We 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. Our results suggest that our symbolic regression robustly discovers accurate models with succinct mathematic expressions in invariant-based, stretch-based, and strain-based scenarios. Strikingly, the strain-based model exhibits superior accuracy, while both stretch- and strain-based models effectively capture the nonlinearity and tension-compression asymmetry inherent to human brain tissue. Polyconvexity examinations affirm the rigor of convexity within the training regime and demonstrate excellent extrapolation capabilities beyond this regime for all three models. However, the stretch-based models raise concerns regarding potential convexity loss under large deformations. Finally, robustness tests on noise-embedded data underscore the reliability of our symbolic regression algorithms. Our study confirms the applicability and accuracy of symbolic regression in the automated discovery of hyperelastic models for the human brain and gives rise to a wide variety of applications in other soft matter systems.

翻译：我们提出了一种数据驱动框架，用于从稀疏数据中自动识别可解释且物理意义明确的超弹性本构模型。该方法以基于遗传编程的符号回归算法为核心，能够生成形式简洁、拟合精准的超弹性模型，同时严格满足多凸性等超弹性约束条件。本研究涵盖三类具有代表性的超弹性模型——基于不变量、基于主拉伸和基于正应变——并充分展示了符号回归的普适性。我们利用五种经典超弹性模型的合成数据以及人脑组织的实验数据验证了新方法的有效性。结果表明，符号回归能够稳健地发现具有简洁数学表达式的精确模型，适用于不变量、拉伸和应变三种场景。值得注意的是，基于应变的模型展现出最优的精度，而拉伸和应变模型均能有效捕捉人脑组织固有的非线性和拉压不对称特性。多凸性检验验证了训练域内所有模型均保持严格的凸性，并展现出超越该域的优异外推能力。然而，基于拉伸的模型在大变形下存在凸性丧失的潜在风险。最后，针对含噪数据的鲁棒性测试进一步证实了符号回归算法的可靠性。本研究证实了符号回归在自动发现人脑超弹性模型中的适用性与准确性，并为其他软物质系统的广泛工程应用提供了新范式。