bifurcation in neural network training?

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We use a simple network to approximate a single variable function to be used for trending. Everyday we have new measurement data to train the network so it will follow the trend. This same model is applied to multiple data streams to generate trend function for each data stream. We found a strange phenomenon: Originally all trends looked similar and follow raw data. However when the raw data began to show some fine detail features (oscillation), The majority of trend function follow the raw data to have oscillation. There were a few of the trend functions began to cut corners (some top half, some bottom half, some in first half cycle, some in second half cycle). When you look at the raw data, they still all look similar and visually, those cuts should not be made. It is like a "bifurcation". After a month, the ones with error still keep the same error even after every day's training while the ones follow raw data still keep following raw data. Have anyone encountered similar situation? Any suggestion?