This paper addresses the inverse problem of evolutionary and metaheuristic optimization: given the outcome of a process, what can be recovered about the fitness landscape, state space, and selection pressures that produced it? We first establish a negative result: observing a single optimized point x* is hopelessly underdetermined - infinitely many environments share the same optimum, and the inverse class E^-1 (x*) is too large to be informative. The central contribution is a positive identification theorem for the distributional regime. When the observation is a stationary population P* arising from a process with a Gibbsian invariant measure (e. g. , Metropolis–Hastings, fixed-temperature simulated annealing, or overdamped Langevin dynamics), the fitness function F is identifiable up to a positive affine reparametrization F -> a F + c. The residual ambiguity is exactly the symmetry group Sym (P*) of the observed distribution. Furthermore, the local covariance of P* estimates the Hessian of F, so flat directions in the population directly reveal the continuous symmetries of the underlying environment - without ever observing F directly. This framework unifies inverse reinforcement learning, Boltzmann inversion in statistical mechanics, and neural-network interpretability under a common mathematical lens. Standard metaheuristics such as genetic algorithms, particle swarm optimization, and ant colony optimization are discussed as motivating examples, with explicit acknowledgment that they fall outside the Gibbsian class and hence remain genuinely underdetermined. The results are formalized categorically via a forward functor E and its inverse fibres, which are shown to be parametrized by the affine scale and Sym (P*).
Aleksandar Bakalov (Fri,) studied this question.