Flow matching with gappy proper orthogonal decomposition guidance for balanced global field reconstruction

Reconstructing physical fields from sensor measurements is a challenging task in scientific and engineering applications, especially when sensors are sparse and unevenly distributed. Existing generative models directly incorporate point observations into the generation process to guide the generation of the full field. However, they did not provide equitable guidance for relatively sparsely observed regions, resulting in an imbalanced reconstruction error. To mitigate this imbalance, we propose proper orthogonal decomposition conditioned flow matching (POD-FM), a framework that synergistically integrates gappy POD's global physical consistency propagation to provide reconstruction guidance for poorly observed regions. Specifically, we take the POD modal coefficients inverted from observations as physically interpretable global feature guidance, then jointly encode them with the point observations to guide the generative model to reconstruct the full field. These coefficients provide a low-dimensional global prior that anchors the probability flow to the dominant modes of the field, which supplements implicit physical consistency for sparsely observed areas, and thus alleviates the accuracy imbalance caused by uneven sensor distribution. The efficacy of POD-FM has been verified on three progressive POD modal complexity datasets. Experiments have shown that POD-FM achieved higher accuracy and could effectively suppress the influence of sensor noise. The code will be available at https://github.com/XinWang199/POD-FM.