Viscoelasticty with physics-augmented neural networks: Model formulation and training methods without prescribed internal variables

TL;DR

This work tackles data-driven viscoelastic constitutive modeling by embedding physics into neural networks through generalized standard materials. It develops a dual-potential QQ model with a convex free-energy ψ and a convex-dissipation φ learned via fully input convex (FICNN) and partially input convex (PICNN) networks, ensuring thermodynamic consistency and isotropy. A key contribution is a recurrent-cell training approach (LSTM-based) that generates internal variables on the fly, enabling calibration from stress–strain paths alone and avoiding explicit internal-variable labels. Across invariant-based formulations, it demonstrates superior extrapolation and data efficiency relative to coordinate-based inputs, and shows that an auxiliary RNN offers the best balance of accuracy and scalability for large datasets. The method broadens the applicability of NN-based constitutive modeling to viscoelastic materials and potentially to elastoplastic and multiscale problems, with clear pathways for extending to more complex internal-variable sets or finite-strain regimes.

Abstract

We present an approach for the data-driven modeling of nonlinear viscoelastic materials at small strains which is based on physics-augmented neural networks (NNs) and requires only stress and strain paths for training. The model is built on the concept of generalized standard materials and is therefore thermodynamically consistent by construction. It consists of a free energy and a dissipation potential, which can be either expressed by the components of their tensor arguments or by a suitable set of invariants. The two potentials are described by fully/partially input convex neural networks. For training of the NN model by paths of stress and strain, an efficient and flexible training method based on a recurrent cell, particularly a long short-term memory cell, is developed to automatically generate the internal variable(s) during the training process. The proposed method is benchmarked and thoroughly compared with existing approaches. These include a method that obtains the internal variable by integrating the evolution equation over the entire sequence, while the other method uses an an auxiliary feedforward neural network for the internal variable(s). Databases for training are generated by using a conventional nonlinear viscoelastic reference model, where 3D and 2D plane strain data with either ideal or noisy stresses are generated. The coordinate-based and the invariant-based formulation are compared and the advantages of the latter are demonstrated. Afterwards, the invariant-based model is calibrated by applying the three training methods using ideal or noisy stress data. All methods yield good results, but differ in computation time and usability for large data sets. The presented training method based on a recurrent cell turns out to be particularly robust and widely applicable and thus represents a promising approach for the calibration of other types of models as well.

Figures (16)

• Figure 1: Structure of the invariant-based two-potential model and prediction process. The rate of the internal variables is determined iteratively so that the internal stress $\boldsymbol{\uptau}$ calculated from the free energy and the internal stress $\hat{\boldsymbol{\uptau}}$ calculated from the dissipation potential are equal within a specified tolerance $e$.
• Figure 2: Schematic representation of the training process using the integration strategy. In each time step, the new material state is obtained iteratively with the procedure given in Fig. \ref{['fig:FNN']}. The internal variable is passed on to every time step as starting point for the next integration step until the last time step of the sequence is reached. Consequently, the calculation of the stress for time step $n$ requires the evaluation of all time steps $1,2,\ldots, n-1$ in advance.
• Figure 3: Schematic representation of the training process using an FNN as auxiliary network for the internal variable. This FNN receives a single input, the time ${}^{n}t$, and outputs the six independent entries of ${}^{n}\mathbf{q}$. To calculate the rate ${}^{n}\dot\mathbf{q} \approx {}^{n}\mathbf{q} - {}^{n-1}\mathbf{q} ) / {}^{n}\Delta t$, ${}^{n-1}\mathbf{q}$ is obtained by evaluating this FNN with ${}^{n-1}t$ as input. Note that each time step can be evaluated detached from the others, which allows fast training and the creation of minibatches within a sequence, if required.
• Figure 4: Schematic representation of the proposed training process using an RNN as auxiliary network for the internal variable. In each time step $n$, this RNN receives three inputs: the strain ${}^{n}\boldsymbol{\upvarepsilon}$, the stress ${}^{n}\boldsymbol{\upsigma}$ and the time increment ${}^{n}\Delta t$. Together with the hidden state ${}^{n-1}\boldsymbol{\mathscr{h}}$ and cell state ${}^{n-1}\boldsymbol{\mathscr{c}}$ from the previous time step, the RNN-cell processes these data into a new hidden state, that contains the information about the new internal variable. This new hidden state is forwarded to an FNN, that reduces the dimensionality to six for the six independent entries of ${}^{n}\mathbf{q}$. The new hidden state and cell state are passed on to the next time step, until the final time step of the sequence is reached.
• Figure 5: Rheological model of the viscoelastic reference solid with a single internal variable $\mathbf{q}=\boldsymbol{\upvarepsilon}^{\text{in}}$.