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

John J. Vastola

TL;DR

The paper extends Noether's theorem to fluctuation-driven, non-variational dynamics by adopting the Onsager-Machlup path integral as a stochastic variational principle. It derives analogues of energy, momentum, and angular momentum conservation for noisy dynamical systems and demonstrates how these conserved quantities shape the most likely transition paths in drift-diffusion, attractor-based decision memories, and diffusion-based generative models. The approach provides a principled link between system symmetries and constrained stochastic trajectories, with potential to reveal learnable symmetries and guide the design of equivariant AI components. This framework offers a new lens to analyze decision dynamics and generative processes under noise, with implications for neuroscience and machine learning.

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.

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

TL;DR

The paper extends Noether's theorem to fluctuation-driven, non-variational dynamics by adopting the Onsager-Machlup path integral as a stochastic variational principle. It derives analogues of energy, momentum, and angular momentum conservation for noisy dynamical systems and demonstrates how these conserved quantities shape the most likely transition paths in drift-diffusion, attractor-based decision memories, and diffusion-based generative models. The approach provides a principled link between system symmetries and constrained stochastic trajectories, with potential to reveal learnable symmetries and guide the design of equivariant AI components. This framework offers a new lens to analyze decision dynamics and generative processes under noise, with implications for neuroscience and machine learning.

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.

Paper Structure

This paper contains 12 sections, 1 theorem, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Noether's theorem, symmetries, and conserved quantities
  3. Applying Noether's theorem to stochastic dynamical systems
  4. Energy, momentum, and angular momentum conservation
  5. Energy conservation
  6. Momentum conservation
  7. Angular momentum conservation
  8. Case studies relevant to neuroscience and AI
  9. Drift-diffusion models of decision-making
  10. Changes of mind in an attractor network model of decision memory
  11. Data manifold symmetries and reverse diffusion dynamics
  12. Conclusion

Key Result

Theorem 1

Let $\delta t \geq 0$ and $\delta \boldsymbol{x} \in \mathbb{R}^N$, and consider a transformation that takes $t \to t' := t + \epsilon \ \delta t$ and $\boldsymbol{x} \to \boldsymbol{x}' := \boldsymbol{x} + \epsilon \ \delta \boldsymbol{x}$. If the Lagrangian $L$ satisfies for all $\epsilon > 0$ sufficiently small, where $K$ is some function, then the quantity is conserved in the sense that $dJ/

Figures (2)

  • Figure 1: Conservation laws relevant to simple decision-making and decision memory models. a. For a drift-diffusion model with fixed decision bounds at $\pm 1$, the most likely path from any evidence state to the opposite boundary is (approximately) a straight line (blue, 'LAP'). Raw paths ($n = 981$) with same start state, end state, and transition time also shown. b. Transitions between two states of a single-attractor model (heatmap: steady state distribution) are direct when energy and angular momentum are high, and involve a diversion to the attractor state when they are low. c. Transitions between two attractor basins in a model with three attractors (heatmap: steady state distribution) are more direct when energy is high, and involve visiting an intermediate attractor when energy is low.
  • Figure 2: Angular momentum conservation in reverse diffusion. Left: depiction of reverse diffusion for a rotationally-symmetric data distribution. Right: most likely transition paths given a fixed starting point (black dot) and different angular momentum values (green is lowest, red is highest). Dashed black line is the PF-ODE trajectory.

Theorems & Definitions (1)

  • Theorem 1: Noether's theorem