标题： 使用显式指数积分方法训练刚性神经微分方程 显示英文标题

标题： Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methods

摘要： 刚性常微分方程（ODE）在许多科学和工程领域中很常见，但标准的神经ODE方法难以准确学习这些刚性系统，这成为神经ODE广泛采用的重大障碍。 在我们之前的工作中，我们通过利用单步隐式方法来解决刚性神经ODE的问题。 虽然有效，但这些隐式方法计算成本高且实现复杂。 本文扩展了我们之前的工作，探讨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，旨在提高它们在更广泛的科学和工程问题中的适用性。 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出色。 尽管隐式方案无法训练刚性范德波尔振子，但IF Euler方法即使在大步长下也能成功。 然而，IF Euler的一阶精度限制了其应用，开发适用于刚性神经ODE的高阶方法仍然是一个开放的研究问题。 显示英文摘要

摘要： 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.