Momentum Point-Perplexity Mechanics in Large Language Models

TL;DR

This paper introduces a physics-inspired framework to analyze and control large language models' internal states, leading to improved interpretability, alignment, and output quality through a novel Jacobian steering method.

Contribution

It proposes a new physics-based perspective on transformer dynamics and develops Jacobian steering for targeted, minimal perturbations to guide model outputs.

Findings

Energy-like quantity remains nearly constant during inference

Random-weight models conserve this energy more tightly

Jacobian steering improves output quality and interpretability

Abstract

We take a physics-based approach to studying how the internal hidden states of large language models change from token to token during inference. Across 20 open-source transformer models (135M-3B parameters), we find that a quantity combining the rate of change in hidden states and the model's next-token certainty, analogous to energy in physics, remains nearly constant. Random-weight models conserve this "energy" more tightly than pre-trained ones, while training shifts models into a faster, more decisive regime with greater variability. Using this "log-Lagrangian" view, we derive a control method called Jacobian steering, which perturbs hidden states in the minimal way needed to favor a target token. This approach maintained near-constant energy in two tested models and produced continuations rated higher in semantic quality than the models' natural outputs. Viewing transformers through this mechanics lens offers a principled basis for interpretability, anomaly detection, and low-risk steering. This could help make powerful models more predictable and aligned with human intent.