Are there any studies on finding cellular automata rules for modeling specific systems?

Question

While playing around I managed to design a Cellular Automaton (CA) that behaves like a fluid. Very basic behavior, like diffusion, obstacle avoidance, pressure etc. And that got me wondering about it. Designing by hand doesn't seem like the way to go; is there any work that tried to explore the rule space using other methods? Like, combining different rules in different blocks, or maybe use machine learning to tweak the parameters of a set of rules while comparing with a traditional simulation.

A couple of disclaimers: I noticed there are different types of cellular automata, and they can be vastly different from each other. For sake of generality, consider this question without restricting to any particular type of CA. I also understand that many CA rules are Turing complete, meaning they you could model anything that can run on a Turing computer, in principle. So, for the sake of this question, consider that we want the dynamics to show up as directly as possible.

Answer 1

You should take a look at the book (or, really, "tome") called "A new kind of science" by Steven Wolfram. It is about many questions like the one you state.

But, more concretely, think of starting with, say, the Navier-Stokes equations and then discretizing it in space and time on a uniform mesh. So $U(i,j,k)$ represents, for example, the state of the fluid at location $(i,j)$ on the grid and at the $k$th time step. If you had used an explicit time stepping scheme, then this state clearly depends on the state of the fluid in the previous time step. If, furthermore, you use a simple nearest neighbor finite difference stencil for the spatial derivatives, then $U(i,j,k)$ also only depends on the states $U(i\pm 1, j\pm 1, k-1)$.

So, voila, you've just got yourself that looks a lot like a cellular automaton!

Answer 2

"A New Kind of Science" (NKS) has been brought up, in this case for a very legitimate reason; this book provides a lot of information on the actual rules generation & analysis, universal computation, etc. However, I feel the need to add a link to the famous Review of "A New Kind of Science" by C. Shalizi.

However, I bring it up not for the reason of yet another debate on the NKS (only partially to add a perspective), but for the links to other (actually Wolfram's early papers are very valuable as well) people's works that are mentioned in it and should be valuable, most importantly:

• D. Rothman and S. Zaleski, Lattice Gas Cellular Automata, Cambridge Univ. Press, 1997