Diffusion-regularized PINNs for Poisson inverse source reconstruction of charge dynamics from noisy potential measurements

Reconstructing spatiotemporal charge density from noisy potential measurements (e.g., Kelvin probe force microscopy) is a fundamental inverse problem in semiconductor characterization, yet it is severely ill-posed because charge is proportional to the Laplacian of the potential, which amplifies high-frequency noise. Conventional approaches typically require smoothing or explicit initial/boundary conditions (IC/BC) to stabilize solutions; however, such priors are often unavailable in realistic experiments. Here, we propose a Poisson–diffusion inverse physics-informed neural network (PD-iPINN) that couples the quasi-static Poisson equation with a diffusion–decay dynamics constraint for charge transport. Crucially, we train PD-iPINN without IC/BC loss terms, using only potential observations over space–time and PDE residuals. Across systematic synthetic test cases (1D sinusoidal, 1D Gaussian, and 2D sinusoidal charge dynamics) with relative Gaussian noise levels up to 100%, PD-iPINN achieves substantially improved charge reconstruction compared to a Poisson inverse PINN (P-iPINN), reducing relative L2 error by 1.8–2.4× at 50% noise and 3.0–4.8× at 100% noise while maintaining comparable potential fitting. PD-iPINN also exhibits strong grid robustness, retaining relative L2 error below 0.3 at 50% noise even on sparse spatial grids (Nx=21). In parameter estimation tests, the decay rate k is reliably identified (≤8% error at 100% noise), whereas the diffusion coefficient D is more noise-sensitive. We interpret the improved stability as diffusion-induced temporal regularization that suppresses high-frequency components in Fourier space, synergizing with neural networks’ spectral bias. The proposed framework provides a practical route to IC/BC-free charge dynamics reconstruction from noisy potential data.