Triple periodic minimal surfaces (TPMS) have garnered significant interest due to their structural efficiency and controllable geometry, making them suitable for a wide range of applications. This paper investigates the relationships between porosity and persistence entropy with the shape factor of TPMS. We propose conjectures suggesting that these relationships are polynomial in nature, derived through the application of machine learning techniques. This study exemplifies the integration of machine learning methodologies in pure mathematical research. Besides the conjectures, we provide the mathematical models that might have the potential implications for the design and modeling of TPMS structures in various practical applications.

翻译：三周期极小曲面（TPMS）因其结构高效性和几何可控性而备受关注，使其适用于广泛的应用领域。本文研究了TPMS孔隙率、持续熵与形状因子之间的关系。我们提出了这些关系本质上是多项式形式的猜想，这些猜想是通过机器学习技术推导得出的。本研究展示了机器学习方法在纯数学研究中的整合应用。除猜想外，我们还提供了可能对TPMS结构在各种实际应用中的设计与建模具有潜在意义的数学模型。