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

arXiv: https://arxiv.org/abs/2504.09761 · HTML: https://mlsanity.com/paper/2504.09761/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 34 references that are in the map.
- First posted on arXiv 13 Apr 2025.

## Builds on (most cited first, 10 of 10)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Noether's Learning Dynamics: Role of Symmetry Breaking in Neural Networks](https://mlsanity.com/paper/2105.02716/) (2021)
- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)
- [Non-normal Recurrent Neural Network (nnRNN): learning long time dependencies while improving expressivity with transient dynamics](https://mlsanity.com/paper/1905.12080/) (2019)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Nature's Cost Function: Simulating Physics by Minimizing the Action](https://mlsanity.com/paper/2303.02115/) (2023)

## Built on by (most cited first, 2 of 2)

- [Gradient Descent as Loss Landscape Navigation: a Normative Framework for Deriving Learning Rules](https://mlsanity.com/paper/2510.26997/) (2025)
- [A Variational Manifold Embedding Framework for Nonlinear Dimensionality Reduction](https://mlsanity.com/paper/2511.22128/) (2025)

## Abstract (arXiv)

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.