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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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When a structural member undergoes plastic deformation due to bending, it is crucial to understand the position of the neutral axis and the stress distribution. This member, characterized by a single plane of symmetry, exhibits a uniform stress distribution, with negative stress above the neutral axis and positive stress below. Notably, the neutral axis does not align with the centroid of the cross-section. This misalignment is typical in cases where the cross-section is not rectangular or...
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Analyzing Mitochondrial Morphology Through Simulation Supervised Learning
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Machine Learning Hidden Symmetries.

Ziming Liu1, Max Tegmark1

  • 1Department of Physics, Institute for AI and Fundamental Interactions, and Center for Brains, Minds and Machines, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.

Physical Review Letters
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Summary
This summary is machine-generated.

This study introduces an automated method to discover hidden symmetries in physical systems by minimizing asymmetry using neural networks. It successfully identified symmetries in black hole metrics and other complex properties.

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Area of Science:

  • Physics
  • Computational Mathematics
  • Machine Learning

Background:

  • Identifying symmetries is crucial for simplifying complex physical systems and understanding their properties.
  • Traditional methods for symmetry detection are often limited to known coordinate systems.
  • Hidden symmetries, which appear in undiscovered coordinate systems, pose a significant challenge.

Purpose of the Study:

  • To develop an automated method for discovering hidden symmetries in physical systems.
  • To quantify asymmetry as the violation of partial differential equations.
  • To explore the application of invertible neural networks for discovering new coordinate systems.

Main Methods:

  • Quantifying asymmetry by measuring the violation of specific partial differential equations.
  • Employing invertible neural networks to represent and search over the space of all invertible transformations.
  • Numerically minimizing the quantified asymmetry to find coordinate transformations that reveal hidden symmetries.

Main Results:

  • The method successfully rediscovers the Gullstrand-Painlevé metric, revealing hidden translational symmetry in the Schwarzschild metric.
  • Demonstrates the ability to identify non-traditional symmetries such as Hamiltonicity and modularity.
  • Validates the effectiveness of the automated approach in uncovering complex symmetries.

Conclusions:

  • The automated method provides a powerful new tool for symmetry discovery in physics and mathematics.
  • This approach can uncover simplifying traits that are not conventionally recognized as symmetries.
  • The technique holds potential for applications in various scientific domains requiring the analysis of complex systems.