Inferring interaction structure from steady-state observations is a central inverse problem when transient trajectories are unavailable. Here we formulate this problem as simultaneous compatibility of a single interaction operator with equilibrium constraints generated by heterogeneous perturbations. We introduce a variational physics-informed ansatz that represents the unknown operator as a trainable object and minimizes the resulting steady-state residuals across experiments. In the affine-interaction setting, the stacked equilibrium equations yield explicit finite-sample identifiability conditions: unique recovery is controlled by the rank of the compatibility matrix after elimination of experiment-wise gauge freedom. Synthetic benchmarks on pairwise, directed, weighted, empirical-topology, and selected higher-order systems illustrate this identifiability picture and show how additional heterogeneous steady states improve structural discrimination under the stated assumptions. The results clarify a concrete steady-state reconstruction regime in which equilibrium observations alone can determine hidden interaction operators when the governing dynamics are known and node-level equilibria are fully observed.

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