The phenomenon of capillary rise is commonly observed in both nature and engineering fields. There has been growing interest in the study of the geometry effect on the capillary rising. In this work, we investigate the effect of tube geometry on the capillary rise, and explore the optimal tube geometry that corresponds to the maximum equilibrium height. To this end, we present two machine learning approaches that are able to optimize the tube shape to obtain maximum equilibrium height given a material. Specifically, the first one is based on the Gaussian process regression (GPR) in a purely data-driven manner, and the second one utilizes deep neural networks (DNNs) which are capable of encoding the physical constraint, e.g., the governing equation for the capillary rise in a tube. We apply both methods to study the capillary rise in the uniform cylindrical, trigonometric-shaped and quadratic-shaped tubes, and then determine the optimal geometries that achieve the maximum equilibrium heights. Our results demonstrate that the both approaches can effectively optimize complex capillary channels corresponding to the equilibrium height. Further, GPR is computationally more efficient than DNNs, while the DNNs with physical constraints are more promising for solving problems in which the tube geometry is parameterized at high dimensions.
Xu et al. (Mon,) studied this question.