  We study the approximation of functions which are invariant with respect to
certain permutations of the input indices using flow maps of dynamical systems.
Such invariant functions includes the much studied translation-invariant ones
involving image tasks, but also encompasses many permutation-invariant
functions that finds emerging applications in science and engineering. We prove
sufficient conditions for universal approximation of these functions by a
controlled equivariant dynamical system, which can be viewed as a general
abstraction of deep residual networks with symmetry constraints. These results
not only imply the universal approximation for a variety of commonly employed
neural network architectures for symmetric function approximation, but also
guide the design of architectures with approximation guarantees for
applications involving new symmetry requirements.
