识别梯度流的凸非参数公式
Convex Nonparametric Formulation for Identification of Gradient Flows
摘要：
本文针对非线性梯度流动力学，提出了一种非参数系统辨识方法。在这些系统中，向量场是势能函数的梯度场。这个关于系统动态的基本事实在所提出的识别方法中起着结构先验知识和约束的作用。虽然识别问题的性质是在函数空间中的估计，我们推导出一个等价的有限维公式，这是一个二次规划形式的凸优化。这使得问题的可扩展性，并提供了利用最近开发的大规模优化求解器的机会。该方法的核心思想是将能量函数表示为两个凸函数的差，并联合估计这些凸函数。基于函数凸性的充要条件，给出了识别问题的形式，并讨论了其解的存在性、唯一性和光滑性。我们还说明了数值方法的一个示范性的例子。
英文摘要：
In this paper, we develop a nonparametric system identification method for the nonlinear gradient-flow dynamics. In these systems, the vector field is the gradient field of a potential energy function. This fundamental fact about the dynamics of system plays the role of a structural prior knowledge as well as a constraint in the proposed identification method. While the nature of the identification problem is an estimation in the space of functions, we derive an equivalent finite dimensional formulation, which is a convex optimization in form of a quadratic program. This gives scalability of the problem and provides the opportunity for utilizing recently developed large-scale optimization solvers. The central idea in the proposed method is representing the energy function as a difference of two convex functions and estimating these convex functions jointly. Based on necessary and sufficient conditions for function convexity, the identification problem is formulated, and then, the existence, uniqueness and smoothness of the solution is addressed. We also illustrate the method numerically for a demonstrative example.
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