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相关概念视频

Linear Approximation in Frequency Domain01:26

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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Differential Form of Maxwell's Equations01:17

Differential Form of Maxwell's Equations

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James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and...
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The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
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For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
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Computational Modeling of Retinal Neurons for Visual Prosthesis Research - Fundamental Approaches
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基于物理的内核功能神经网络用于解决部分微分方程.

Zhuojia Fu1, Wenzhi Xu2, Shuainan Liu2

  • 1Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nanjing 210098, China; College of Mechanics and Materials, Hohai University, Nanjing 211100, China.

Neural networks : the official journal of the International Neural Network Society
|January 10, 2024
PubMed
概括
此摘要是机器生成的。

基于物理学的核心函数神经网络 (PIKFNNs) 提供了一种解决部分微分方程 (PDEs) 的新方法. 这种方法将物理信息直接嵌入到神经网络激活功能中,提高了准确性和可行性.

关键词:
激活功能 激活功能没有网格的无网格网.基于物理的内核功能.基于物理学的神经网络.辐射基础函数神经网络神经网络

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科学领域:

  • 计算数学 计算数学 计算数学
  • 机器学习 机器学习
  • 数字分析 数字分析

背景情况:

  • 部分微分方程 (PDEs) 是复杂物理现象的建模的基础.
  • 对于PDEs的传统数值方法可能是计算密集的.
  • 基于物理学的神经网络 (PINNs) 将物理定律集成到神经网络中,但依赖于损失函数约束.

研究的目的:

  • 引入基于物理的内核功能神经网络 (PIKFNNs),作为比标准PINN的进步.
  • 为了证明将PDE信息直接嵌入到激活函数中的有效性.
  • 为解决线性和特定非线性PDEs提供一种新的神经网络架构.

主要方法:

  • 开发了PIKFNNs,使用一个浅层神经网络和一个隐藏层.
  • 使用物理信息的内核函数 (PIKFs) 作为定制的激活函数.
  • PIKF包含PDE信息,例如基本解决方案或Green的函数.

主要成果:

  • PIKFNNs成功地解决了各种线性和特定的非线性PDEs.
  • 该方法将PDE信息嵌入到激活函数中,与PINNs的损失函数约束不同.
  • 通过基准示例验证了PIKFNNs的可行性和准确性.

结论:

  • PIKFNNs代表了一种新且有效的神经网络架构,用于解决PDEs.
  • 将物理信息嵌入到激活函数中,为传统的PINN方法提供了一个有希望的替代方案.
  • 拟议的PIKFNNs在一系列PDE问题上表现出高精度和可行性.