https://link.springer.com/chapter/10.1007%2F11494645_21 seemingly states ODEs can simulate turing machines. Is the simulation enough proof ofit's turing completeness, and in what sense can PDEs be called a language?
In mathematics (and computer science) a language is just a set of strings, and a machine is just a language attached to a semantics. I.e. whatever symbols you need to represent PDEs, inductively generated by the axioms and inference rules of mathematics and theoretical physics (like for lambda-calculus: lambda, x0,x1,... '(', ')' and reduction and construction rules)
Note that this is a very handwavy definition. The "types" of input, output, and what it means to "compute" will depend on your language, semantics and encoding. (E.g. for lambda calcus "compute" means appying reduction rules until the machine halts, if ever)
Turing machines, neural networks, cellular automata, computer programs and (apparently) ODEs can execute each other's "programs" (among any others).