WHAT CAN A REPRESENTATION IDENTIFY? SPECTRAL LIMITS OF LEARNING PDE DYNAMICS FROM SPATIAL DATA

Inferring dynamical laws from spatial observations is fundamentally limited by the information preserved in the representation supplied to the inference method. We study this limitation for PDE identification from one or a few spatial observations through a representation-dependent view of identifiability. For linear translationinvariant PDEs, we show that power spectral density (PSD) captures only the real part of the Fourier symbol, while phase-sensitive temporal information is required to recover its imaginary part. Consequently, mechanisms such as advection and odd-order dispersion are invisible to magnitude-only spectral representations. We characterize when PDE parameters can be recovered from spectral data and show how identifiability depends jointly on the governing dynamics, observation protocol, and representation used for inference. When elapsed time is unknown, only parameter ratios, parameter combinations, or spectral shape information may remain identifiable, while a single spatial snapshot generally confounds the initial condition with subsequent dynamics. Numerical experiments distinguish structural non-identifiability from practical estimation error and representation-induced information loss. These results motivate a representation-first approach to learning dynamics from spatial data: before selecting an inference architecture, one should determine which dynamical distinctions the chosen representation makes learnable.