Machine learning the real discriminant locus
This work addresses the challenge of analyzing real solutions in parameterized systems for applications in science and engineering, representing an incremental improvement through a hybrid approach.
The authors tackled the problem of locating the real discriminant locus in parameterized polynomial systems by framing it as a supervised classification task, using a novel sampling method combined with homotopy continuation and machine learning techniques to efficiently approximate boundaries, with examples demonstrating effectiveness on complex models like the Kuramoto model.
Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose boundary forms the real discriminant locus. This article views locating the real discriminant locus as a supervised classification problem in machine learning where the goal is to determine classification boundaries over the parameter space, with the classes being the number of real solutions. For multidimensional parameter spaces, this article presents a novel sampling method which carefully samples the parameter space. At each sample point, homotopy continuation is used to obtain the number of real solutions to the corresponding polynomial system. Machine learning techniques including nearest neighbor and deep learning are used to efficiently approximate the real discriminant locus. One application of having learned the real discriminant locus is to develop a real homotopy method that only tracks the real solution paths unlike traditional methods which track all~complex~solution~paths. Examples show that the proposed approach can efficiently approximate complicated solution boundaries such as those arising from the equilibria of the Kuramoto model.