Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methods

TL;DR

This paper investigates explicit exponential integration methods, particularly the integrating factor Euler method, for training stiff neural ODEs, demonstrating improved stability and efficiency over implicit methods, especially for challenging stiff systems.

Contribution

It introduces explicit exponential integration methods as efficient alternatives for training stiff neural ODEs, highlighting the effectiveness of the IF Euler method in stability and training success.

Findings

IF Euler outperforms implicit methods in stability and efficiency

Implicit methods fail to train the stiff Van der Pol oscillator

IF Euler's first-order accuracy limits its applicability

Abstract

Stiff ordinary differential equations (ODEs) are common in many science and engineering fields, but standard neural ODE approaches struggle to accurately learn these stiff systems, posing a significant barrier to widespread adoption of neural ODEs. In our earlier work, we addressed this challenge by utilizing single-step implicit methods for solving stiff neural ODEs. While effective, these implicit methods are computationally costly and can be complex to implement. This paper expands on our earlier work by exploring explicit exponential integration methods as a more efficient alternative. We evaluate the potential of these explicit methods to handle stiff dynamics in neural ODEs, aiming to enhance their applicability to a broader range of scientific and engineering problems. We found the integrating factor Euler (IF Euler) method to excel in stability and efficiency. While implicit schemes failed to train the stiff Van der Pol oscillator, the IF Euler method succeeded, even with large step sizes. However, IF Euler's first-order accuracy limits its use, leaving the development of higher-order methods for stiff neural ODEs an open research problem.