TERM: Enriching Representations in PDE Foundation Models with Term-wise Physical Responses

Partial Differential Equation (PDE) foundation models aim to learn transferable solution priors across heterogeneous physical systems, yet existing methods typically represent governing equations as static symbolic structures or combine equations and trajectories only at a global level. Such representations capture which physical terms constitute a PDE, but rarely how each term acts in the current state. Although physical terms have stable mathematical identities, their strength, spatial distribution, and dynamical roles vary along physical trajectories. Motivated by this observation, we propose the Term-wise Enriched Representation Model (TERM), which organizes PDE information at the level of individual physical terms when the governing equation is available. TERM first normalizes the equation into a structured term graph, then computes state-dependent responses for each physical term and explicitly binds them to their symbolic identities. A structure-aware Transformer jointly models these response-enriched term representations and compact trajectory information, while a continuous query decoder enables prediction across equations, resolutions, and continuous coordinates. During autoregressive rollout, the equation structure remains fixed while term responses are recomputed from the evolving predicted state. Experiments across multi-PDE pretraining, equation-specified zero-shot transfer, compositional generalization, cross-resolution prediction, and long-horizon rollout show consistent improvements over the evaluated PDE foundation-model baselines and representation controls. The results suggest that state-dependent, term-level physical representations provide a useful interface for scalable and compositional PDE foundation modeling.