PINN for stiff moving-boundary PDE to predict the locking point in superheated steam drying

Abstract

Abstract Physics-informed neural networks (PINNs) often struggle to solve stiff partial differential equations (PDEs) with moving boundaries, such as convection-dominated droplet drying with shrinking domains and related Stefan-type phase-change problems. This study investigates a physics-guided PINN formulation for nanosuspension droplet drying in superheated steam, where an accurate prediction of the locking point is important, as it marks the onset of structure formation and significantly influences particle morphology and final powder characteristics. The formulation retains the standard PINN backbone and combines two problem-specific modifications: a logarithmic state-variable transformation to compress the solution dynamic range and improve the representation of steep near-surface gradients, and a Péclet number-based scaling of the surface residual to better balance the boundary and PDE training signals across the stiffness regimes. Compared with a numerically verified Crank–Nicolson (CN) reference solution, the proposed formulation showed the greatest advantage in the high-Péclet regime, where the boundary layer steepening and training difficulty were most pronounced. In the stiffest case considered, it substantially reduced the discrepancy from the numerically verified CN reference solution compared with both the baseline PINN and the unscaled log-transformed formulation, while also improving the locking-time prediction. These results show that a physics-guided state transformation, combined with residual scaling, can improve the robustness of PINNs for this class of stiff, moving-boundary drying problems.