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.

翻译：刚性常微分方程（ODEs）在许多科学与工程领域中普遍存在，但标准的神经ODE方法难以准确学习此类刚性系统，这构成了神经ODE广泛采用的一个主要障碍。在我们先前的工作中，我们通过采用单步隐式方法求解刚性神经ODE来应对这一挑战。尽管有效，这些隐式方法计算成本高昂且实现可能较为复杂。本文在我们先前工作的基础上，探索了显式指数积分方法作为一种更高效的替代方案。我们评估了这些显式方法处理神经ODE中刚性动力学的潜力，旨在增强其对更广泛科学与工程问题的适用性。我们发现积分因子欧拉（IF Euler）方法在稳定性和效率方面表现优异。当隐式格式无法训练刚性的Van der Pol振子时，IF Euler方法即使采用大步长也能成功训练。然而，IF Euler的一阶精度限制了其应用，因此针对刚性神经ODE开发更高阶方法仍是一个开放的研究问题。