We don't know where noodles were invented. Noodles do not preserve well archaeologically, their invention could well be prehistoric, and early written sources don't usually pay much attention to details of food. What we can say for certain is that the oldest known noodles have been found in China (in the northwest, in Qinghai Province (next to Xinjiang and Tibet)). We know of no European pasta until about the 11th century AD (and that's from literature, not archaeology).
The oldest known noodles were made of millet in about 2000BC. Whether an offering, leftovers, or whatever, they were found in an upside-down bowl. They appear to be pulled noodles.
For the original short paper reporting the find: Houyuan Lu et al., "Millet noodles in Late Neolithic China", Nature 437, 967–968 (2005) https://doi.org/10.1038/437967a
For a more detailed paper on the history of noodles in China: Y. Wei et al., "Origin and evolution of Chinese noodles", Cereal Foods World 62(2), 44-51 (2017).
Some type of noodles might have been eaten in the 5th century AD in Syria; we have references to itrium, which might have been some kind of pasta/noodles. At least, it was a wheat dough product, cooked in a pot. Unambiguous wheat pasta is described in the Mediterranean world (specifically, in Egypt) in the 10th century, and we have European descriptions after that. In the 13th century, pasta had appear in Italy, and was commercially important (thus, pasta was certainly in Italy before Marco Polo's return).
For more detail on the history of pasta in Europe and the Mediterranean world, see Silvano Serventi and Françoise Sabban, Pasta: The Story of a Universal Food, Columbia University Press, 2002.
Apparently, there was a 4000 year-old bowl of noodles unearthed in China made from millet. A brief National Geographic and BBC article regarding the discovery can be found here and here, respectively. There's some debate regarding the composition of the noodles, as millet is gluten free and therefore the process of making Chinese lamian which is the basis of some of the most popular noodle dishes in the world (and which the ancient noodles closely resemble) would have been near impossible. But general consensus seems to agree that the find was indeed, of noodles. The researchers did get published in Nature after all. Noodles have made repeat appearances in Chinese literature since the Han dynasty, but writings about noodles don't really have much continuity from earlier eras because they're noodles, not some divine object or holy request. Since the earliest corpus of preserved Chinese writing are the oracle bones which were of religious value, it's not too hard to imagine why the lowly noodle escaped mention for tens of centuries, right up until the Han dynasty. I must say though, that millet noodles are shit and should be avoided. Eat wheat noodles or rice noodles.
Pasta's archaeological record seems to be a lot younger. My brief skim through JSTOR and wikipeida seems to note that the earliest mention of what we'd recognize as pasta and not fried dough dates to around the 12th century, CE. But honestly, I don't think we'll ever really know. It's not too hard to imagine some bored ancient housewife mixing some sort of flour and water to make a dough, then accidentally stumbling upon some form of rudimentary pasta.