Merriam-Webster gave the following rules that will help us learn how to use ‘keep up with’ and ‘keep up’
keep up with
1: to go or make progress at the same rate as (others) :to stay even with (others) in a race, competition, etc.
Example: The other runners struggled to keep up with the leader.
2: to continue to know the newest information about (something)
Example: I find it hard to keep up with the news.
3: to maintain contact or relations with someone
Example: keep up with old friends
keep up
1: to persist or persevere in
Example: kept up the good work
2: to keep adequately informed or up-to-date
Example: keep up on international affairs
3: to continue without interruption
Example: rain kept up all night
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