Flow-RI: Residual Inference with Flow Matching for Wireless Spatial Field Reconstruction

Inverse problems recover an unknown signal or physical field from incomplete and noisy observations. In wireless spatial field reconstruction, measurements are often available at only a small fraction of locations, making the problem particularly difficult at low sampling rates. A pretrained generative model can provide a strong prior for the complete field, while the local spatial correlation in wireless fields provides an additional structural prior that can be exploited through spatial regularization. A straightforward approach is to apply this regularization to the complete reconstructed field. However, doing so may alter structures that are already captured by the learned prior. To address this, we introduce Flow-RI, a residual inference framework built on a pretrained Flow Matching prior. Instead of regularizing the complete field, Flow-RI applies spatial regularization only to the prediction residual. At each flow step, the mismatch between the observed measurements and the predicted values at the sampled locations provides sparse observations of this residual. A spatial regularizer estimates the residual over the complete grid from these sparse observations, and the resulting residual estimate is used to correct the Flow Matching prediction before forming the next intermediate state. We further analyze the effect of applying spatial regularization to the prediction residual rather than to the complete field. At a fixed flow step, under the assumption that the reference prediction equals the conditional mean and that the measurement noise is conditionally zero-mean, residual regularization preserves conditional unbiasedness, whereas regularizing the complete field generally introduces an additional smoothing bias. Experiments on ray-traced wireless fields show that Flow-RI achieves the lowest mean absolute error (MAE) across all evaluated wireless reconstruction settings. The method also generalizes to different transmitter deployments and previously unseen urban scenes without retraining.