Non-Equilibrium Stochastic Dynamics as a Unified Framework for Insight and Repetitive Learning: A Kramers Escape Approach to Continual Learning

2026-04-05

AI Summary

This work addresses the stability-plasticity dilemma in continual learning and the lack of a unified theoretical framework for insight (sudden learning) and rehearsal-based mechanisms. Drawing on nonequilibrium statistical physics, the authors model learning dynamics as Langevin motion on a double-well energy landscape, where a time-varying effective temperature modulates noise levels. By leveraging the Fokker–Planck equation and Kramers’ escape rate theory, they unify insight—modeled as transient high-temperature pulses—and repetitive learning—represented by sustained moderate heating—within a single formalism. Notably, the Elastic Weight Consolidation (EWC) penalty is reinterpreted as an energy barrier that grows linearly with the number of encountered tasks. The theory predicts, and simulations confirm, an exponential decay of plasticity with task count, establishing a physically grounded continual learning framework and proposing an adaptive noise scheduling criterion.

Technology Category

• Machine Learning: Life-Long and Continual Learning
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Continual learning in artificial neural networks is fundamentally limited by the stability--plasticity dilemma: systems that retain prior knowledge tend to resist acquiring new knowledge, and vice versa. Existing approaches, most notably elastic weight consolidation~(EWC), address this empirically without a physical account of why plasticity eventually collapses as tasks accumulate. Separately, the distinction between sudden insight and gradual skill acquisition through repetitive practice has lacked a unified theoretical description. Here, we show that both problems admit a common resolution within non-equilibrium statistical physics. We model the state of a learning system as a particle evolving under Langevin dynamics on a double-well energy landscape, with the noise amplitude governed by a time-dependent effective temperature $T(t)$. The probability density obeys a Fokker--Planck equation, and transitions between metastable states are governed by the Kramers escape rate $k = (ω_0ω_b/2π)\,e^{-ΔE/T}$. We make two contributions. First, we identify the EWC penalty term as an energy barrier whose height grows linearly with the number of accumulated tasks, yielding an exponential collapse of the transition rate predicted analytically and confirmed numerically. Second, we show that insight and repetitive learning correspond to two qualitatively distinct temperature protocols within the same Fokker--Planck equation: insight events produce transient spikes in $T(t)$ that drive rapid barrier crossing, whereas repetitive practice operates at a modestly elevated but fixed temperature, achieving transitions through sustained stochastic diffusion. These results establish a physically grounded framework for understanding plasticity and its failure in continual learning systems, and suggest principled design criteria for adaptive noise schedules in artificial intelligence.

Problem

• continual learning
• stability-plasticity dilemma
• insight
• repetitive learning
• Kramers escape

Innovation

• non-equilibrium stochastic dynamics
• Kramers escape rate
• continual learning
• Fokker–Planck equation
• effective temperature