Fitting the sum of broad, featureless, functions to broad, featureless, data is prone to a lot of error, especially when the functions are all of the same form. There may be ways to improve the situation, but I would need to know a lot more about your data. For example, if you can obtain multiple sets of data where you know, for physical reasons, that some (but not all; the data sets must be distinct in some way) of the parameters should have the same value in all data sets, you may be able to reduce the errors in the estimated parameters considerably.
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