Learn English – Do mathematical equations require punctuation

Bryan Garner, Garner's Modern American Usage, second edition (2003), offers a succinct and reasonable discussion of thus in its various senses from an American perspective:

thus. A. General Senses. Thus has four meanings: (1) in this or that manner {one does it thus}; (2) so {thus far}; (3) hence, consequently; and (4) as an example. In senses 3 and 4, thus, when it begins a clause, should usually have a comma after it.

The sentence that you ask about—

It exploits the knowledge of the problem to create new and thus better solutions from the combination of existing ones.

—uses thus in Garner's sense 3, as an equivalent of hence, consequently, or therefore. But instead of appearing at the beginning of a major clause, it appears in the midst of the phrase "new and better," where both new and better are adjectives modifying the same noun (solutions); in such a position, Garner implies, thus should not have a comma after it. According to Garner's analysis, the case would be different if the sentence read as follows:

Thus, it exploits the knowledge of the problem to create new and better solutions from the combination of existing ones.

Of course, the meaning of that sentence differs significantly from the meaning of the sentence that you ask about. But while we're on the subject of meaning, I must echo the observation in ugajin's answer that including thus in your original sentence makes sense only if you truly intend to say that, simply by virtue of being new, new solutions are better than older ones; as a matter of logic, that's a problematic assertion to make, since it's easy to imagine new attempted solutions that turn out to be worse than the status quo.

In my view, whether a comma is appropriate after thus even at the beginning of a clause is ultimately a matter of style preference; but I agree with Garner's implicit view that, when thus appears in the middle of a clause, it normally does better without a comma.