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.

翻译：刚性常微分方程组广泛出现于科学与工程领域，传统上因其稳定性和高效性而采用隐式积分方法求解。然而，这些方法计算成本高昂，尤其对于需要重复积分的应用场景，如参数估计、贝叶斯推断、神经ODE、物理信息神经网络以及MeshGraphNets。显式指数积分方法作为一种潜在的替代方案被提出，其利用矩阵指数处理刚性而无需非线性求解器。本研究在基准刚性ODE问题上评估了多种先进的显式单步指数格式与经典隐式方法的性能，分析了它们在步长变化下的精度、稳定性和可扩展性。尽管显式指数方法初看具有吸引力，但我们的结果表明，对于刚性ODE问题，显式指数方法在精度和可扩展性上显著落后于隐式格式。在较小步长下，后向欧拉法在精度上始终优于高阶指数方法，且无一能超越一阶积分因子欧拉法的精度。指数方法未能改进一阶精度，这表明积分因子欧拉法是神经ODE和参数估计等应用中唯一可靠且计算成本较低的重复积分选择。本研究揭示了显式指数方法的局限性，并呼吁开发改进的算法。