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神经网络中退化的随机微分方程的稳定和高效学习
Zhiding Wang1, Ming Yi1, Ting Gao2
1School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China.
Chaos (Woodbury, N.Y.)
|March 3, 2026
概括
我们开发了三种新的算法,用于学习具有困难,退化的噪音的随机微分方程 (SDEs). 这些方法提高了复杂动态系统建模的稳定性和准确性,优于传统方法.
科学领域:
- 动态系统和控制理论.
- 计算数学 计算数学 计算数学
- 机器学习 机器学习
背景情况:
- 随机微分方程 (SDEs) 对于随机噪声系统的建模至关重要.
- 从数据中学习SDEs是很困难的,特别是与退化的扩散矩阵 (不可逆转).
- 在这些场景中,现有的方法面临数值不稳定性和高计算成本.
研究的目的:
- 开发新的,高效和稳定的算法来学习退化扩散的SDEs.
- 克服传统方法在处理不可逆转的噪声条件方面的局限性.
- 从经验数据中增强复杂动态系统建模的可靠性.
主要方法:
- 算法I:将二级SDE解决器技术集成到2D退化SDEs的学习框架中.
- 算法II:引入了一个针对退化扩散矩阵量身定制的混合损失函数 (MSE和MLE).
- 算法III:在神经网络训练期间采用稳定辅助噪声机制,以防止梯度不稳定.
主要成果:
- 与传统的最大概率估计相比,提出的算法显示出更高的训练稳定性.
- 在新方法中观察到更快的收率.
- 在学习退化的SDEs方面,获得了更高的准确性,超过了现有的技术.
- 数字实验证实了开发的算法的有效性和稳定性.
结论:
- 这些新的算法为学习退化扩散的SDEs提供了高效和稳定的解决方案.
- 这些进步使得高维度随机系统的建模更加可靠.
- 这些方法为机器学习任务提供了有价值的工具,这些任务需要对噪声结构进行强大的适应.
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