PDE-LENS: Scaling End-to-End PDE Discovery through Spectral Representation Learning

End-to-end PDE discovery maps an observed solution field to its governing equation. We study the end-to-end models that learn this map and find a recurring failure: they capture the slowly varying background of the field but miss the sharp fronts, narrow peaks, and oscillations that separate one candidate equation from another. We call this high-frequency neglect and locate the failure in the field encoder. PDE-LENS addresses it with PRISM, an encoder that operates directly on scattered observations in the spectral domain, using non-equispaced Fourier analysis. To train at scale, we build a quality-controlled dataset of 1.19 million numerically solved PDE trajectories from a library of 59 atomic terms, with LLM-assisted solver generation handling stiff or ill-posed cases. On unseen four-term equations, PDE-LENS reaches a median NMSE more than sixfold lower than PDE-FIND, the strongest non-pretrained baseline. Across 99 unseen trajectories over ten equation structures, PRISM recovers 65 within NMSE $\le 0.01$, against 50 for NeSymReS and 34 for a ViT encoder. For conventional symbolic regression, our encoder also achieved high performance. On the Feynman benchmark it also raises conventional symbolic-regression accuracy and high-frequency fidelity over E2E-SR and NeSymReS. PRISM's 256-dimensional latent representations also predict four dynamical properties with the best $R^2$ on three of four. Equation discovery thus yields not only the governing equation but a reusable, structured PDE latent space.