A Link between Shock-wave Theory and Symmetry-reduced Stochastic Gradient Descent for Artificial Neural Networks
This paper establishes a mathematically explicit link between shock-wave theory and the symmetry-quotiented learning dynamics of stochastic gradient descent, using differential geometry, Lie group theory, and fluid mechanics. After quotienting parameter symmetries and applying local-entropy coarse-graining, the effective dynamics satisfy a viscous Hamilton-Jacobi equation on the quotient manifold. Under the assumption that raw parameter dynamics can be summarized by a gradient field on the quotiented space, the gradient of the coarse-grained loss function obeys a Burgers-type equation, and shock formation can be rigorously established. The theory is applied to multilayer perceptrons, CNNs, Transformers, and mean-field networks, which all obey the Hamilton-Jacobi or Burgers-type equations. The author conjectures this framework yields practical diagnostics for deep learning, as raw parameter norms are often distorted by symmetry redundancy, while symmetry-corrected quotient observables provide a principled basis for monitoring, forecasting, and controlling training-phase transitions.
[2606.18303] A Link between Shock-wave Theory and Symmetry-reduced Stochastic Gradient Descent for Artificial Neural Networks
[Submitted on 16 Jun 2026]
Title:A Link between Shock-wave Theory and Symmetry-reduced Stochastic Gradient Descent for Artificial Neural Networks
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Abstract:We develop a mathematically explicit link between shock-wave theory and the symmetry-quotiented learning dynamics of stochastic gradient descent, drawing on differential geometry, Lie group theory, and fluid mechanics. Specifically, after quotienting parameter symmetries and applying local-entropy coarse-graining, the effective dynamics satisfy a viscous Hamilton--Jacobi equation on the quotient manifold. Moreover, under the assumption that the raw parameter dynamics can be summarized by a gradient field on the quotiented space, the gradient of the coarse-grained loss function obeys a Burgers-type equation, and shock formation can be established rigorously. We apply our theory to multilayer perceptrons, convolutional neural networks, Transformers, and mean-field networks, and show that they obey the Hamilton--Jacobi or Burgers-type equations. We conjecture that this framework also yields practical diagnostics for deep learning. In architectures such as Transformers, raw parameter norms are often distorted by symmetry redundancy and may therefore be misleading, whereas symmetry-corrected quotient observables provide a principled basis for monitoring, forecasting, and controlling training-phase transitions.
Comments: Accepted to the 35th International Conference on Artificial Neural Networks (ICANN) 2026
Subjects:
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
Cite as: arXiv:2606.18303 [cs.LG]
(or arXiv:2606.18303v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.18303
arXiv-issued DOI via DataCite
Submission history
From: Taiki Miyagawa [view email] [v1] Tue, 16 Jun 2026 06:20:52 UTC (300 KB)
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