{"paper":{"title":"A Statistical Formulation Gap for Nonlinear Multiscale Physics-Informed Learning","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.AP"],"primary_cat":"math.NA","authors_text":"Ronald Katende","submitted_at":"2026-07-17T07:26:29Z","abstract_excerpt":"We prove a finite-sample formulation gap for physics-informed learning of nonlinear multiscale elliptic equations. For a uniformly monotone divergence-form class with coefficients oscillating at scale $\\epsilon$, we derive a finite-width, finite-sample, and finite-iteration error bound for a boundary-compatible variational neural solver. Its stability, sampling, and optimization constants are independent of $\\epsilon$, while all unresolved scale dependence is isolated in the best-approximation error.\n  We then construct a minimal obstruction witness for a one-dimensional periodic diffusion equ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15702","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.15702/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}