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Finding diagrams that contain a specific part or a similar part is important in many engineering tasks. In this search task, the query part is expected to match only a small region in a complex image.This paper investigates several local matching networks that explicitly model local region-to-region similarities. Deep convolutional neural networks extract local features and model local matching patterns. Spatial convolution is employed to cross-match local regions at different scale levels, addressing cases where the target part appears at a different scale, position, and/or angle. A gating network automatically learns region importance, removing noise from sparse areas and visual metadata in engineering diagrams. Experimental results show that local matching approaches are more effective for engineering diagram search than global matching approaches. Suppressing unimportant regions via the gating net-work enhances accuracy. Matching across different scales via spatial convolution substantially improves robustness to scale and rotation changes. A pipelined architecture efficiently searches a large collection of diagrams by using a simple local matching network to identify a small set of candidate images and a more sophisticated network with convolutional cross-scale matching to re-rank candidates.