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

초록

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 L-2 error by 1.8-2.4 & times; at 50% noise and 3.0-4.8 & times; at 100% noise while maintaining comparable potential fitting. PD-iPINN also exhibits strong grid robustness, retaining relative L-2 error below 0.3 at 50% noise even on sparse spatial grids (N-x=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.

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