Enabling PINNs for stiff moving-boundary PDEs: Locking-point prediction in superheated steam drying

Physics-informed neural networks (PINNs) typically fail in solving stiff PDEs with moving boundary conditions such as in convection-dominated droplet drying with shrinking domains and in Stefan-type phase-change problems. This study introduces a step-wise approach (scaled logarithmically transformed PINN, LT-PINN) to resolve this issue. The approach is applied to the process of nanosuspension droplet drying in superheated steam exhibiting severe stiffness due to exponential spatial concentration increase and high drying rates. Scaled LT-PINN introduces two synergistic innovations: 1) a logarithmic state-space transformation that compresses the dynamic range and linearizes gradients, and 2) an inverse Péclet -number boundary loss scaling, ensuring a balanced training signal across all stiffness regimes. This approach eliminates the loss imbalance that causes error degradation at high Péclet numbers (∆T = 100 K) reducing the mean relative L ₂ error from 97.05 ± 1.22% (Baseline) and 5.44 ± 1.49% (LT-PINN) to 2.59 ± 0.77% (Scaled LT-PINN). Scaled LT-PINN was validated against Crank-Nicolson benchmarks across a wide range of superheating temperatures (∆T = 10–100 K) and achieved a mean relative L ₂ error below 2.97 ± 0.49% and a locking time error under 8.52 ± 3.01% even at the most extreme drying rates compared to the reference Crank-Nicolson solution. This framework offers a robust approach for solving stiff PDEs with steep gradients and moving boundaries.