{"paper":{"title":"Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The H^{-1} norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain.","cross_cats":["cs.LG","cs.NA"],"primary_cat":"math.NA","authors_text":"Diego Marcondes","submitted_at":"2026-05-05T09:14:38Z","abstract_excerpt":"The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of the residual requires a supremum over an infinite-dimensional test space. We prove that the $H^{-1}$ norm of any functional is equivalent to its expected squared evaluation against a random test function whose probability distribution depends only on the domain. Crucially, realisations of this random test function have negative Sobolev regularity for $d \\geq 2$, yet this roughness is not an obs"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the H^{-1} norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That there exists a random test function distribution depending only on the domain such that averaging squared evaluations exactly recovers the H^{-1} norm and the weak topology independently of the differential operator.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"H^{-1} norm equivalence to expected squared evaluations on domain-dependent random test functions enables SV-PINNs that recover accurate solutions to challenging second-order elliptic PDEs faster than standard PINNs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The H^{-1} norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"03173123c6d08d28bcfde79bb0ebef13a4cce30edc9335c9421b2f182a2609fd"},"source":{"id":"2605.03542","kind":"arxiv","version":2},"verdict":{"id":"49d74b17-d8ff-40cd-98bd-01b981d1e63e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T14:49:20.046692Z","strongest_claim":"We prove that the H^{-1} norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain.","one_line_summary":"H^{-1} norm equivalence to expected squared evaluations on domain-dependent random test functions enables SV-PINNs that recover accurate solutions to challenging second-order elliptic PDEs faster than standard PINNs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That there exists a random test function distribution depending only on the domain such that averaging squared evaluations exactly recovers the H^{-1} norm and the weak topology independently of the differential operator.","pith_extraction_headline":"The H^{-1} norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.03542/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T13:39:13.516521Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T01:01:21.401842Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T15:15:37.164209Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"d8462a657f45b1d03b5e851b76820aeac7fece48939f18eabbda6fd8c2b338c2"},"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"}