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

• URL: http://arxiv.org/abs/2606.18303v1
• Date: Tue, 16 Jun 2026 06:20:52 GMT
• Title: A Link between Shock-wave Theory and Symmetry-reduced Stochastic Gradient Descent for Artificial Neural Networks
• Abstract summary: We develop a mathematically explicit link between shock-wave theory and the symmetry-quotiented learning dynamics of gradient descent.<n>We show that multilayer perceptrons, convolutional neural networks, Transformers, and mean-field networks obey the Hamilton--Jacobi or Burgers-type equations.
• Score: 10.076691842234002
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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