Dynamical symmetries in the fluctuation-driven regime: an application of Noether's theorem to noisy dynamical systems
This work provides a theoretical framework for analyzing symmetries in noisy systems relevant to neuroscience and AI, though it appears incremental as it extends existing principles to new contexts.
The paper tackled the problem of applying Noether's theorem to noisy dynamical systems, which lack traditional variational principles, by using a nonequilibrium physics principle to link continuous symmetries to conserved quantities in likely trajectories, resulting in the identification of analogues for energy, momentum, and angular momentum in models like decision-making and neural networks.
Noether's theorem provides a powerful link between continuous symmetries and conserved quantities for systems governed by some variational principle. Perhaps unfortunately, most dynamical systems of interest in neuroscience and artificial intelligence cannot be described by any such principle. On the other hand, nonequilibrium physics provides a variational principle that describes how fairly generic noisy dynamical systems are most likely to transition between two states; in this work, we exploit this principle to apply Noether's theorem, and hence learn about how the continuous symmetries of dynamical systems constrain their most likely trajectories. We identify analogues of the conservation of energy, momentum, and angular momentum, and briefly discuss examples of each in the context of models of decision-making, recurrent neural networks, and diffusion generative models.