Noether's learning dynamics: physicist brings Lagrangian mechanics to neural network theory

Hidenori8Tanaka · x · 2026-09-10

Researcher Hidenori Tanaka (with Daniel Kunin) shared his line of work studying machine learning through the lens of physics, motivated by one question: will macroscopic control variables emerge — variables we can use to monitor and steer swarms of agents?

Key work: Noether's Learning Dynamics (arXiv:2105.02716) models gradient descent with a continuous-time Lagrangian formulation, where the learning rule is kinetic energy and the loss is potential energy. Introducing "kinetic symmetry breaking" (KSB), they generalize Noether's theorem to neural networks and show that normalization layers induce an "implicit adaptive optimization" mechanism analogous to RMSProp.

Also cited: a Landau-textbook-style Lagrangian formulation of learning dynamics, and a Landau free-energy formulation of loss landscapes in self-supervised learning — aiming to bring "the joy of physics back into AI research."

Related event: Physicists apply Landau-style theory to understand agent collectives(2 posts)→

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