly popular in mechanics to accelerate computations with model order reduction techniques and as universal models for a wide variety of materials. However, the major disadvantage of neural networks remains: their numerous parameters are challenging to interpret and explain. Thus, neural networks are often labeled as black boxes, and their results often elude human interpretation. In mechanics, the new and active field of physics-informed neural networks attempts to mitigate this disadvantage by designing deep neural networks on the basis of mechanical knowledge. By using this a priori knowledge, deeper and more complex neural networks became feasible, since the mechanical assumptions could be explained. However, the internal reasoning and explanation of neural network parameters remain mysterious. Complementary to the physics-informed approach, we propose a first step towards a physics-informing approach, which explains neural networks trained on mechanical data a posteriori. This novel explainable artificial intelligence approach aims at elucidating the black box of neural networks and their high-dimensional representations. Therein, the principal component analysis decorrelates the distributed representations in cell states of RNNs and allows the comparison to known and fundamental functions. The novel approach is supported by a systematic hyperparameter search strategy that identifies the best neural network architectures and training parameters. The findings of three case studies on fundamental constitutive models (hyperelasticity, elastoplasticity, and viscoelasticity) imply that the proposed strategy can help identify numerical and analytical closed-form solutions to characterize new materials.