{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:BY7YXBVB5ZGLJEP43Q6ZG736CA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f783b8b6e8038fc465647f4ad33b724d3e504bc3aa16e231b0deac7ed4186968","cross_cats_sorted":["cs.LG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-09T14:10:26Z","title_canon_sha256":"d2d9f8d8d453503a22a2e5501890be1bc60becd3177e0b603047fb7ba0478af5"},"schema_version":"1.0","source":{"id":"1904.04685","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.04685","created_at":"2026-05-17T23:48:58Z"},{"alias_kind":"arxiv_version","alias_value":"1904.04685v1","created_at":"2026-05-17T23:48:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.04685","created_at":"2026-05-17T23:48:58Z"},{"alias_kind":"pith_short_12","alias_value":"BY7YXBVB5ZGL","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"BY7YXBVB5ZGLJEP4","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"BY7YXBVB","created_at":"2026-05-18T12:33:12Z"}],"graph_snapshots":[{"event_id":"sha256:d16f56f12ed30ca8c3434f3c1e6fc8efdfde70be6d76ca2552f4b4fbaa2266d4","target":"graph","created_at":"2026-05-17T23:48:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper is concerned with the approximation of the solution of partial differential equations by means of artificial neural networks. Here a feedforward neural network is used to approximate the solution of the partial differential equation. The learning problem is formulated as a least squares problem, choosing the residual of the partial differential equation as a loss function, whereas a multilevel Levenberg-Marquardt method is employed as a training method. This setting allows us to get further insight into the potential of multilevel methods. Indeed, when the least squares problem aris","authors_text":"Elisa Riccietti, Henri Calandra, Serge Gratton, Xavier Vasseur","cross_cats":["cs.LG","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-09T14:10:26Z","title":"On the approximation of the solution of partial differential equations by artificial neural networks trained by a multilevel Levenberg-Marquardt method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.04685","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5a9861804f541a1b55cb418a4c604e73255408a53f46f5fe32f8a6ecf8e64426","target":"record","created_at":"2026-05-17T23:48:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f783b8b6e8038fc465647f4ad33b724d3e504bc3aa16e231b0deac7ed4186968","cross_cats_sorted":["cs.LG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-09T14:10:26Z","title_canon_sha256":"d2d9f8d8d453503a22a2e5501890be1bc60becd3177e0b603047fb7ba0478af5"},"schema_version":"1.0","source":{"id":"1904.04685","kind":"arxiv","version":1}},"canonical_sha256":"0e3f8b86a1ee4cb491fcdc3d937f7e10039f8cee8b34ab265b678515bfbac93d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e3f8b86a1ee4cb491fcdc3d937f7e10039f8cee8b34ab265b678515bfbac93d","first_computed_at":"2026-05-17T23:48:58.430453Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:58.430453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A8PLhBKyRULEcl/ZU/B0SFC0bLrXgh+MVX6aoZX1CHc+hxLXzfBd98/WxR9g6O1U4QbLH8ojo6m3mQz2uFK5Ag==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:58.430970Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.04685","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a9861804f541a1b55cb418a4c604e73255408a53f46f5fe32f8a6ecf8e64426","sha256:d16f56f12ed30ca8c3434f3c1e6fc8efdfde70be6d76ca2552f4b4fbaa2266d4"],"state_sha256":"073814f74b76fd05421bb33b2366d68bbcfa133ec9bd2d2a768299ba4c0f88a9"}