{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:245IDMAYSVK6X5STH4PZQFYJSS","short_pith_number":"pith:245IDMAY","schema_version":"1.0","canonical_sha256":"d73a81b0189555ebf6533f1f981709948c02f3272266a7cfe2f641b28b847f96","source":{"kind":"arxiv","id":"1912.03937","version":2},"attestation_state":"computed","paper":{"title":"Deep Ritz revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","cs.NE","math.AP"],"primary_cat":"math.NA","authors_text":"Johannes M\\\"uller, Marius Zeinhofer","submitted_at":"2019-12-09T09:59:38Z","abstract_excerpt":"Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations (PDEs). In the latter the variational formulation of the Poisson problem is used in order to obtain an objective function - a regularised Dirichlet energy - that was used for the optimisation of some neural networks. In this notes we use the notion of $\\Gamma$-convergence to show that ReLU networks of growing architecture that are trained with respect to suitably regularised Dirichlet energies converge to the true solution of the Poisson problem. We discuss how thi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1912.03937","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-12-09T09:59:38Z","cross_cats_sorted":["cs.LG","cs.NA","cs.NE","math.AP"],"title_canon_sha256":"bf0782d66cd2e36f04f299a40c97f6cf308baa6e43081602dcb6d0d0a8d8f093","abstract_canon_sha256":"8aa6da728bfbea5e0587473de74c28cc7d6f9b992b4ecd5b7576e9b05663a56c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:32:53.302456Z","signature_b64":"4MYOY3F0DiDmwzX43fdYXwyUObGRNjMx02sXEmW5EdqMciqBYNLtJHVXnGQVqSpGGmkBl1Qw+8W9HJvzKXRJCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d73a81b0189555ebf6533f1f981709948c02f3272266a7cfe2f641b28b847f96","last_reissued_at":"2026-07-05T00:32:53.301972Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:32:53.301972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Deep Ritz revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","cs.NE","math.AP"],"primary_cat":"math.NA","authors_text":"Johannes M\\\"uller, Marius Zeinhofer","submitted_at":"2019-12-09T09:59:38Z","abstract_excerpt":"Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations (PDEs). In the latter the variational formulation of the Poisson problem is used in order to obtain an objective function - a regularised Dirichlet energy - that was used for the optimisation of some neural networks. In this notes we use the notion of $\\Gamma$-convergence to show that ReLU networks of growing architecture that are trained with respect to suitably regularised Dirichlet energies converge to the true solution of the Poisson problem. We discuss how thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.03937","kind":"arxiv","version":2},"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/1912.03937/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1912.03937","created_at":"2026-07-05T00:32:53.302027+00:00"},{"alias_kind":"arxiv_version","alias_value":"1912.03937v2","created_at":"2026-07-05T00:32:53.302027+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1912.03937","created_at":"2026-07-05T00:32:53.302027+00:00"},{"alias_kind":"pith_short_12","alias_value":"245IDMAYSVK6","created_at":"2026-07-05T00:32:53.302027+00:00"},{"alias_kind":"pith_short_16","alias_value":"245IDMAYSVK6X5ST","created_at":"2026-07-05T00:32:53.302027+00:00"},{"alias_kind":"pith_short_8","alias_value":"245IDMAY","created_at":"2026-07-05T00:32:53.302027+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.04018","citing_title":"The Coercivity Gap in Neural PDE Solvers: Parameter Escape and Functional Convergence","ref_index":29,"is_internal_anchor":false},{"citing_arxiv_id":"2603.20120","citing_title":"Deep learning-based phase-field modelling of brittle fracture in anisotropic media","ref_index":48,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/245IDMAYSVK6X5STH4PZQFYJSS","json":"https://pith.science/pith/245IDMAYSVK6X5STH4PZQFYJSS.json","graph_json":"https://pith.science/api/pith-number/245IDMAYSVK6X5STH4PZQFYJSS/graph.json","events_json":"https://pith.science/api/pith-number/245IDMAYSVK6X5STH4PZQFYJSS/events.json","paper":"https://pith.science/paper/245IDMAY"},"agent_actions":{"view_html":"https://pith.science/pith/245IDMAYSVK6X5STH4PZQFYJSS","download_json":"https://pith.science/pith/245IDMAYSVK6X5STH4PZQFYJSS.json","view_paper":"https://pith.science/paper/245IDMAY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1912.03937&json=true","fetch_graph":"https://pith.science/api/pith-number/245IDMAYSVK6X5STH4PZQFYJSS/graph.json","fetch_events":"https://pith.science/api/pith-number/245IDMAYSVK6X5STH4PZQFYJSS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/245IDMAYSVK6X5STH4PZQFYJSS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/245IDMAYSVK6X5STH4PZQFYJSS/action/storage_attestation","attest_author":"https://pith.science/pith/245IDMAYSVK6X5STH4PZQFYJSS/action/author_attestation","sign_citation":"https://pith.science/pith/245IDMAYSVK6X5STH4PZQFYJSS/action/citation_signature","submit_replication":"https://pith.science/pith/245IDMAYSVK6X5STH4PZQFYJSS/action/replication_record"}},"created_at":"2026-07-05T00:32:53.302027+00:00","updated_at":"2026-07-05T00:32:53.302027+00:00"}