  Modern pendant drop tensiometry relies on numerical solution of the
Young-Laplace equation and allow to determine the surface tension from a single
picture of a pendant drop with high precision. Most of these techniques solve
the Young-Laplace equation many times over to find the material parameters that
provide a fit to a supplied image of a real droplet. Here we introduce a
machine learning approach to solve this problem in a computationally more
efficient way. We train a deep neural network to determine the surface tension
of a given droplet shape using a large training set of numerically generated
droplet shapes. We show that the deep learning approach is superior to the
current state of the art shape fitting approach in speed and precision, in
particular if shapes in the training set reflect the sensitivity of the droplet
shape with respect to surface tension. In order to derive such an optimized
training set we clarify the role of the Worthington number as quality indicator
in conventional shape fitting and in the machine learning approach. Our
approach demonstrates the capabilities of deep neural networks in the material
parameter determination from rheological deformation experiments in general.
