Physics-informed neural networks with relaxed alternating learning for image inpainting

Reconstructing missing image content is especially challenging when the damaged region extends to the image boundary, rendering a portion of the boundary data inaccessible and turning the underlying partial differential equation (PDE) into an ill-posed elliptic Cauchy problem. Classical inpainting methods, whether PDE-driven or data-driven, typically assume that complete boundary information is available and therefore degrade significantly in this setting. In this work we propose a novel hybrid framework that unites the mathematical rigor of iterative regularization with the mesh-free flexibility of Physics-Informed Neural Networks (PINNs). The core idea is threefold. First, we decompose the ill-posed Cauchy problem into a sequence of well-posed mixed boundary-value problems via a relaxed Kozlov-Maz'ya-Fomin (KMF) alternating iteration that switches between Dirichlet and Neumann conditions on the inaccessible boundary. Second, each sub-problem is solved by a dedicated lightweight PINN whose loss function encodes the nonlinear edge-enhancing diffusion (EED) operator, thereby enforcing physical consistency; including anisotropic gradient preservation; without mesh generation or matrix assembly. Third, a Bayesian hyperparameter optimization layer automatically selects the relaxation factor, loss-term weights, and network capacity, eliminating tedious manual tuning and ensuring stable convergence across diverse inpainting scenarios. The proposed method achieves considerable improvement in PSNR over conventional PDE solvers and state-of-the-art deep inpainting models, with the most pronounced gains observed for large or boundary-adjacent damaged regions. These results highlight the untapped potential of coupling physics-based modeling with deep learning for principled, reliable, and fully automated image restoration.