When do World Models Successfully Learn Dynamical Systems?
This work addresses the challenge of predicting when world models can simulate physical systems, which is incremental as it builds on existing concepts to provide theoretical insights.
The paper tackled the problem of understanding when world models can learn dynamical systems by proposing a theoretical framework that explains the effectiveness of tokenization and characterizing conditions for successful reconstruction, and validated this with models achieving successful flow recreation on a challenging CFD dataset.
In this work, we explore the use of compact latent representations with learned time dynamics ('World Models') to simulate physical systems. Drawing on concepts from control theory, we propose a theoretical framework that explains why projecting time slices into a low-dimensional space and then concatenating to form a history ('Tokenization') is so effective at learning physics datasets, and characterise when exactly the underlying dynamics admit a reconstruction mapping from the history of previous tokenized frames to the next. To validate these claims, we develop a sequence of models with increasing complexity, starting with least-squares regression and progressing through simple linear layers, shallow adversarial learners, and ultimately full-scale generative adversarial networks (GANs). We evaluate these models on a variety of datasets, including modified forms of the heat and wave equations, the chaotic regime 2D Kuramoto-Sivashinsky equation, and a challenging computational fluid dynamics (CFD) dataset of a 2D Kármán vortex street around a fixed cylinder, where our model is successfully able to recreate the flow.