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arXiv cs.LG
arXiv cs.LG
6/18/2026
A Link between Shock-wave Theory and Symmetry-reduced Stochastic Gradient Descent for Artificial Neural Networks

A Link between Shock-wave Theory and Symmetry-reduced Stochastic Gradient Descent for Artificial Neural Networks

Short summary

Researchers establish a mathematical connection between shock-wave theory and neural network training dynamics through differential geometry and Lie group theory, proving that symmetry-quotiented dynamics satisfy Hamilton-Jacobi and Burgers-type equations. The framework applies across multiple architectures—MLPs, CNNs, and Transformers—and potentially enables better diagnostics for monitoring and controlling training phase transitions. Symmetry-corrected observables could provide more reliable signals than raw parameter norms, especially in Transformers.

  • •Shock-wave theory linked to neural network training via symmetry-quotiented dynamics
  • •Proves dynamics follow Hamilton-Jacobi and Burgers-type equations across architectures
  • •Offers new diagnostics for monitoring training using symmetry-corrected observables

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