This was one of the ideas which eventually led to the resampling-based approach to natural abstraction. You can see the relevant generalization of Noether’s Theorem in this Baez & Fong paper—I read the paper about a month or two before the resampling approach clicked, and it was one of the things on my mind at the time. Basically, for a Markov process, the conserved quantities correspond to eigenvectors with eigenvalue 1, which we can probe by looking for operators which commute with the transition matrix (the commutation is the “symmetry”). The main tricky part, at the time, was to figure out the right kind of “dynamics” such that the conserved quantities would be the natural abstractions; it wasn’t obvious ahead of time that “just use a typical MCMC sampler” was the right answer.
Have you looked into “conditionally conserved” quantities/symmetries here? Most macroscopic properties fall into this category—e.g. the color of a particular material is conserved so long as it doesn’t change phase or transmute (i.e. it stays within a particular energy range). This is associated with a (spontaneously-broken) symmetry, since the absorption spectrum of a material can be uniquely determined from its space group. I’d be willing to bet that the only information accessible at a distance (up to a change of variables) are these conditionally conserved quantities, but I’ve had a hard time rigorously proving it. (I believe it requires KAM theory).
Also, have you looked into Koopman spectral theory? It’s for deterministic systems, but seems quite relevant—the Koopman operator maps observables at one time to observables at another, telling you how they change over time. You can relate its spectrum to underlying geometrical properties, and it’s a linear operator even if your dynamics are nonlinear.
This was one of the ideas which eventually led to the resampling-based approach to natural abstraction. You can see the relevant generalization of Noether’s Theorem in this Baez & Fong paper—I read the paper about a month or two before the resampling approach clicked, and it was one of the things on my mind at the time. Basically, for a Markov process, the conserved quantities correspond to eigenvectors with eigenvalue 1, which we can probe by looking for operators which commute with the transition matrix (the commutation is the “symmetry”). The main tricky part, at the time, was to figure out the right kind of “dynamics” such that the conserved quantities would be the natural abstractions; it wasn’t obvious ahead of time that “just use a typical MCMC sampler” was the right answer.
Have you looked into “conditionally conserved” quantities/symmetries here? Most macroscopic properties fall into this category—e.g. the color of a particular material is conserved so long as it doesn’t change phase or transmute (i.e. it stays within a particular energy range). This is associated with a (spontaneously-broken) symmetry, since the absorption spectrum of a material can be uniquely determined from its space group. I’d be willing to bet that the only information accessible at a distance (up to a change of variables) are these conditionally conserved quantities, but I’ve had a hard time rigorously proving it. (I believe it requires KAM theory).
Also, have you looked into Koopman spectral theory? It’s for deterministic systems, but seems quite relevant—the Koopman operator maps observables at one time to observables at another, telling you how they change over time. You can relate its spectrum to underlying geometrical properties, and it’s a linear operator even if your dynamics are nonlinear.