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 104 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：我们提出了一种基于格子玻尔兹曼驱动的动理学物理信息神经网络（K-PINN），用于结构化表面上液滴动力学的预测性建模，其中离散的Boltzmann-BGK方程被纳入学习框架。与受限于宏观连续方程的传统PINN不同，K-PINN框架建立在介观动理学层次上，在数据高效的神经网络中保留了基本的格子玻尔兹曼物理机理。该K-PINN已成功应用于模拟各种非平凡液滴现象，包括接触钉扎、各向异性铺展以及在不同形态基底（从随机粗糙度到周期性柱状结构）上的毛细滞后。此外，K-PINN框架确保了严格的物理一致性，例如质量守恒误差在1.5%以内。进一步地，K-PINN中基于U-Net的编码器-解码器结构相比传统神经网络实现了50-75%的误差降低，与高分辨率格子玻尔兹曼模拟几乎完美吻合（$L_2$ ~ 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对不同表面形态具有稳健的收敛性。收敛后，K-PINN可实现每秒超过10^4次评估的实时预测。通过动理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、物理一致性建模建立了新范式。