Toroidal Approximate Identity Neural Networks Are Universal Approximators

The approximation of a continuous function on the torus T-2 is an important problem in approximation theory of artificial neural networks. In this work, we investigate the universal approximation capability of one-hidden layer feedforward toroidal approximate identity neural networks. To this end, we present notions of toroidal convolution and toroidal approximate identity. Using these notions, we apply a convolution linear operator approach to prove uniform converges in terms of continuous functions on the torus T-2. Using this result, we also prove a main theorem. The main theorem shows that one-hidden layer feedforward toroidal approximate identity neural networks are universal approximators in the space of continuous functions on the torus T-2.