Text similarity for badly written text Consider the following scenario:
Suppose two lists of words $L_{1}$ and $L_{2}$ are given. $L_{1}$ contains just bad-written phrases (like 'age' instead of '4ge' or 'blwe' instead of 'blue' etc.). On the other hand, each element of $L_{2}$ is a well-written version of each element of $L_{1}$.
Here is an example:
$$L_{1}=[...,dqta \ 5ciencc,...,s7ack \ exch9nge,...],$$
$$L_{2}=[...,stack \ exchange,...,data \ science,...].$$
Problem: Is there any strategy to try to predict which element $w^{\prime}$ in $L_{2}$ is the syntactically correct counterpart of a given bad-written element $w$ of $L_{1}$?
By 'strategy' I mean some sort of syntactic word embeddings (that allow us to compare texts by using cosine similarity), any "syntactic" Word2Vec or a probabilistic model that could compute $P(w^{\prime} \mid  w)$ (how likely is that $w^{\prime}$ could be the well-written version of $w$) etc.
Note: To be concrete, I'm asking for a measure of syntactic similarity among two pieces of text.

Yes, there’s a lot of research done on recovering from misspellings.
The seminal probabilistic framework is Probability scoring for spelling correction by Church and Gale, which takes a Bayesian noisy channel approach. It’s really very straightforward to implement. (In your case, you’d only allow substitutions errors, not insertions or deletions.)

More things to consider...
Word embeddings are not a reasonable foundation for a strategy of recovering from spelling errors of the kind you’ve shown.
An alternative way of framing the problem is as a “c-test”, which has a long history in psychometric literature: building a model of the individual character gaps.
As a final avenue to explore, notice that your misspellings always superficially resemble the right letters. OCR (optical character recognition) has to deal with this problem all the time: picking the right letter based on the scanned page, but also based on the surrounding context.
