SDD-PINN: Physics-informed neural network for single droplet drying

Single droplet drying, a fundamental process in spray drying, presents a challenging nonlinear moving boundary diffusion problem. This process is described by a parabolic partial differential equation in a shrinking spherical domain with a Robin mass-transfer boundary condition. Despite extensive recent applications of physics-informed neural networks (PINNs) to solve PDEs, vanilla PINNs often struggle with time-dependent, moving boundary transport problems. This study develops SDD-PINN, a compact and reproducible PINN framework that remains reliable on such evolving domains. To decouple the training from domain shrinkage, we first map the problem to a fixed unit domain via a radius transformation, ξ = r / a ( t ) , yielding a nonlinear advection-diffusion PDE that preserves mass conservation (baseline PINN). Building on this, a compact, physics-motivated PINN recipe is presented comprising: (i) hard enforcement of the initial condition, (ii) a squared transformed radius ( ξ 2 ) as a symmetry-consistent coordinate input to promote regularity near the spherical center, (iii) a logarithmic time reparameterization θ from non-dimensional time τ , (iv) smart collocation sampling, and (v) a combined Adam + L-BFGS optimization schedule. A 2 5 full-factorial experimental design (soft vs. hard initial condition, ξ vs . ξ 2 , τ vs . θ , uniform vs. smart sampling, Adam vs. Adam + L-BFGS) is first conducted on a reference regime to identify a progressive enhancement path. The selected configuration, SDD-PINN (hard initial condition, ξ 2 feature , and θ as input, uniform sampling, and a combined Adam + L-BFGS optimizer), resulted in a mean relative L 2 error 0.021 ± 0.010 against a traditional Crank–Nicolson (CN) reference. Validation across multiple drying conditions spanning shrinkage intensity, diffusivity, and particle size with multi-seed repeats showed that relative to the CN reference, the unified recipe yields a mean relative L 2 error ranging from 2.69×10 −3 to 3.30×10 −2 . These results provide a reproducible, PINN recipe for single droplet drying.