Dynamical symmetries in the fluctuation-driven regime: an application of Noether's theorem to noisy dynamical systems

Department of Neurobiology, Harvard Medical School, Boston, MA, USA Editors: Simone Azeglio, Christian Shewmake, Bahareh Tolooshams, Sophia Sanborn, Chase van de Geijin, Nina Miolane Abstract 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. Keywords: symmetry, invariance, Noether's theorem, stochastic processes, diffusion 1. Introduction In physics, Noether's theorem provides a fundamental link between the symmetries of physical systems on the one hand, and conserved quantities like energy and momentum on the other hand (Noether, 1918; Kosmann-Schwarzbach et al., 2011; Neuenschwander, 2017). In its modern form, it uniquely associates (equivalence classes of) independent continuous symmetries, which can be formalized in terms of Lie groups and algebras, with (equivalence classes of) independent conserved quantities (Martinez Alonso, 1979; Olver, 1986, 1993; Brown, 2020).