Solving the Young-Laplace Equation with Physics-Informed Neural Networks

We present a unified Physics-Informed Neural Network (PINN) framework to solve the Young-Laplace equation for diverse free-surface problems that involve the interplay of surface tension, gravity, and surface wetting properties. The methodology is first established and validated with a canonical geometry, the shape of a meniscus on the outside of a vertical circular cylinder. The predictions on the shape and height of the meniscus from our model agree well with the available analytical theory in the surface tension-dominated regime and the approximate formulae in the gravity-dominated regime. We further demonstrate the framework's versatility by extending it to predict the meniscus around a sphere at a liquid-gas interface and the capillary bridge between two parallel horizontal plates. The framework is finally applied to a significantly more complex problem, the shape of a floating surface bubble. In this case, our framework simultaneously solves the Young-Laplace equation for both inner and outer menisci. The prediction of the bubble shape is formulated as a forward problem, while the determination of the meeting point of the menisci is treated as an inverse problem, both of which are solved self-consistently. The framework shows high accuracy across the full range of Bond numbers and successfully overcomes the challenges faced by previous numerical approaches at low Bond numbers. The PINN-based framework is established as a robust, computationally efficient, and general-purpose tool for a wide class of wetting problems.