概括
神经常规微分方程 (神经ODE) 现在在无限时间内近似复杂,多稳定的动态. 这一突破解决了多个稳定状态或振荡行为的近似系统的局限性.
科学领域:
- 动态系统理论 动态系统理论
- 机器学习 机器学习
- 神经网络的神经网络的神经网络
背景情况:
- 全球近似定理证明了神经网络的表达力.
- 现有的动态系统定理仅限于有限的时间或全球稳定的平衡.
- 无限时间动态系统中的多稳定性和极限周期仍然没有得到当前理论的解决.
研究的目的:
- 将神经ODEs的普遍近似能力扩展到无限的时间地平线.
- 解决多态动态系统的近似问题,包括具有极限周期的系统.
- 将理论保障与神经网络动态的实践培训指标相结合.
主要方法:
- 在无限时间地平线 $[0,\infty) $ 上,证明了神经 ODE 的 $\varepsilon$-$δ$ 接近性.
- 他将方法应用于三个类:具有超标固定点的摩尔斯-斯梅尔系统,具有超标极限周期的摩尔斯-斯梅尔系统 (使用周期匹配),以及具有通常超标吸引器的系统 (通过离散).
- 建立了一个时间概括界限,将 $\varepsilon$-$δ$ 接近度与 $L^p$ 错误度联系起来.
主要成果:
- 神经ODEs在指定的类中实现无限地平线动态的 $\varepsilon$-$δ$ 接近.
- 证明了具有多稳定性和极限周期的系统的近似性.
- 导出一个界面显示 $\varepsilon$-$δ$ 接近意味着所有 $t \geq 0$ 的界面 $L^p$ 错误.
结论:
- 这项工作为使用神经ODEs的多稳定无限地平线动态提供了第一个通用近似框架.
- 这些发现扩大了对复杂系统神经网络能力的理论理解.
- 时间概括界限将理论近似质量与实际错误指标相连接.
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