Convex Nonparametric Formulation for Identification of Gradient Flows
Mohammad Khosravi and Roy S. Smith Corresponding authorThis research project is part of the Swiss Competence Center for Energy Research SCCER FEEB、D of the Swiss Innovation Agency Innosuisse.The authors are with Automatic Control Lab, ETH Zurich, Switzerland
Abstract
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.
中文速览
梯度流动力系统（gradient-flow dynamics）广泛存在于物理、化学和生物等领域，但从观测数据中辨识其向量场时，如何将"向量场必须是某个势能函数的梯度"这一结构性先验知识嵌入模型，一直是个难题。本文提出了一种非参数系统辨识方法，核心思路是把未知势能函数表示为两个凸函数之差（DC分解），再利用凸函数的充要条件将原本定义在无穷维函数空间上的估计问题等价转化为有限维的二次规划（凸优化），从而可以直接调用成熟的大规模优化求解器。理论上，作者严格证明了该方法解的存在性、唯一性和光滑性，数值实验也验证了方法的有效性。这项工作的意义在于：它把"梯度流"这一物理约束以凸优化的形式精确编码进辨识过程，既保证了结构正确性，又具备良好的计算可扩展性，为保结构的非线性动力系统辨识提供了一套严谨且实用的理论框架。
原文 arXiv:2003.12336;中英对照 + 大白话阅读 https://aha.fim.ai/paper/2003.12336v1