  We introduce polynomial couplings, a generalization of probabilistic
couplings, to develop an algorithm for the computation of equivalence relations
which can be interpreted as a lifting of probabilistic bisimulation to
polynomial differential equations, a ubiquitous model of dynamical systems
across science and engineering. The algorithm enjoys polynomial time complexity
and complements classical partition-refinement approaches because: (a) it
implements a local exploration of the system, possibly yielding equivalences
that do not necessarily involve the inspection of the whole system of
differential equations; (b) it can be enhanced by up-to techniques; and (c) it
allows the specification of pairs which ought not to be included in the output.
Using a prototype, these advantages are demonstrated on case studies from
systems biology for applications to model reduction and comparison. Notably, we
report four orders of magnitude smaller runtimes than partition-refinement
approaches when disproving equivalences between Markov chains.
