VPNets: Volume-preserving neural networks for learning source-free dynamics

TL;DR

This paper introduces volume-preserving neural networks (VPNets) designed to learn unknown source-free dynamical systems from trajectory data, ensuring intrinsic volume preservation and theoretical expressivity guarantees.

Contribution

The paper presents novel volume-preserving neural network architectures, R-VPNet and LA-VPNet, with proven approximation theorems for source-free dynamics learning.

Findings

VPNets effectively learn source-free dynamics from data.

VPNets demonstrate strong generalization in experiments.

VPNets preserve volume structure in learned dynamics.

Abstract

We propose volume-preserving networks (VPNets) for learning unknown source-free dynamical systems using trajectory data. We propose three modules and combine them to obtain two network architectures, coined R-VPNet and LA-VPNet. The distinct feature of the proposed models is that they are intrinsic volume-preserving. In addition, the corresponding approximation theorems are proved, which theoretically guarantee the expressivity of the proposed VPNets to learn source-free dynamics. The effectiveness, generalization ability and structure-preserving property of the VP-Nets are demonstrated by numerical experiments.