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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.<\/jats:p>","DOI":"10.1063\/5.0352595","type":"journal-article","created":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T13:05:08Z","timestamp":1792587908000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Variational physics-informed ansatz for reconstructing hidden interaction networks from steady states"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-3727-4187","authenticated-orcid":false,"given":"Kaiming","family":"Luo","sequence":"first","affiliation":[{"name":"College of Future Information Technology, Fudan University , Shanghai 200438,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,10,21]]},"reference":[{"key":"2026102109050344900_c1","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/j.physrep.2008.09.002","volume":"469","year":"2008","journal-title":"Phys. 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