Jixin Hou, Xianyan Chen, Taotao Wu, Ellen Kuhl, Xianqiao Wang AbstractWe introduce a data-driven framework to automatically identify interpretable and physically meaningful hyperelastic constitutive models from sparse data. Leveraging symbolic regression, an algorithm based on genetic programming, our approach generates elegant hyperelastic models that achieve accurate data fitting through parsimonious mathematic formulae, while strictly adhering to hyperelasticity constraints such as polyconvexity. Our investigation spans three distinct hyperelastic models—invariant-based, principal stretch-based, and normal strain-based—and highlights the versatility of symbolic regression. We validate our new approach using synthetic data from five classic hyperelastic models and experimental data from the human brain to demonstrate algorithmic efficacy.
) and ([2 3] 2 2 ). This observation suggests the potential (exp([2 3]) 1 existence of multiple optima for the current optimization problem. For further exploration, it is (exp([2 3]) 1 (exp([2 3] ) 1 crucial to consider either an enriched function space or more diverse loading modes when conducting symbolic regression or CANN. Among the four models we discovered, the third model exhibits the highest predictive accuracy with the simplest = 0.017(exp(27.91[2 3]) 1) form. Therefore, it serves as the optimal model discovered by the Invariant-based Symbolic
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Regression algorithm. Figure 4. Four distinct hyperelastic models discovered with invariant-based algorithm. Models are trained simultaneously with data from three loading modes, and tested with tension, compression and, shear data individually. Dots illustrate the experimental data of the human brain indicates the goodness of fit. Corresponding mathematical expressions of strain energy cortex. function are presented at the bottom of the figure. 2
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Within the framework of invariants, Figure 5 provides a comparison on the predictive performance of the optimal hyperelastic model discovered by symbolic regression, multiple regression, and artificial neural networks. The latter two models are derived from our recent paper , where artificial neural networks followed the idea of CANNs , but utilized a different loss function, the mean absolute percentage error, MAPE. From the comparison, no significant difference can be observed for these three models, except for a smaller underestimation in tension for the symbolic regression model, which yields a slightly higher value (0.908), as shown in 2 Figure 5a. Notably, the model obtained from multiple regression takes the same format as the
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= 0.552[2 3] 2.858 ln(1 2.483[2 3] 2 ) Figure 5. Symbolic regression vs Multiple regression and artificial neural network. Comparison on predictive performance of invariant-based hyperelastic models derived from symbolic regression (SR), multiple regression (MR), and neural network (NN). Models are trained simultaneously with data from three loading modes, and tested with tension (a), compression (b), and shear data (c), individually. Dots illustrate the experimental data of the human brain cortex. indicates the goodness of fit. Corresponding mathematical expressions of strain energy function are presented at the bottom of the figure. 2 For invariant-based hyperelastic models, critical physical admissible conditions elucidated in Section 2.2, such as the polyconvexity, have been predefined and embedded within the symbolic
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