在非平衡流动中存在复杂的物理化学过程，导致NS方程中基于经验假设的应力和热流本构关系失效，如何构建非平衡条件下的本构关系和控制方程，是有挑战性的工作。课题组将基于物理的分子模拟和基于数据的机器学习方法相结合，通过数据驱动发现隐藏在分子模拟背后的流体力学控制方程，如图5所示，为微观和宏观耦合提供了新思路[12]。近年来，课题组发展了满足量纲齐次性约束的基因表达式编程算法，结合分子模拟产生的数据，可用于非平衡条件下的本构关系和宏观方程预测，显著提升了宏观方程描述非平衡特征的能力[13,14]。

Fig. 5. Data-driven discovery of governing equations based on molecular simulations.

[12] J. Zhang*, and W. Ma, "Data-driven discovery of governing equations for fluid dynamics based on molecular simulation," Journal of Fluid Mechanics 892, A5 (2020).

[13] H. Xing, J. Zhang*, W. Ma, and D. Wen, "Using gene expression programming to discover macroscopic governing equations hidden in the data of molecular simulations," Physics of Fluids 34, 057109 (2022).

[14] W. Ma, J. Zhang*, et al., "Dimensional homogeneity constrained gene expression programming for discovering governing equations," Journal of Fluid Mechanics 985, A12 (2024).