用诺特定理分析噪声系统中的对称性，发现能量动量等守恒规律。
Dynamical symmetries in the fluctuation-driven regime: an application of Noether's theorem to noisy dynamical systems
- 基于非平衡物理的变分原理，将诺特定理推广到噪声动力系统
- 发现系统在最可能路径上存在能量、动量等守恒量
- 适用于决策模型、循环神经网络和生成扩散模型的研究
诺特定理建立了连续对称性与守恒量之间的强关联，但其适用前提要求系统遵循某种变分原理。然而，神经科学与人工智能中多数动力系统无法满足此条件。本文利用非平衡物理学提供的变分原理——描述一般噪声动力系统在两状态间最可能的演化路径——将诺特定理应用于此类系统，从而揭示连续对称性如何约束系统的最可能轨迹。研究识别出能量、动量及角动量的类守恒形式，并简要讨论其在决策模型、递归神经网络及扩散生成模型中的应用实例。
原文摘要 · Abstract (English)
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.
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