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Gauss's Law: Problem-Solving

Gauss's Law: Problem-Solving

Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area...

Poisson's And Laplace's Equation

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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.

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...

Gauss's Law

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this...

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Consider a polar dielectric placed in an external field. In such a dielectric, opposite charges on adjacent dipoles neutralize each other, such that the net charge within the dielectric is zero. When a polar dielectric is inserted in between the capacitor plates, an electric field is generated due to the presence of net charges near the edge of the dielectric and the metal plates interface. Since the external electrical field merely aligns the dipoles, the dielectric as a whole is neutral. An...

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Deep Neural Networks for Image-Based Dietary Assessment

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Gauss Newton Method for Solving Variational Problems of PDEs with Neural Network Discretizaitons.

Wenrui Hao¹, Qingguo Hong², Xianlin Jin³

¹Department of Mathematics, Pennsylvania State University, State College, USA.

Journal of Scientific Computing

|December 27, 2024

Summary

This study introduces a Gauss-Newton method for solving differential equations numerically using machine learning. The proposed method demonstrates efficient superlinear convergence for neural network-based solutions.

Keywords:

Convergence analysis Gauss-Newton method Neural network discretization Partial differential equations Variational form

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Area of Science:

Computational mathematics
Machine learning
Numerical analysis

Background:

Machine learning approaches, particularly neural network-based discretization, are increasingly used for solving differential equations.
Existing methods like the deep Ritz method and physics-informed neural networks employ various training algorithms.

Purpose of the Study:

To propose and analyze a Gauss-Newton method for the numerical solution of differential equations based on their variational formulation.
To investigate the superlinear convergence properties of the proposed method.

Main Methods:

Focuses on the variational formulation of differential equations.
Proposes a Gauss-Newton method for numerical computation.
Analyzes superlinear convergence and discusses semi-regular zeros of the vanishing gradient.

Main Results:

The proposed Gauss-Newton method offers an efficient approach for numerical solutions.
Superlinear convergence properties of the method are theoretically analyzed.
Numerical examples validate the method's efficiency.

Conclusions:

The Gauss-Newton method provides an effective and efficient tool for solving differential equations via machine learning.
The theoretical analysis supports the practical performance demonstrated by numerical examples.