Hypothesize, Evaluate, Refine: A Scientific Agent for Open-Form PDE Discovery with Unknown Spatial Coefficient Fields

Understanding dynamics in heterogeneous media requires identifying both the governing differential operator and the spatial properties that determine its local action. Discovering these jointly is difficult: operator scope changes the law, while flexible coefficient fields can mask structural error. Existing open-form methods generally assume scalar or explicitly parameterized coefficients; nonparametric field estimators typically operate within prescribed operator libraries. We introduce Hypothesize, Evaluate, Refine for PDE Discovery (HER-PDE), coupling a scientific Agent with the Hypothesis Evaluation Interface (HEI). The Agent composes operators and field nodes into complete expression trees; HEI fits their time-invariant spatial fields and scores short-horizon prediction across two excitations within its supported evolution class. This separates structural reasoning from continuous estimation while preserving operator scope and shared-field identity. On seven two-dimensional heterogeneous PDE families with 5% state noise, including nonlinear cubic absorption, HER-PDE recovers 70/105 exact structures and achieves 94.3% Pass@3, compared with 39/105 exact recoveries for recursive typed evolution using the same HEI. Registered-structure fits attain median field correlation 0.891 across 65 fields; targeted controls test noise, solver, and morphology shifts. In a real spatial-drug case study, an Agent-proposed density-inhibitory effective model reduces continuous-prediction error by 15.7% relative to a refitted linear reference in post-discovery technical-repeat validation on measured microbial trajectories.