[论文解读] A Unified Dynamical Field Theory of Learning, Inference, and Emergence

Byung Gyu Chae|arXiv (Cornell University)|Jan 15, 2026

Neural dynamics and brain function被引用 0

一句话总结

本论文提出一个统一的随机动力场理论，用于学习、推断与涌现，显示推断作为鞍点轨迹，涌现通过产生慢的参与模态的循环校正实现。它在一个几何框架下将能量基模型、递归神经网络、变换器与自稳态动力学统一起来。

ABSTRACT

Learning, inference, and emergence in biological and artificial systems are often studied within disparate theoretical frameworks, ranging from energy-based models to recurrent and attention-based architectures. Here we develop a unified dynamical field theory in which learning and inference are governed by a minimal stochastic dynamical equation admitting a Martin--Siggia--Rose--Janssen--de Dominicis formulation. Within this framework, inference corresponds to saddle-point trajectories of the associated action, while fluctuation-induced loop corrections render collective modes dynamically emergent and generate nontrivial dynamical time scales. A central result of this work is that cognitive function is controlled not by microscopic units or precise activity patterns, but by the collective organization of dynamical time scales. We introduce the \emph{time-scale density of states} (TDOS) as a compact diagnostic of the distribution of collective relaxation modes governing inference dynamics. Learning and homeostatic regulation are naturally interpreted as processes that reshape both the effective potential and the underlying state-space geometry, thereby reorganizing the TDOS and selectively stabilizing slow collective modes that support stable inference, memory, and context-dependent computation despite stochasticity and structural irregularity. This framework unifies energy-based models, recurrent neural networks, transformer architectures, and biologically motivated homeostatic dynamics within a single physical description, and provides a principled route toward understanding cognition as an emergent dynamical phenomenon.

研究动机与目标

• 为生物系统与人工系统中的学习、推断与涌现提供一个 principled、统一的框架的动机。
• 描述控制高维集体神经状态的最小随机动力方程。
• 展示在MSRJD路径积分形式下，推断如何作为鞍点轨迹出现。
• 证明循环校正如何生成涌现的集体模态并重新组织时间尺度。
• 展示学习如何重塑势能景观和状态空间几何，以稳定慢模态。

提出的方法

• 提出一个集体状态的最小随机动力方程：ẋ = -G^{-1}(x) ∇Φ(x) + R(x) + ξ(t)。
• 将动力学写成Martin–Siggia–Rose–Janssen–de Dominicis (MSRJD) 路径积分，其作用量为 S[x, x̃]。
• 从MSRJD作用量中导出鞍点（推断）轨迹，其中 ∂S/∂x̃ = 0。
• 通过对鞍点周围的循环校正分析涨落，以揭示涌现的集体模态。
• 引入时间尺度密度态TDOS，用以诊断放松模态的分布。
• 显示学习作为对Φ、G、R的缓慢结构性自适应，从而重新组织TDOS。

Figure 1: Conceptual overview of the unified dynamical field theory. (a) Collective neural states evolve on a learned state-space geometry, shaped by an effective potential $\Phi(x)$ , a state-dependent metric $G(x)$ , non-conservative reentrant flows $R(x)$ , and stochastic fluctuations $\xi(t)$ .

实验结果

研究问题

• RQ1 如何在单一随机动力学框架中描述学习、推断与涌现？
• RQ2 在此框架中，推断如何对应鞍点轨迹，涨落如何产生涌现的时间尺度？
• RQ3 有效势、几何与再进入流在塑造集体动力学和TDOS中扮演何种角色？
• RQ4 是否可以将若干典型模型（Hopfield网络、RNN、Transformer、稳态再进入网络）作为统一理论的极限推导？
• RQ5 学习如何重构时间尺度谱以支持稳定记忆、情境相关计算与鲁棒推断？

主要发现

• 推断作为MSRJD作用量的鞍点轨迹出现，即学习动态下的最可能路径。
• 涨落循环校正通过重新归一化自能产生慢的集体模态与时间尺度。
• TDOS提供一个紧凑的诊断工具，显示学习如何重塑集体放松模态并稳定慢动力学。
• 学习作为Φ、G、R的缓慢结构自适应，重新组织时间尺度分布以促进鲁棒推断。
• 特殊极限下可推导出Hopfield网络、RNN、Transformer与自稳态再进入网络，作为统一理论的结果。
• 提出神经音（neuroton）作为自生成的集体放松模态，承载涌现计算与记忆。

Figure 2: Recurrent versus reentrant neural architectures. (a) Recurrent architectures update a latent hidden state sequentially in time. The recurrence operates through discrete or implicit temporal state updates, while the underlying representation space and geometry remain fixed. Temporal depende