ResNet After All? Neural ODEs and Their Numerical Solution

TL;DR

This paper investigates the dependency of trained Neural ODE models on the numerical solver used during training, revealing limitations in the continuous-time interpretation and proposing an adaptive step size method to ensure valid ODE behavior.

Contribution

It demonstrates the solver dependency issue in Neural ODEs and introduces an adaptive step size algorithm to maintain valid ODE properties during training.

Findings

Training with coarse solvers leads to accuracy drops with different solvers.

A critical step size exists beyond which the model behaves as a true ODE.

The proposed adaptive method improves model validity without excessive computation.

Abstract

A key appeal of the recently proposed Neural Ordinary Differential Equation (ODE) framework is that it seems to provide a continuous-time extension of discrete residual neural networks. As we show herein, though, trained Neural ODE models actually depend on the specific numerical method used during training. If the trained model is supposed to be a flow generated from an ODE, it should be possible to choose another numerical solver with equal or smaller numerical error without loss of performance. We observe that if training relies on a solver with overly coarse discretization, then testing with another solver of equal or smaller numerical error results in a sharp drop in accuracy. In such cases, the combination of vector field and numerical method cannot be interpreted as a flow generated from an ODE, which arguably poses a fatal breakdown of the Neural ODE concept. We observe, however, that there exists a critical step size beyond which the training yields a valid ODE vector field. We propose a method that monitors the behavior of the ODE solver during training to adapt its step size, aiming to ensure a valid ODE without unnecessarily increasing computational cost. We verify this adaptation algorithm on a common bench mark dataset as well as a synthetic dataset.