Molecular dynamics inferred transfer learning models for finite‐strain hyperelasticity of monoclinic crystals: Sobolev training and validations against physical constraints

Abstract

We present a machine learning framework to train and validate neural networks to predict the anisotropic elastic response of a monoclinic organic molecular crystal known as β$$ \beta $$‐HMX in the geometrical nonlinear regime. A filtered molecular dynamic (MD) simulations database is used to train neural networks with a Sobolev norm that uses the stress measure and a reference configuration to deduce the elastic stored free energy functional. To improve the accuracy of the elasticity tangent predictions originating from the learned stored free energy, a transfer learning technique is used to introduce additional tangential constraints from the data while necessary conditions (e.g., strong ellipticity, crystallographic symmetry) for the correctness of the model are either introduced as additional physical constraints or incorporated in the validation tests. Assessment of the neural networks is based on (1) the accuracy with which they reproduce the bottom‐line constitutive responses predicted by MD, (2) the robustness of the models measured by detailed examination of their stability and uniqueness, and (3) the admissibility of the predicted responses with respect to mechanics principles in the finite‐deformation regime. We compare the training efficiency of the neural networks under different Sobolev constraints and assess the accuracy and robustness of the models against MD benchmarks for β$$ \beta $$‐HMX.

FE$${}^\textrm{ANN}$$: an efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining

“…In order to train the ANN with respect to the stresses, gradients of the output with respect to the input are inserted into the loss [12,45,46,52,53]. This technique is also named as Sobolev training in [77,80]. Furthermore, physical knowledge can be inserted via constraint training processes [81].…”

“…An extension of the deep material network approach to fully coupled thermo-mechanical multiscale simulations of composite materials is given in [26]. The application of ANNs used as a surrogate for molecular dynamics simulations is discussed in [7,80].…”

“…In the work [45], the growth condition is fulfilled in a similar way by adding an additional energy term which is not directly included in ψANN . In [80], a post-training validation test is suggested to check the growth condition. Thereby, deformation gradients with det F → 0 + are prescribed to the network and it is observed whether the resultant energy is monotonically increasing.…”

A comparative study on different neural network architectures to model inelasticity

“…[8][9][10][14][15][16][17][18] Thereby, an improved training is applied that allows calibration of the network directly by tuples of stress and strain, that is, the derivative of energy with respect to the deformation is included into the loss, which is also called Sobolev training. 19,20 Alternatively, a network previously trained to predict stress coefficients can be used to construct a pseudopotential, thus ensuring thermodynamic consistency of NN-based elastic models a posteriori. 21 Compared to elasticity, the modeling of path-dependent, that is, inelastic, constitutive behavior by NN-based approaches is more complex.…”

“…Meanwhile, FNNs using invariants as input and the hyperelastic potential as output are a very well established approach 8‐10,14‐18 . Thereby, an improved training is applied that allows calibration of the network directly by tuples of stress and strain, that is, the derivative of energy with respect to the deformation is included into the loss, which is also called Sobolev training 19,20 . Alternatively, a network previously trained to predict stress coefficients can be used to construct a pseudopotential, thus ensuring thermodynamic consistency of NN‐based elastic models a posteriori 21 …”