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The phrase "advances over the initial" is correct and usable in written English.
It can be used when discussing improvements or progress made beyond an initial stage or starting point.
Example: "The project has seen significant advances over the initial proposal, leading to better outcomes."
Alternatives: "progress beyond the initial" or "improvements over the initial".
Exact(1)
It appears that this heap advances over the initial uniform layer which induces a shear in a band between two almost undeformed blocks.
Similar(59)
"The season is over!" was the initial lament.
Armed with a Lewis Gun, Peeler advanced with the initial wave.
Those who advance from the initial group stages compete in knockout rounds, and the eventual victors are rewarded for their efforts with elegant medals and trophies.
We've gotten over the initial shock.
— By now, voters here are over the initial shock.
Getting over the initial price is hard.
The farthest advance achieved on the initial day of the offensive was achieved by the 4th Guards Brigade, which advanced 5 km.
Advances over the room in the burl of his worth.
"Yes, medical treatment has made astonishing advances over the years.
After the war, advances came rapidly, with the initial emphasis on the correction or amelioration of congenital defects.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com