Teaching Vision-Language Models to Use the Scale They Are Given: Label-Free Equivariance Training for Metric Physical Reasoning

Abstract

Metric questions about video, such as the speed of a moving object, require a vision-language model to convert visual measurements into physical units using a real-world reference supplied in the prompt. We find that current models use this reference only partially. When every world-space quantity in the prompt is multiplied by a common factor, the prompt still describes the same video and the correct answer changes by exactly that factor, but the predictions of eight models change by less, and their accuracy stays concentrated near the scale that the depicted objects usually have. Asked the same physics in a scale-free form, the two models we test recover the closed-form scaling laws on most items, which indicates that the deficit lies in metric grounding and not in knowledge of the physical mechanism. Because the scaling relation is exact, it can serve as supervision without metric annotations. Equivariance Self-Distillation (EquiSD) projects a model's own prediction onto the functions that satisfy this relation and fine-tunes the model on the resulting targets, with one query per training question and no ground truth. Trained on synthetic video only, EquiSD brings a 3B model close to the exact relation on held-out simulated videos, also at scales not seen in training, and improves its accuracy across scales. Without adaptation, it also improves accuracy across scales on the QuantiPhy benchmark, where its gain reaches 93% of that obtained by supervision with exact simulator answers.