Experimental study of Neural ODE training with adaptive solver for dynamical systems modeling

TL;DR

This paper investigates the challenges of using adaptive ODE solvers in Neural ODE training for dynamical systems, demonstrating limitations of naive approaches and proposing a more integrated solution, with experiments on the Lorenz'63 system.

Contribution

It reveals the limitations of applying adaptive solvers naively in Neural ODE training and proposes a method for tighter solver-training integration.

Findings

Naive adaptive solver application fails on Lorenz'63 system.

A tighter solver-training interaction improves results.

Code is available for reproducibility.

Abstract

Neural Ordinary Differential Equations (ODEs) was recently introduced as a new family of neural network models, which relies on black-box ODE solvers for inference and training. Some ODE solvers called adaptive can adapt their evaluation strategy depending on the complexity of the problem at hand, opening great perspectives in machine learning. However, this paper describes a simple set of experiments to show why adaptive solvers cannot be seamlessly leveraged as a black-box for dynamical systems modelling. By taking the Lorenz'63 system as a showcase, we show that a naive application of the Fehlberg's method does not yield the expected results. Moreover, a simple workaround is proposed that assumes a tighter interaction between the solver and the training strategy. The code is available on github: https://github.com/Allauzen/adaptive-step-size-neural-ode