Data-driven discovery of governing equations is of great significance for helping us understand intrinsic mechanisms and build physical models. Recently, numerous highly innovative algorithms have emerged, aimed at inversely discovering the underlying governing equations from data, such as sparse regression-based methods and symbolic regression-based methods. Along this direction, a novel dimensional homogeneity constrained gene expression programming (DHC-GEP) method is proposed in this work. The DHC-GEP simultaneously discovers the forms and coefficients of functions using basic mathematical operators and physical variables, without requiring preassumed candidate functions. The constraint of dimensional homogeneity is capable of filtering out the overfitting equations effectively. The key advantages of DHC-GEP compared with the original gene expression programming, including being more robust to hyperparameters, the noise level and the size of datasets, are demonstrated on two benchmark studies. Furthermore, DHC-GEP is employed to discover the unknown constitutive relations of two representative non-equilibrium flows. Galilean invariance and the second law of thermodynamics are imposed as constraints to enhance the reliability of the discovered constitutive relations. Comparisons, both quantitative and qualitative, indicate that the derived constitutive relations are more accurate than the conventional Burnett equations in a wide range of Knudsen numbers and Mach numbers, and are also applicable to the cases beyond the parameter space of the training data.

中文翻译：

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用于发现控制方程的维度同质性约束基因表达编程

数据驱动的控制方程发现对于帮助我们理解内在机制和建立物理模型具有重要意义。最近，出现了许多高度创新的算法，旨在从数据中逆向发现潜在的控制方程，例如基于稀疏回归的方法和基于符号回归的方法。沿着这个方向，本文提出了一种新的维度同质性约束基因表达编程（DHC-GEP）方法。 DHC-GEP 使用基本数学运算符和物理变量同时发现函数的形式和系数，而不需要预先假设的候选函数。维度同质性约束能够有效滤除过拟合方程。两项基准研究证明了 DHC-GEP 与原始基因表达编程相比的主要优势，包括对超参数、噪声水平和数据集大小更稳健。此外，DHC-GEP 用于发现两个代表性非平衡流的未知本构关系。伽利略不变性和热力学第二定律被强加作为约束，以增强所发现的本构关系的可靠性。定量和定性比较表明，在较宽的努森数和马赫数范围内，推导的本构关系比传统的伯内特方程更准确，并且也适用于超出训练数据参数空间的情况。