Phaedra: Learning High-Fidelity Discrete Tokenization for the Physical Science

Tokens are discrete representations that allow modern deep learning to scale by transforming high-dimensional data into sequences that can be efficiently learned, generated, and generalized to new tasks. While foundational for image and video generation, the application of tokens to physical simulation remains nascent. Because existing tokenizers are designed for the perceptual requirements of natural images, they struggle with scientific data, which exhibits large dynamic ranges and requires exact preservation of physical and spectral properties. In this work, we investigate the performance of a suite of image tokenizers across metrics designed to measure PDE fidelity. Observing that these baselines struggle to simultaneously capture fine geometric details and precise physical magnitudes, we propose Phaedra, a novel tokenizer inspired by classical shape-gain quantization and the paradigm of basis functions coupled with continuous coefficients. Phaedra acts as a highly effective nonlinear compression algorithm, massively reducing dataset footprints while maintaining physical fidelity. We demonstrate that Phaedra consistently improves reconstruction across diverse 2D gridded PDE solutions, generalizes robustly to unseen PDE types and real-world Earth observation data, and is competitive with continuous models in downstream proof-of-concept operator learning and masked autoencoding tasks.