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We study the approximation of functions which are invariant with respect tocertain permutations of the input indices using flow maps of dynamical systems.Such invariant functions includes the much studied translation-invariant onesinvolving image tasks, but also encompasses many permutation-invariantfunctions that finds emerging applications in science and engineering. We provesufficient conditions for universal approximation of these functions by acontrolled equivariant dynamical system, which can be viewed as a generalabstraction of deep residual networks with symmetry constraints. These resultsnot only imply the universal approximation for a variety of commonly employedneural network architectures for symmetric function approximation, but alsoguide the design of architectures with approximation guarantees forapplications involving new symmetry requirements.

Deep Neural Network Approximation of Invariant Functions through Dynamical Systems