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.

中文翻译：

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机器学习真正的判别轨迹

多项式方程的参数化系统出现在科学和工程中的许多应用中，其真实解描述了例如动态系统的平衡、满足设计约束的链接以及计算机视觉中的场景重建。由于不同的参数值可以有不同数量的实解，参数空间被分解为边界形成实判别轨迹的区域。本文将定位真实判别轨迹视为机器学习中的一个监督分类问题，其目标是确定参数空间上的分类边界，类别是真实解的数量。对于多维参数空间，本文提出了一种新颖的采样方法，该方法对参数空间进行仔细采样​​。在每个样本点，同伦延拓用于获得相应多项式系统的实解数。包括最近邻和深度学习在内的机器学习技术用于有效地逼近真实判别轨迹。学习真实判别轨迹的一个应用是开发一种真实同伦方法，该方法仅跟踪真实解路径，这与跟踪所有~复杂~解~路径的传统方法不同。示例表明，所提出的方法可以有效地逼近复杂的解边界，例如由 Kuramoto 模型的平衡引起的边界。学习真实判别轨迹的一个应用是开发一种真实同伦方法，该方法仅跟踪真实解路径，这与跟踪所有~复杂~解~路径的传统方法不同。示例表明，所提出的方法可以有效地逼近复杂的解边界，例如由 Kuramoto 模型的平衡引起的边界。学习真实判别轨迹的一个应用是开发一种真实同伦方法，该方法仅跟踪真实解路径，这与跟踪所有~复杂~解~路径的传统方法不同。示例表明，所提出的方法可以有效地逼近复杂的解边界，例如由 Kuramoto 模型的平衡引起的边界。