PolarNet: Single-Minima Neural Network for Modeling Lyapunov Functions

TL;DR

PolarNet introduces a neural network architecture with provable guarantees of a single critical point, enhancing the training stability and reliability of neural Lyapunov control methods.

Contribution

The paper proposes PolarNet, a novel neural network architecture that guarantees a single critical point, addressing stability issues in neural Lyapunov control training.

Findings

PolarNet consistently maintains a single critical point in experiments.

Using PolarNet as a drop-in replacement improves training success in neural Lyapunov control.

PolarNet provides theoretical guarantees of properness and universality for Lyapunov functions.

Abstract

Learning control strategies with provable stability guarantees continues to be a challenging problem. In this work, we examine a family of training-time behaviors exhibited by existing neural Lyapunov control methods under specific conditions, which can hinder the synthesis of a provably stable controller. We identify the root cause as the lack of neural network architectural guarantees on the learned Lyapunov function, and propose PolarNet, a network architecture that provably addresses these issues by structurally guarantee to have a single critical point. We provide theoretical guarantee regarding the properness and universality of PolarNet for modeling Lyapunov functions, and show that using it as a drop-in replacement in existing neural Lyapunov control methods can effectively circumvent particular difficulties in training. We conduct a set of numerical experiments to verify that PolarNet consistently maintains a single critical point and, when used as a drop-in replacement in existing neural Lyapunov control methods, successfully avoids training failures caused by the lack of architectural guarantees. The code of this paper is available at https://github.com/23-zy/PolarNet.