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

2025-04-13

AI Summary

Classical Noether’s theorem—linking continuous symmetries to conservation laws—relies on a Lagrangian formulation and thus fails for noise-driven, far-from-equilibrium stochastic systems (e.g., decision-making models, recurrent neural networks, diffusion generative models) lacking a conventional action functional. Method: We develop a variational framework grounded in large deviation theory, introducing the path-space rate functional as a generalized action. This enables rigorous derivation of symmetry–conservation correspondences—yielding “quasi-conserved” quantities (e.g., quasi-energy, quasi-momentum, quasi-angular momentum)—without assuming equilibrium, reversibility, or detailed balance. Contribution/Results: Our framework constitutes the first extension of Noether’s theorem to non-Lagrangian stochastic dynamics. It characterizes symmetry constraints on most-probable trajectories in the fluctuation-dominated regime. Empirical validation across three AI paradigms confirms the existence of analogous conservation laws, establishing the first unified theoretical framework for non-equilibrium symmetries in neural and artificial intelligence systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Neuro-Symbolic Learning
• Philosophy and Ethics of AI: Artificial General Intelligence

Application Category

• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Extends Noether's theorem to noisy dynamical systems
• Identifies conserved quantities in neuroscience and AI models
• Analyzes symmetries in decision-making and neural networks

Innovation

• Applies Noether's theorem to noisy systems
• Uses nonequilibrium physics variational principle
• Identifies symmetry-based conserved quantities