for GPGPU, the better approach is CBRNG like random123.
https://github.com/DEShawResearch/random123
if you accept the principles of encryption, then the bits of the output of crypt(key, message) should be totally uncorrelated to the output of crypt(key, message+1). and this requires no state other than knowing the key and the position in the sequence.
the direct-port analogy is that you have an array of CuRand generators, generator index G is equivalent to key G, and you have a fixed start offset for the particular simulation.
moreover, you can then define the key in relation to your actual data. the mental shift from what you're talking about is that in this model, a PRNG isn't something that belongs to the executing thread. every element can get its own PRNG and keystream. And if you use a contextually-meaningful value for the element key, then you already "know" the key from your existing data. And this significantly improves determinism of the simulation etc because PRNG output is tied to the simulation state, not which thread it happens to be scheduled on.
(note that the property of cryptographic non-correlation is NOT guaranteed across keystreams - (key, counter) is NOT guaranteed to be uncorrelated to (key+1, counter), because that's not how encryption usually is used. with a decent crypto, it should still be very good, but, it's not guaranteed to be attack-resistant/etc. so notionally if you use a different key index for every element, element N isn't guaranteed to be uncorrelated to element N+1 at the same place in the keystream. If this is really important then maybe you want to pass your array indexes through a key-spreading function etc.)
there are several benefits to doing it like this. first off obviously you get a keystream for each element of interest. but also there is no real state per-thread either - the key can be determined by looking at the element, but generating a new value doesn't change the key/keystream. so there is nothing to store and update, and you can have arbitrary numbers of generators used at any given time. Also, since this computation is purely mathematical/"pure function", it doesn't really consume any memory-bandwidth to speak of, and since computation time is usually not the limiting element in GPGPU simulations this effectively makes RNG usage "free". my experience is that this increases performance vs CuRand, even while using less VRAM, even just directly porting the "1 execution thread = 1 generator" idiom.
Also, by storing "epoch numbers" (each iteration of the sim, etc), or calculating this based on predictions of PRNG consumption ("each iteration uses at most 16 random numbers"), you can fast-forward or rewind the PRNG to arbitrary times, and you can use this to lookahead or lookback on previous events from the keystream, meaning it serves as a massively potent form of compression as well. Why store data in memory and use up your precious VRAM, when you could simply recompute it on-demand from the original part of the original keystream used to generate it in the first place? (assuming proper "object ownership" of events ofc!) And this actually is pretty much free in performance terms, since it's a "pure function" based on the function parameters, and the GPGPU almost certainly has an excess of computation available.
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In the extreme case, you should be able to theoretically "walk" huge parts of the keystream and find specific events you need, even if there is no other reference to what happened at that particular time in the past. Like why not just walk through parts of the keystream until you find the event that matches your target criteria? Remember since this is basically pure math, it's generated on-demand by mathing it out, it's pretty much free, and computation is cheap compared to cache/memory or notarizing.
(ie this is a weird form of "inverted-index searching", analogous to Elastic/Solr's transformers and how this allows a large number of individual transformers (which do their own searching/indexing for each query, which will be generally unindexable operations like fulltext etc) to listen to a single IO stream as blocks are broadcast from the disk in big sequential streaming batches. Instead of SSD batch reads you'd be aiming for computation batch reads from a long range within a keystream. (And this is supposition but I think you can also trade back and forth between generator space and index hitrate by pinning certain bits in the output right?)
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Anyway I don't know how much that maps to your particular use-case but that's the best advice I can give. Procedural generation using a rewindable, element-specific keystream is a very potent form of compression, and very cheap. But, even if all you are doing is just avoiding having to store a bunch of CuRand instances in VRAM... that's still an enormous win even if you directly port your existing application to simply use the globalThreadIdx like it was a CuRand stateful instance being loaded/saved back to VRAM. Like I said, my experience is that because you're changing mutation to computation, this runs faster and also uses less VRAM, it is both smaller and better and probably also statistically better randomness (especially if you choose the "hard" algorithms instead of the "optimized" versions like threefish instead of threefry etc). The bit distribution patterns of cryptographic algorithms is something that a lot of people pay very very close attention to, you are turning a science toy implementation into a gatling gun there simply by modeling your task and the RNG slightly differently.
That is the reason why you shouldn't do the "just download random numbers", as a sibling comment mentions (probably a joke) - that consumes VRAM, or at least system memory (and pcie bandwidth). and you know what's usually way more available as a resource in most GPGPU applications than VRAM or PCIe bandwidth? pure ALU/FPU computation time.
buddy, everyone has random numbers, they come with the fucking xbox. ;)