Symbolic Regression-Based Models for Hyperelastic Material

Understanding and predicting material behavior is crucial for material design and optimization. Traditional phenomenological models, while simpler, often lack physical accuracy. Machine learning (ML) tools, such as symbolic regression (SR), offer a promising solution by generating human-readable mathematical expressions that fit given datasets. This study investigates the application of symbolic regression (SR) for modeling hyperelastic material behavior. We explore the effectiveness of an error metric based on implicit derivatives compared to the standard inverse normalized root mean squared error (NRMSE). Synthetic data generated from the Mooney-Rivlin model was used for training and evaluation. The results demonstrate that SR can capture the qualitative behavior of the material with moderate accuracy (R-squared values up to 0.86). The proposed implicit derivative metric (E<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>) did not consistently outperform the standard NRMSE metric (E<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf>). However, in specific cases (single hysteresis loop without constant search), E2 offered a visually better fit. The study highlights the challenges of defining suitable error metrics for SR in material behavior modeling. Further research is needed to refine the E2 metric and explore alternative approaches for improved accuracy and generalizability.