Transient diffusion-controlled droplet growth beyond self-similarity: Insights from physics-informed neural networks

Droplet growth and evaporation are ubiquitous in nature and technology. In this work, we focus on diffusion-controlled condensational droplet growth under isothermal conditions. Obtaining a closed-form solution to this problem is challenging because of the moving-boundary condition at the droplet surface (the Stefan condition). However, in the limiting case of a vanishing initial radius and a constant diffusion coefficient, a self-similar solution exists, predicting that the droplet radius grows as the square root of time. The Stefan condition also complicates numerical solution using conventional methods. Here, we employ a physics-informed neural network (PINN) framework to solve the governing diffusion equation coupled with the interfacial mass balance, treating the droplet radius as an additional trainable function of time. The PINN accurately reproduces the analytical self-similar solution, including both the droplet growth law and the concentration profiles. We then apply the framework to transient diffusion problems for which self-similar solutions are unavailable by considering droplet growth from a finite initial radius and introducing a concentration-dependent diffusion coefficient. In both cases, the numerical solutions converge toward the classical self-similar growth regime. These results demonstrate that the PINN framework provides an effective tool for solving moving-boundary diffusion problems beyond the reach of analytical methods, while also supporting the robustness of the classical self-similar solution.