LARC-K: K-step learned adjoint-inspired residual correction for sparse-sensor spatiotemporal source reconstruction in a two-dimensional convection-diffusion equation

Abstract Sparse-sensor spatiotemporal source reconstruction for a two-dimensional convection–diffusion partial differential equation (PDE) seeks to recover a dense time-dependent source field from time series recorded by fixed sensors. The inverse problem is ill-posed: advection displaces source evidence, diffusion attenuates localized peaks, and different source histories can produce similar sparse observations. We propose K-step learned adjoint-inspired residual correction (LARC-K), which starts from a neural source prior and repeatedly refines the source with the residual of the current PDE rollout. At each correction step, LARC-K recomputes the sensor residual, backprojects it to source space through an adjoint-style sensitivity cue, and applies a learned source-space update. On a controlled synthetic multi-regime evaluation covering diverse source families and variations in sensor sparsity, observation noise, and PDE parameters, LARC-K5 achieves the lowest source root-mean-square error (RMSE) of 0.7036, as well as the lowest center distance and mass gap among the representative baselines.