PAC bounds of continuous Linear Parameter-Varying systems related to neural ODEs

TL;DR

This paper derives PAC bounds for continuous-time neural ODEs modeled as LPV systems, providing stability-based guarantees that are independent of the integration interval, advancing theoretical understanding of neural ODEs.

Contribution

It introduces PAC bounds for LPV systems related to neural ODEs, with stability considerations and interval-independent guarantees, a novel theoretical contribution.

Findings

PAC bounds are established for neural ODEs within LPV systems.

Bounds are independent of the integration interval.

The approach leverages stability properties of LPV systems.

Abstract

We consider the problem of learning Neural Ordinary Differential Equations (neural ODEs) within the context of Linear Parameter-Varying (LPV) systems in continuous-time. LPV systems contain bilinear systems which are known to be universal approximators for non-linear systems. Moreover, a large class of neural ODEs can be embedded into LPV systems. As our main contribution we provide Probably Approximately Correct (PAC) bounds under stability for LPV systems related to neural ODEs. The resulting bounds have the advantage that they do not depend on the integration interval.