We study LLMs’ ability to extrapolate the behavior of various dynamical systems, including stochastic, chaotic, continuous, and discrete systems, whose evolution is governed by principles of physical interest. Our results show that LLaMA-2, a language model trained on text, achieves accurate predictions of dynamical system time series without fine-tuning or prompt engineering. Moreover, the accuracy of the learned physical rules increases with the length of the input context window, revealing an in-context version of a neural scaling law. Along the way, we present a flexible and efficient algorithm for extracting probability density functions of multi-digit numbers directly from LLMs.