# Recurrent Neural Networks with Integrated Gradients Explanation for Predicting the Hysteresis Behavior of Shape Memory Alloys

## TL;DR

This paper uses recurrent neural networks to predict the hysteresis behavior of shape memory alloys, achieving high accuracy and interpretability through the Integrated Gradients method.

## Contribution

The novel contribution is the application of LSTM and GRU networks with Integrated Gradients to model SMA hysteresis behavior and interpret key influencing factors.

## Key findings

- LSTM networks showed the highest accuracy and stable extrapolation across most loading frequencies.
- Stress was identified as the dominant factor influencing predicted strain through the Integrated Gradients analysis.
- The Cycle feature's influence increased with cycle number, reflecting material fatigue accumulation.

## Abstract

The study presents an approach to predicting the hysteresis behavior of shape memory alloys (SMAs) using recurrent neural networks, including SimpleRNN, LSTM, and GRU architectures. The experimental dataset was constructed from 100 to 250 loading–unloading cycles collected at seven loading frequencies (0.1, 0.3, 0.5, 1, 3, 5, and 10 Hz). The input features included the applied stress σ (MPa), the cycle number N (the Cycle parameter), and the indicator of the loading–unloading stage (UpDown). The output variable was the material strain ε (%). Data for training, validation, and testing were split according to the group-based principle using the Cycle parameter. Eighty percent of cycles were used for model training, while the remaining 20% were reserved for independent assessment of generalization performance. Additionally, 10% of the training portion was reserved for internal validation during training. Model accuracy was evaluated using MAE, MSE, MAPE, and the coefficient of determination R2. All architectures achieved R2 > 0.999 on the test sets. Generalization capability was further assessed on fully independent cycles 251, 260, 300, 350, 400, 450, and 500. Among all architectures, the LSTM network showed the highest accuracy and the most stable extrapolation, consistently reproducing hysteresis loops across frequencies 0.1–3 Hz and 10 Hz, whereas the GRU network showed the best performance at 5 Hz. Model interpretability using the Integrated Gradient (IG) method revealed that Stress is the dominant factor influencing the predicted strain, contributing the largest proportion to the overall feature importance. The UpDown parameter has a stable but secondary role, reflecting transitions between loading and unloading phases. The influence of the Cycle feature gradually increases with the cycle number, indicating the model’s ability to account for the accumulation of material fatigue effects. The obtained interpretability results confirm the physical plausibility of the model and enhance confidence in its predictions.