What you are looking for is the lottery ticket hypothesis for neural networks. Hit a search engine with those words you will find examples.

https://arxiv.org/abs/1803.03635 ( you can follow up on semantic scholar for more)

Selecting which weights to discard seems as hard as the original problem. But random decimation, sometimes barely informed decimation have been observed to be effective.

On the theory side now it's understood that in the thicket of weights, lurk a much much smaller subset that can have nearly the same output.

These observations are for DNNs in general. For time series specifically I don't know what the state of the art is. In general NNs are still catching up with traditional stats approaches in this domain. There are a few examples where traditional approaches have been beaten, but only a few.

One good source to watch are the M series of competitions.