通过正则化雅可比矩阵，提升神经微分方程长期积分稳定性。
Jacobian Regularization Stabilizes Long-Term Integration of Neural Differential Equations
- 用方向导数正则化NDE的雅可比矩阵，避免长期积分发散。
- 短训练轨迹下仍能实现稳定长时模拟，误差显著降低。
- 计算成本远低于长轨迹训练，适合大规模系统建模。
混合模型与神经微分方程（NDE）在物理系统建模中日益重要，但在长期积分中常面临稳定性与精度问题。传统基于展开轨迹的训练虽能缓解发散，但因需对迭代过程求梯度，计算开销过大。本文提出通过方向导数正则化NDE的雅可比矩阵来稳定长期积分，针对已知动力学设计精确导数方法，未知动力学则用有限差分近似。两种方法在训练成本远低于长轨迹训练的前提下，成功提升了多个常微分方程与偏微分方程的长期模拟稳定性，为大规模系统长期建模提供了新路径。
原文摘要 · Abstract (English)
Hybrid models and Neural Differential Equations (NDE) are getting increasingly important for the modeling of physical systems, however they often encounter stability and accuracy issues during long-term integration. Training on unrolled trajectories is known to limit these divergences but quickly becomes too expensive due to the need for computing gradients over an iterative process. In this paper, we demonstrate that regularizing the Jacobian of the NDE model via its directional derivatives during training stabilizes long-term integration in the challenging context of short training rollouts. We design two regularizations, one for the case of known dynamics where we can directly derive the directional derivatives of the dynamic and one for the case of unknown dynamics where they are approximated using finite differences. Both methods, while having a far lower cost compared to long rollouts during training, are successful in improving the stability of long-term simulations for several ordinary and partial differential equations, opening up the door to training NDE methods for long-term integration of large scale systems.
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