Performance Evaluation of Single-step Explicit Exponential Integration Methods on Stiff Ordinary Differential Equations

Abstract

Stiff systems of ordinary differential equations (ODEs) arise in a wide range of scientific and engineering disciplines and are traditionally solved using implicit integration methods due to their stability and efficiency. However, these methods are computationally expensive, particularly for applications requiring repeated integration, such as parameter estimation, Bayesian inference, neural ODEs, physics-informed neural networks, and MeshGraphNets. Explicit exponential integration methods have been proposed as a potential alternative, leveraging the matrix exponential to address stiffness without requiring nonlinear solvers. This study evaluates several state-of-the-art explicit single-step exponential schemes against classical implicit methods on benchmark stiff ODE problems, analyzing their accuracy, stability, and scalability with step size. Despite their initial appeal, our results reveal that explicit exponential methods significantly lag behind implicit schemes in accuracy and scalability for stiff ODEs. The backward Euler method consistently outperformed higher-order exponential methods in accuracy at small step sizes, with none surpassing the accuracy of the first-order integrating factor Euler method. Exponential methods fail to improve upon first-order accuracy, revealing the integrating factor Euler method as the only reliable choice for repeated, inexpensive integration in applications such as neural ODEs and parameter estimation. This study exposes the limitations of explicit exponential methods and calls for the development of improved algorithms.