Natual isomorphisms with FDHilb are very difficult to get here: Different flavors of Petri nets present different flavors of FREE monoidal categories. "Free" here means that we have categories that satisfy exactly the equations that are needed to be (symmetric, commutative) monoidal, nothing more. Instead, FDHilb is compact closed, and even more, hypergraph. This means that it has a lot more structure beyond monoidality: It has products (that are actually biproducts), cups and caps (because it is compact closed), etc. So there is no way to generate this kind of stuff from one of our nets: FDHilb has waaay more equations than just a monoidal cat. What you can get, tho, is functors from our categories to FDHilb. This is what "freeness" means. :)
In https://arxiv.org/abs/1805.05988 we were able to tweak the definition of Petri net a bit to let it generate free compact closed categories, and I feel this is the best we can do.
The kind of graphical gadget that generates FDHilb (in the sense that the graphical calculus is sound and complete wrt FDHilb) is called ZX calculus (or one of its equivalent variants, such as ZW). It took roughly 10 years to prove that ZX is complete wrt FDHilb! In any case, a string diagram in ZX calculus looks like a hypegraph with extra properties and equations. But you lose the dynamic interpretation of tokens moving in the net, there are no tokens in ZX!
The kind of graphical gadget that generates FDHilb (in the sense that the graphical calculus is sound and complete wrt FDHilb) is called ZX calculus (or one of its equivalent variants, such as ZW). It took roughly 10 years to prove that ZX is complete wrt FDHilb! In any case, a string diagram in ZX calculus looks like a hypegraph with extra properties and equations. But you lose the dynamic interpretation of tokens moving in the net, there are no tokens in ZX!