Generalization Bound for a General Class of Neural Ordinary Differential Equations

TL;DR

This paper derives the first generalization bounds for neural ODEs with broad nonlinear dynamics, providing insights into their performance on unseen data and the effects of overparameterization.

Contribution

It introduces a novel theoretical analysis establishing generalization bounds for a wide class of nonlinear neural ODEs, extending beyond linear cases.

Findings

Generalization bounds depend on Lipschitz continuity and overparameterization.

Solutions to neural ODEs with nonlinear dynamics have bounded variations.

Domain constraints influence the generalization performance.

Abstract

Neural ordinary differential equations (neural ODEs) are a popular type of deep learning model that operate with continuous-depth architectures. To assess how well such models perform on unseen data, it is crucial to understand their generalization error bounds. Previous research primarily focused on the linear case for the dynamics function in neural ODEs - Marion, P. (2023), or provided bounds for Neural Controlled ODEs that depend on the sampling interval Bleistein et al. (2023). In this work, we analyze a broader class of neural ODEs where the dynamics function is a general nonlinear function, either time dependent or time independent, and is Lipschitz continuous with respect to the state variables. We showed that under this Lipschitz condition, the solutions to neural ODEs have solutions with bounded variations. Based on this observation, we establish generalization bounds for both time-dependent and time-independent cases and investigate how overparameterization and domain constraints influence these bounds. To our knowledge, this is the first derivation of generalization bounds for neural ODEs with general nonlinear dynamics.