Adaptive domain-decomposition physics-informed neural networks for interface problems

Inverse interface problems are particularly challenging when the internal interface is unknown a priori, because the physical subdomains, regional parameters, and PDE residuals cannot be assigned in advance. Under this setting, conventional domain-decomposition methods and interface-aware physics-informed neural networks (PINNs) that rely on prescribed subdomain partitions are not directly applicable. Standard PINNs are then forced to approximate piecewise-smooth or discontinuous behaviors with globally smooth neural surrogates, which often leads to residual mixing across subdomains, gradient singularities near interfaces, and optimization stiffness. To address this difficulty, we propose adaptive domain-decomposition physics-informed neural networks (ADD-PINNs) for inverse PDE problems with unknown interfaces. The method represents the hidden interface through a learnable neural geometric partition and combines hard subdomain assignment with differentiable geometric weighting, so that state approximation and PDE residual evaluation are restricted to locally smooth subdomains. To update the interface without labeled geometric data or prescribed shape priors, we further introduce a physics-feedback interface-evolution mechanism driven by the two-sided difference in local physical consistency across the evolving interface. In this way, the geometric partition, regional parameters, and state fields are identified in a coupled manner from sparse observations and governing equations. Numerical experiments on strongly discontinuous two-dimensional hidden inclusions, high-order beam inversion, elastic weak-interlayer identification, weak-discontinuity heat-source reconstruction, and three-dimensional hidden-inclusion recovery show that ADD-PINNs improve interface reconstruction, field accuracy, and PDE consistency relative to global PINNs, soft-partitioning baselines, and conventional numerical inversion. These results demonstrate that adaptive domain decomposition provides an effective computational framework for hidden-interface inverse problems.