  In certain situations, neural networks are trained upon data that obey
underlying symmetries. However, the predictions do not respect the symmetries
exactly unless embedded in the network structure. In this work, we introduce
architectures that embed a special kind of symmetry namely, invariance with
respect to involutory linear/affine transformations up to parity $p=\pm 1$. We
provide rigorous theorems to show that the proposed network ensures such an
invariance and present qualitative arguments for a special universal
approximation theorem. An adaption of our techniques to CNN tasks for datasets
with inherent horizontal/vertical reflection symmetry is demonstrated.
Extensive experiments indicate that the proposed model outperforms baseline
feed-forward and physics-informed neural networks while identically respecting
the underlying symmetry.
