I'm sorry but you're confusing "problem statement" with "solution".
The Navier-Stokes equations are a set of partial differential equations - they are the problem statement. Given some initial and boundary conditions, we can find (approximate or exact) solutions, which are functions. But we don't know that these solutions are always Lebesgue integrable, and if they are not, neural nets will not be able to approximate them.
This is just a simple example from well-understood physics that we know neural nets won't always be able to give approximate descriptions of reality.
There are even strong inapproximability results for some problems, like set cover.
"Neural networks are universal approximators" is a fairly meaningless sound bite. It just means that given enough parameters and/or the right activation function, a neural network, which is itself a function, can approximate other functions. But "enough" and "right" are doing a lot of work here, and pragmatically the answer to "how approximate?" can be "not very".
The Navier-Stokes equations are a set of partial differential equations - they are the problem statement. Given some initial and boundary conditions, we can find (approximate or exact) solutions, which are functions. But we don't know that these solutions are always Lebesgue integrable, and if they are not, neural nets will not be able to approximate them.
This is just a simple example from well-understood physics that we know neural nets won't always be able to give approximate descriptions of reality.