Abstract
(Artificial) neural networks have become increasingly popular in mechanics and materials sciences 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. 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 can 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-explaining approach, which interprets neural networks trained on mechanical data a posteriori. This proof-of-concept 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.
Highlights
Data-driven models trained with deep learning algorithms have achieved tremendous successes in many research fields (LeCun et al, 2015)
Since time-variant problems in mechanics are often modeled by training Recurrent Neural Networks (RNNs) on time-variant and path-dependent mechanical data (Freitag et al, 2011; Cao et al, 2016; Koeppe et al, 2019; Wu L. et al, 2020), we propose to use the Principal Component Analysis (PCA) to investigate recurrent cell states, e.g., in Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) cells
We proposed a step towards physics-explaining neural networks, which inductively complement existing deductive approaches for physics-informed and physics-guided neural networks
Summary
Data-driven models trained with deep learning algorithms have achieved tremendous successes in many research fields (LeCun et al, 2015). As the archetypical deep learning model, (artificial) neural networks and their variants are powerful predictors exceptionally well-suited for spatio-temporal data, such as mechanical tensor fields (Koeppe et al, 2020a). Each successive layer of a deep neural network learns to extract higher-level representations of the input and creates a data-driven model by Explainable Artificial Intelligence for Mechanics supervised learning. As one of the first works in mechanics, Ghaboussi et al (1991) proposed a unified constitutive model with shallow neural networks that learn from experimental data. To bridge the gap between atomistic and continuum mechanics, Teichert et al (2019) trained integrable deep neural networks on atomistic scale models and successively approximated free energy functions. Various data science methods have been successfully applied to simulation data and synthesized microstructures to analyze, characterize, and quantifify (e.g., Zhao et al (2020); Altschuh et al (2017))
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