Merriam-Webster gave the following rules that will help us learn how to use ‘keep up with’ and ‘keep up’

(1) 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.

Ex. The other runners struggled to keep up with the leader.

2: to continue to know the newest information about (something)

Ex. I find it hard to keep up with the news.

3: to maintain contact or relations with someone

Ex. keep up with old friends

Source 

(2) keep up

1: to persist or persevere in

Ex. kept up the good work

2: to keep adequately informed or up-to-date

Ex. keep upon international affairs

3: to continue without interruption

Ex. rain kept up all night

Source

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