Physics-Informed Neural Networks for Predicting the Shape of a Bubble Floating on a Liquid Surface

Bubbles floating on a liquid surface exhibit complex behaviors due to the interplay of surface tension, buoyancy, and pressure differentials. The present study aims to predict the shape of a floating surface bubble by solving the corresponding Young-Laplace equation of both inner and outer menisci with a Physics-Informed Neural Network (PINN). In our machine-learning framework, the task of predicting the bubble shape is formulated as a forward problem, while determining the meeting point of the inner and outer menisci is treated as an inverse problem. Our framework treats both problems simultaneously and self-consistently. The framework has shown high accuracy in predicting bubble shapes across the full range of Bond numbers, from small to large, that are physically relevant. Particularly, we find a way to correctly formulate the PINN in the low Bond number region that overcomes the challenges faced by previously reported numerical approaches to obtain bubble shapes at low Bond numbers.

Furthermore, we find that directly embedding the Young-Laplace equation and the associated boundary conditions into the loss function of the PINN significantly enhances computational efficiency and facilitates real-time predictions of bubble shapes. The capability of quickly and accurately predicting bubble shapes may find applications in understanding a wide range of natural phenomena and industrial processes involving bubbles. The work may also lend insights into the treatment of other differential equations frequently encountered in science with PINNs.