The Operator Atlas: Learning the Geometry of PDE Solution Manifolds

Existing PDE learning methods treat each equation in isolation and lack a representation that captures relations between equations. We introduce STAR, a contrastive encoder trained on the local differential features of solution fields. Three objectives jointly shape the latent space: a contrastive loss recovers the topological relations of the solution manifold, a reconstruction loss makes the coordinates physically decodable, and a coefficient loss anchors the shared representation to physical parameters. Trained on only three equation families, the model produces an Operator Atlas over twelve families in a train-free manner. The atlas organizes families by solution morphology, unfolds within-family parameters into continuous axes, and reveals a differential core shared by all twelve families ($p<0.001$) together with two cross-family relations: Poisson and Biharmonic, which differ in order, are mutual nearest neighbours under both metrics, and Helmholtz and mKdV are mutual nearest neighbours under centroid distance. Once frozen, the atlas solves four forward and inverse problems without per-query training: geodesic interpolation reconstructs unseen fields (median $L_2$ of $1.6\times 10^{-2}$), patch voting identifies the equation family ($0.889$), manifold position infers parameters (absolute error $0.081$), and departure from the manifold detects anomalies (AUROC $0.811$). The first three tasks are further paired with ground-truth-free uncertainty metrics, enabling reliable deployment without labelled validation data.