Convex Nonparametric Formulation for Identification of Gradient Flows

We develop a nonparametric identification method for 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 is a structural prior knowledge and 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 the form of a quadratic program. This provides scalability and the opportunity for utilizing recently developed large-scale optimization solvers. The central idea is representing the energy function as a difference of two convex functions and learning 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 and evaluate the method numerically with two demonstrative examples.