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相关概念视频

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To calculate the flow rate for a trapezoidal channel, first, identify the bottom width, side slope, and flow depth of the channel. The cross-sectional area (A) corresponding to the depth of flow (y), channel bottom width (B), and side slope (θ) is determined by:Next, calculate the wetted perimeter, which includes the bottom width and the sloped side lengths in contact with the water. Using the values of the cross-sectional area and the wetted perimeter, determine the hydraulic radius by...
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The important convolution properties include width, area, differentiation, and integration properties.
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Uniform depth channel flow keeps fluid depth consistent along channels such as irrigation canals. In natural channels, such as rivers, approximate uniform flow is often assumed. This condition occurs when the channel’s bottom slope matches the energy slope, balancing potential energy lost from gravity with head loss due to shear stress. This balance prevents depth changes along the channel length, resulting in a steady, uniform flow.Uniform flow in open channels with a constant cross-section...
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深度和宽度之间的相互作用用于神经ODEs中的插曲.

Antonio Álvarez-López1, Arselane Hadj Slimane2, Enrique Zuazua3

  • 1Universidad Autónoma de Madrid, Departamento de Matemáticas, C. Francisco Tomás y Valiente, 7, Madrid, 28049, Spain; Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Mathematics, Chair for Dynamics, Control, Machine Learning, and Numerics (Alexander von Humboldt Professorship), Cauerstraße, 11, Erlangen, 91058, Germany.

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概括

神经常规微分方程 (NODE) 为监督学习提供了基于控制的方法. 这项研究揭示了网络宽度和深度之间的权衡,用于数据插入和测量近似,对自主系统有影响.

关键词:
在深度深度的深度.神经的ODE是神经的ODE.同时可控性的同时可控性运输控制控制运输的控制.瓦斯斯坦的距离是瓦斯斯坦的距离宽度 宽度 宽度 宽度

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科学领域:

  • 机器学习 机器学习
  • 动态系统 动态系统
  • 控制理论 控制理论

背景情况:

  • 神经常规微分方程 (NODE) 越来越多地用于监督学习,与控制理论相似.
  • NODE的架构影响,特别是宽度 (p) 和深度 (L),对他们的学习能力需要进一步阐明.

研究的目的:

  • 在神经常规微分方程 (NODE) 中研究网络宽度 (p) 和深度 (L) 之间的关系.
  • 分析这些架构参数如何影响有限数据集和概率测量的插值.
  • 探索NODE的自主模式 (L=0) 中的数据插值.

主要方法:

  • 构建明确的控制来插入有限的数据集 (D) 和概率测量.
  • 对宽度 (p) 和数据集大小 (N) 或错误率 (ɛ) 的深度 (L) 的缩放进行分析.
  • 在使用概率控制和放松条件的自主模式 (L=0) 中开发数据插值策略.

主要成果:

  • 宽度 (p) 和深度 (L) 之间存在一个权衡:L尺度为1+O(N/p) 用于数据插值和1+Op−1+(1+p)−1ɛ−d用于测量插值.
  • 在高维,宽设置 (d,p>N) 中,深度L=0是可以实现的.
  • 对于L=0,确定了概率控制策略 (p=N时) 和显式错误衰减率 (通过通用近似定理).

结论:

  • 网络架构 (宽度和深度) 极大地影响了 NODE 在监督学习任务中的性能.
  • 这些发现为设计高效的NODE架构提供了洞察力,以应对特定的插值挑战.
  • 该研究促进了对NODE能力的理解,特别是在自主模式下,为新型应用铺平了道路.