[Paper Review] A Unified Dynamical Field Theory of Learning, Inference, and Emergence

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

Neural dynamics and brain function0 citations

TL;DR

This paper develops a unified stochastic dynamical field theory for learning, inference, and emergence, showing inference as saddle-point trajectories and emergence via loop corrections that create slow collective modes. It unifies energy-based models, RNNs, transformers, and homeostatic dynamics under a single geometric framework.

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.

Motivation & Objective

• Motivate a principled, unified framework for learning, inference, and emergence across biological and artificial systems.
• Describe a minimal stochastic dynamical equation governing high-dimensional collective neural states.
• Show how inference arises as saddle-point trajectories within an MSRJD path-integral formulation.
• Demonstrate how loop corrections generate emergent collective modes and reorganize time scales.
• Present how learning reshapes both the potential landscape and the state-space geometry to stabilize slow modes.

Proposed method

• Propose a minimal stochastic dynamical equation for collective states: ẋ = -G^{-1}(x) ∇Φ(x) + R(x) + ξ(t).
• Cast the dynamics into Martin–Siggia–Rose–Janssen–de Dominicis (MSRJD) path integral with action S[x, x̃].
• Derive saddle-point (inferrence) trajectories from the MSRJD action where ∂S/∂x̃ = 0.
• Analyze fluctuations via loop corrections around saddle points to reveal emergent collective modes.
• Introduce the time-scale density of states (TDOS) to diagnose distributions of relaxation modes.
• Show that learning acts as slow structural adaptation reshaping Φ, G, and R, thereby reorganizing 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 $\xi(t)$ .

Experimental results

Research questions

• RQ1 How can learning, inference, and emergence be described within a single stochastic dynamical framework?
• RQ2 In this framework, how does inference correspond to saddle-point trajectories and how do fluctuations produce emergent time scales?
• RQ3 What role do the effective potential, geometry, and reentrant flows play in shaping collective dynamics and TDOS?
• RQ4 Can several canonical models (Hopfield, RNNs, transformers, homeostatic networks) be derived as limits of the unified theory?
• RQ5 How does learning restructure the time-scale spectrum to support stable memory, context-dependent computation, and robust inference?

Key findings

• Inference emerges as the saddle-point trajectory of the MSRJD action, i.e., the most probable path under learned dynamics.
• Fluctuation loop corrections generate slow, collective modes and time scales through renormalized self-energies.
• The TDOS provides a compact diagnostic of how learning reshapes collective relaxation modes and stabilizes slow dynamics.
• Learning acts as slow structural adaptation of Φ, G, and R, reorganizing the time-scale distribution to promote robust inference.
• Special limits recover Hopfield networks, RNNs, Transformers, and homeostatic reentry networks as consequences of the unified theory.
• Neurotons are proposed as self-generated collective relaxation modes that carry emergent computation and memory.

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