Endogenous Regime Switching Driven by Scalar-Irreducible Learning Dynamics

Sheng Ran

cs.LGApr 10, 2026

Endogenous Regime Switching Driven by Scalar-Irreducible Learning Dynamics

Authors: Sheng Ran

Organizations: Department of Physics, Washington University in St. Louis, St. Louis, MO 63130, USA · Reconstructing Future Research Initiative

Abstract

Achieving endogenous regime switching is crucial for the emergence of autonomous intelligence, yet remains a central challenge for existing machine learning frameworks, where such transitions are typically externally imposed. In this work, we introduce a classification that distinguishes scalar-reducible dynamics, which can be expressed as gradient flows driven by a scalar objective, from scalar-irreducible dynamics that cannot be reduced to such a form. While most existing machine learning systems operate within the scalar-reducible class, we demonstrate that scalar-irreducible dynamics naturally enable internally generated regime switching through feedback between fast dynamical variables and slow structural adaptation. Using a minimal dynamical model, we illustrate how this mechanism produces sustained endogenous regime transitions without external scheduling. Our results suggest a new dynamical paradigm for regime exploration and provide a potential route toward autonomous learning systems whose adaptive behavior is organized internally rather than externally prescribed.

Figures & tables

Figure 1: Conceptual illustration of scalar-reducible and scalar-irreducible learning dynamics. In scalar-reducible dynamics (top), the flow follows the gradient of a scalar objective and typically converges into a single basin, locking the system into one regime. In scalar-irreducible dynamics (bottom ), rotational components of the dynamics allow trajectories to repeatedly traverse different regions of the landscape, enabling endogenous regime switching.

Framework	Objective	Dynamics	Mathematical Form	Regime Switching
Supervised learning	Explicit loss	Gradient descent	$\dot{\theta}=-\nabla V(\theta)$	External
Reinforcement learning	Reward maximization	Policy gradient (value updates)	$\dot{\theta}=\nabla J(\theta)$	External
Rule-based adaptive learning	Implicit objective	Local update rules	Effective gradient dynamics	External
Multi-agent learning	Game payoff	Coupled gradient	Gradient game dynamics	External
Scalar-irreducible learning (this work)	None	Intrinsic vector-field dynamics	$\dot{\theta}=F(\theta),\;\nexists V$	Endogenous

Table 1: Classification of learning dynamics in existing machine learning frameworks. Despite their apparent diversity, most approaches can be reduced to dynamics driven by a scalar objective. The framework proposed in this work instead introduces scalar-irreducible dynamics that enable endogenous regime switching.

Figure 2: Comparison between scalar-reducible and scalar-irreducible learning dynamics. The left panel (a–d) shows the scalar-reducible baseline, the middle panel (e–h) corresponds to the scalar-irreducible system, and the right panel (i-l) shows the externally swept case. From top to bottom the figures display the fast dynamical variable $u$ , dynamical activity $R$ of $u$ , the slow structural variables ( $\rho$ and $\phi$ ), and the dynamical badness $B$ .

Regime-Switching Dynamical Systems

Regime-switching dynamical systems model processes that shift among distinct modes, each with its own dynamics and transition behavior.

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Adaptive Learning Self-Evolving Agents

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Despite their apparent diversity, modern machine learning methods can be reduced to a remarkably simple core principle: learning is achieved by continuously optimizing parameters to minimize or maximize a scalar objective function. This paradigm has been extraordinarily successful for well-defined tasks where goals are fixed and evaluation criteria are explicit. However, if artificial systems are to move toward true autonomy-operating over long horizons and across evolving contexts-objectives may become ill-defined, shifting, or entirely absent. In such settings, a fundamental question emerges: in the absence of an explicit objective function, how can a system determine whether its ongoing internal dynamics are productive or pathological? And how should it regulate structural change without external supervision? In this work, we propose a dynamical framework for learning without an explicit objective. Instead of minimizing external error signals, the system evaluates the intrinsic health of its own internal dynamics and regulates structural plasticity accordingly. We introduce a two-timescale architecture that separates fast state evolution from slow structural adaptation, coupled through an internally generated stress variable that accumulates evidence of persistent dynamical dysfunction. Structural modification is then triggered not continuously, but as a state-dependent event. Through a minimal toy model, we demonstrate that this stress-regulated mechanism produces temporally segmented, self-organized learning episodes without reliance on externally defined goals. Our results suggest a possible route toward autonomous learning systems capable of self-assessment and internally regulated structural reorganization.

Abstract

Regime-Switching Dynamical Systems

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