用显式指数方法提升神经微分方程对刚性系统的训练能力
Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methods
- 采用显式指数积分法替代传统隐式方法，提升计算效率
- IF Euler法在大步长下仍能成功训练刚性振子系统
- 适合需要高效求解刚性问题的科学计算与工程建模场景
刚性常微分方程在众多科学与工程领域中普遍存在，但标准神经微分方程方法难以准确学习此类系统，严重制约其广泛应用。此前工作通过使用单步隐式方法解决该问题，虽有效却计算成本高且实现复杂。本文探索显式指数积分方法作为更高效的替代方案。实验表明，积分因子欧拉法（IF Euler）在稳定性和效率上表现优异；当隐式方法无法训练刚性范德波尔振子时，IF Euler法仍能成功，即使使用较大步长。然而，其一阶精度限制了应用范围，因此开发更高阶的刚性神经微分方程求解方法仍是开放问题。
原文摘要 · Abstract (English)
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.
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