Physics-Informed Tensor Completion with Application to Distribution System State Estimation

State estimation in power distribution networks often involves sparse, noisy, and heterogeneous measurements, making classical weighted least-squares methods unreliable in low-observability regimes. To meet this challenge we develop a physics-informed low-rank tensor completion method. The method encodes voltage phasor components, voltage magnitudes, and power injections over time in a low-rank tensor, while incorporating physics through convex residual penalties. For distribution network state estimation, these residuals are instantiated using linearized power-flow equations, yielding a convex optimization problem over a tensor gauge ball, equivalently represented as a polytope implicitly described via integer programming. We solve this problem using globally convergent blended conditional gradients, resulting in a general-purpose algorithmic framework that can be adapted across different domains and applications where low-rank structure and convex physics residuals are available. We also derive a deterministic error bound showing that, under a restricted stability condition, the estimation error is controlled by measurement noise and physics-model mismatch. Numerical experiments show that the proposed method substantially improves state recovery compared with purely data-driven tensor completion and weighted least-squares, especially in low-observability regimes with missing data.