The transition function of the resulting Minimal State Cell (MSC) features d t (transition depth) quadratic layers l i of width w, followed by LSTM-inspired gates. Instead of adopting models that have been developed for nonmechanical applications such as natural language processing ( 16), we design an RNN for material modeling with deep transition and output functions that are decoupled from the number of memory units, enabling the design of RNNs with “minimal state space,” that represent the material with a minimal number of meaningful state variables. We postulate that a universal material model should feature a small, physically meaningful number of state variables but allow for parameter-rich transition and output functions to provide flexibility. As a result, applications of LSTMs and GRUs to material modeling ( 9– 15) involve far more state variables than typical physics-based models.įrom a scientific viewpoint, this excessive use of memory units prevents the identification of interpretable state variables and raises concerns regarding variable redundancy, overfitting, and extrapolation. S1), thereby adding memory units to the network. Additional flexibility is obtained by either widening the layers or stacking ( 8) them (fig. Consequently, LSTM or GRU models with few state variables are necessarily very small (see Supplementary Text), which limits their representational power. However, these transition functions are inherently shallow ( 7).

Typical recurrent networks rely on long short-term memory (LSTM) or gated recurrent unit (GRU) architectures ( 5, 6) to mitigate these problems through gated transition functions (see Table 1). The training of RNNs is notoriously difficult ( 4). At each time step, an input function preprocesses the current inputs, a transition function updates the memory units, and an output function converts the memory units into observable outputs ( 3). RNNs function incrementally and feature memory units, which naturally serve as state variables in the context of material modeling, making them uniquely suited to this problem ( 2). Conceptually, recurrent neural networks (RNNs) ( 1) could alleviate this issue by delivering a universal material model, capable of modeling any material through a simple change of parameters. New classes of materials often require the development of new classes of models with a custom state-space representation (see Fig. Consequently, its formulation is highly specialized. This vector describes succinctly the evolution of the material’s microstructure over time. F Īside from finding the mapping f, a major challenge rests in the choice of the size and contents of the state variable vector χ.

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