We introduce a Lattice-Boltzmann-driven kinetic physics-informed neural network (K-PINN) for predictive modeling of droplet dynamics on structured surfaces, in which the discrete Boltzmann-BGK equation is incorporated into the learning framework. Different from traditional PINNs that are restricted by macroscopic continuum equations, the K-PINN framework is built on the mesoscopic kinetic level, in which the essential Lattice-Boltzmann physics is preserved in the data-efficient neural network. The K-PINN has been successfully employed for modeling non-trivial droplet phenomena such as contact pinning, anisotropic spreading, and capillary hysteresis on substrates of different morphologies, ranging from random roughness to periodic pillar structures. Moreover, strict physical consistency, such as mass conservation within 1.5%, is ensured in the K-PINN framework. Furthermore, the U-Net-based encoder-decoder structure of the K-PINN results in a 50-75% reduction in error compared to traditional neural networks, achieving almost perfect agreement with high-resolution Lattice-Boltzmann simulations $L_2$ ~ 0.021-0.026, $R^2$ ~ 0.999. Robust convergence of the K-PINN to diverse surface morphologies is ensured through curriculum learning and adaptive two-phase optimization. Upon convergence, the K-PINN can perform real-time prediction with over $10^4$ evaluations per second. Through the combination of kinetic theory and physics-informed learning, this work establishes a new paradigm for fast, physically consistent modeling of multiphase flows on complex surfaces.

翻译：我们提出了一种网格玻尔兹曼驱动的动力学物理信息神经网络（K-PINN），用于预测结构化表面上的液滴动力学行为，其中离散的玻尔兹曼-BGK方程被纳入学习框架。与传统受限于宏观连续方程的物理信息神经网络不同，K-PINN框架建立在介观动力学层面，在数据高效神经网络中保留了网格玻尔兹曼的核心物理机制。K-PINN已成功应用于模拟非平凡液滴现象，包括接触钉扎、各向异性铺展以及在不同形貌基底（从随机粗糙表面到周期性柱状结构）上的毛细滞后效应。此外，K-PINN框架确保了严格的物理一致性，例如质量守恒误差控制在1.5%以内。更关键的是，K-PINN中基于U-Net的编码器-解码器结构相比传统神经网络可减少50%-75%的误差，与高分辨率网格玻尔兹曼仿真达到近乎完美的一致性（$L_2$ ~ 0.021-0.026，$R^2$ ~ 0.999）。通过课程学习与自适应两阶段优化，K-PINN对不同表面形貌实现了鲁棒收敛。收敛后，K-PINN可进行实时预测，每秒评估次数超过$10^4$次。通过动力学理论与物理信息学习的结合，本研究为复杂表面上多相流的快速、物理一致建模建立了新的范式。