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Abstract Weather forecasts made with imperfect models contain state‐dependent errors. Data assimilation (DA) partially corrects these errors with new information from observations. As such, the corrections, or “analysis increments,” produced by the DA process embed information about model errors. An attempt is made here to extract that information to improve numerical weather prediction. Neural networks (NNs) are trained to predict corrections to the systematic error in the National Oceanic and Atmospheric Administration's FV3‐GFS model based on a large set of analysis increments. A simple NN focusing on an atmospheric column significantly improves the estimated model error correction relative to a linear baseline. Leveraging large‐scale horizontal flow conditions using a convolutional NN, when compared to the simple column‐oriented NN, does not improve skill in correcting model error. The sensitivity of model error correction to forecast inputs is highly localized by vertical level and by meteorological variable, and the error characteristics vary across vertical levels. Once trained, the NNs are used to apply an online correction to the forecast during model integration. Improvements are evaluated both within a cycled DA system and across a collection of 10‐day forecasts. It is found that applying state‐dependent NN‐predicted corrections to the model forecast improves the overall quality of DA and improves the 10‐day forecast skill at all lead times.