Using machine learning methods to approximate real discriminant loci

In this talk, I will talk about a novel view on the problem of finding discriminant loci of the parametric system of polynomial equations as a machine learning problem, i.e., determining classification boundaries over the parameter space, with the classes being the number of real solutions of the system. The number of real solutions at various parameter points are obtained using the numerical polynomial homotopy continuation method. Machine learning techniques, such as deep learning and nearest neighbor, are employed to efficiently approximate the boundaries which in turn provides a decomposition of the parameter space in terms of the number of real solutions. It is shown that the proposed method can well-approximate complicated discriminant loci of various examples such as the Kuramoto model.