Discontinuous Neural Networks and Discontinuity Learning

In the framework of discontinuous function approximation and discontinuity interface detection, we consider an approach involving Neural Networks. In particular, we define a novel typology of Neural Network layers endowed with new learnable parameters and discontinuities in the space of the activations. These layers allow to create a completely new kind of Neural Networks, characterized to be discontinuous, not only able to approximate discontinuous functions but also to learn and detect the discontinuity interfaces. A sound theoretical analysis concerning the properties of the new discontinuous layers is performed, and some tests on discontinuous functions are proposed, in order to assess the potential of such instruments.