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. These have become foundational for image and video generation and, more recently, physical simulation. As existing tokenizers are designed for the explicit requirements of realistic visual perception of images, it is necessary to ask whether these approaches are optimal for scientific images, which exhibit a large dynamic range and require token embeddings to retain physical and spectral properties. In this work, we investigate the accuracy of a suite of image tokenizers across a range of metrics designed to measure the fidelity of PDE properties in both physical and spectral space. Based on the observation that these struggle to capture both fine details and precise magnitudes, we propose Phaedra, inspired by classical shape-gain quantization and proper orthogonal decomposition. We demonstrate that Phaedra consistently improves reconstruction across a range of PDE datasets. Additionally, our results show strong out-of-distribution generalization capabilities to three tasks of increasing complexity, namely known PDEs with different conditions, unknown PDEs, and real-world Earth observation and weather data.

翻译：标记是离散表示，它通过将高维数据转换为可高效学习、生成并泛化至新任务的序列，使现代深度学习得以规模化。这些标记已成为图像与视频生成的基础，并最近扩展至物理模拟领域。由于现有标记化方法专为满足图像真实视觉感知的显式需求而设计，有必要探究这些方法是否适用于科学图像——后者具有大动态范围，且要求标记嵌入保留物理与光谱特性。本研究通过一系列专为测量物理空间与光谱空间中偏微分方程特性保真度而设计的指标，系统评估了多种图像标记化方法的准确性。基于这些方法在捕捉细节与精确幅值方面均存在不足的观察，我们受经典形状-增益量化与适当正交分解启发，提出了Phaedra。实验证明，Phaedra在多种偏微分方程数据集上持续提升重建质量。此外，我们的结果显示该方法在三个复杂度递增的任务中展现出强大的分布外泛化能力：即已知偏微分方程的不同条件、未知偏微分方程，以及真实世界的地球观测与气象数据。