{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:7HAQL4ZMGKXMOLCTPCJOLR3IF6","short_pith_number":"pith:7HAQL4ZM","canonical_record":{"source":{"id":"2311.00529","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-11-01T14:01:52Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"321f6475e2b35e370374a1280ea5224762ea6f03e0863cc6b1f0beea33fee31d","abstract_canon_sha256":"b800d3471663987aab515e61d99d857f65ba92775a80ba0efb0498247967374a"},"schema_version":"1.0"},"canonical_sha256":"f9c105f32c32aec72c537892e5c7682fb15d096b726ca2bf17758426b08c4077","source":{"kind":"arxiv","id":"2311.00529","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2311.00529","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"arxiv_version","alias_value":"2311.00529v2","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.00529","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"pith_short_12","alias_value":"7HAQL4ZMGKXM","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"pith_short_16","alias_value":"7HAQL4ZMGKXMOLCT","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"pith_short_8","alias_value":"7HAQL4ZM","created_at":"2026-07-05T07:53:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:7HAQL4ZMGKXMOLCTPCJOLR3IF6","target":"record","payload":{"canonical_record":{"source":{"id":"2311.00529","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-11-01T14:01:52Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"321f6475e2b35e370374a1280ea5224762ea6f03e0863cc6b1f0beea33fee31d","abstract_canon_sha256":"b800d3471663987aab515e61d99d857f65ba92775a80ba0efb0498247967374a"},"schema_version":"1.0"},"canonical_sha256":"f9c105f32c32aec72c537892e5c7682fb15d096b726ca2bf17758426b08c4077","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:53:48.751224Z","signature_b64":"ECF8fP1uClx70+8xPJwsQ6fqzrueqwHeyAusOInm59LjM+9vkCQzqYGGjivgqhfM8h7N5ass5ilEEvyh3vr3Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9c105f32c32aec72c537892e5c7682fb15d096b726ca2bf17758426b08c4077","last_reissued_at":"2026-07-05T07:53:48.750824Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:53:48.750824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2311.00529","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:53:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rv4mSv1eA6omchc7ddmrusR9PL/9RmexHed+jLfS3OzLCe1ZYtEfL+qTuW2HdCfrfQpb3XsfgDK+DL1oqztNBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T09:43:35.969314Z"},"content_sha256":"e9085a6c729f324fc74171e7a2d8ea521c4ef2f0f05f9b25a8e8660ed203eca7","schema_version":"1.0","event_id":"sha256:e9085a6c729f324fc74171e7a2d8ea521c4ef2f0f05f9b25a8e8660ed203eca7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:7HAQL4ZMGKXMOLCTPCJOLR3IF6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Unified Framework for the Error Analysis of Physics-Informed Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Kent-Andr\\'e Mardal, Marius Zeinhofer, Rami Masri","submitted_at":"2023-11-01T14:01:52Z","abstract_excerpt":"We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations; and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^2$ penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error deca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.00529","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/2311.00529/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:53:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eEmI7EX3gZSTHMxZonroXirz1UCRNDWvoGNPucEo9hzMjjGrchsfW9KxH4+uHMiUopOERKP303c7k7X/KWJgBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T09:43:35.969825Z"},"content_sha256":"0fcce5b27ddc7b4db796fe037713dbcba1f8abb9388bd179a1515ab523f1c8b5","schema_version":"1.0","event_id":"sha256:0fcce5b27ddc7b4db796fe037713dbcba1f8abb9388bd179a1515ab523f1c8b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7HAQL4ZMGKXMOLCTPCJOLR3IF6/bundle.json","state_url":"https://pith.science/pith/7HAQL4ZMGKXMOLCTPCJOLR3IF6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7HAQL4ZMGKXMOLCTPCJOLR3IF6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-13T09:43:35Z","links":{"resolver":"https://pith.science/pith/7HAQL4ZMGKXMOLCTPCJOLR3IF6","bundle":"https://pith.science/pith/7HAQL4ZMGKXMOLCTPCJOLR3IF6/bundle.json","state":"https://pith.science/pith/7HAQL4ZMGKXMOLCTPCJOLR3IF6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7HAQL4ZMGKXMOLCTPCJOLR3IF6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:7HAQL4ZMGKXMOLCTPCJOLR3IF6","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":"b800d3471663987aab515e61d99d857f65ba92775a80ba0efb0498247967374a","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-11-01T14:01:52Z","title_canon_sha256":"321f6475e2b35e370374a1280ea5224762ea6f03e0863cc6b1f0beea33fee31d"},"schema_version":"1.0","source":{"id":"2311.00529","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2311.00529","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"arxiv_version","alias_value":"2311.00529v2","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.00529","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"pith_short_12","alias_value":"7HAQL4ZMGKXM","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"pith_short_16","alias_value":"7HAQL4ZMGKXMOLCT","created_at":"2026-07-05T07:53:48Z"},{"alias_kind":"pith_short_8","alias_value":"7HAQL4ZM","created_at":"2026-07-05T07:53:48Z"}],"graph_snapshots":[{"event_id":"sha256:0fcce5b27ddc7b4db796fe037713dbcba1f8abb9388bd179a1515ab523f1c8b5","target":"graph","created_at":"2026-07-05T07:53:48Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2311.00529/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations; and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^2$ penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error deca","authors_text":"Kent-Andr\\'e Mardal, Marius Zeinhofer, Rami Masri","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-11-01T14:01:52Z","title":"A Unified Framework for the Error Analysis of Physics-Informed Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.00529","kind":"arxiv","version":2},"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:e9085a6c729f324fc74171e7a2d8ea521c4ef2f0f05f9b25a8e8660ed203eca7","target":"record","created_at":"2026-07-05T07:53:48Z","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":"b800d3471663987aab515e61d99d857f65ba92775a80ba0efb0498247967374a","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-11-01T14:01:52Z","title_canon_sha256":"321f6475e2b35e370374a1280ea5224762ea6f03e0863cc6b1f0beea33fee31d"},"schema_version":"1.0","source":{"id":"2311.00529","kind":"arxiv","version":2}},"canonical_sha256":"f9c105f32c32aec72c537892e5c7682fb15d096b726ca2bf17758426b08c4077","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f9c105f32c32aec72c537892e5c7682fb15d096b726ca2bf17758426b08c4077","first_computed_at":"2026-07-05T07:53:48.750824Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:53:48.750824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ECF8fP1uClx70+8xPJwsQ6fqzrueqwHeyAusOInm59LjM+9vkCQzqYGGjivgqhfM8h7N5ass5ilEEvyh3vr3Cg==","signature_status":"signed_v1","signed_at":"2026-07-05T07:53:48.751224Z","signed_message":"canonical_sha256_bytes"},"source_id":"2311.00529","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9085a6c729f324fc74171e7a2d8ea521c4ef2f0f05f9b25a8e8660ed203eca7","sha256:0fcce5b27ddc7b4db796fe037713dbcba1f8abb9388bd179a1515ab523f1c8b5"],"state_sha256":"a534390c549a9174eea96735c1953d3d29447480785dac5fd6236390a58a7c18"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UDM5tPd/j+ERUB/Tto29/Fy8avWKedxVEykd6rdjANZ6f5VCuRnypiVwNewWrAhSKaSP7GIImanUpoX1zLr4Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T09:43:35.973995Z","bundle_sha256":"3313291654a1ab92e34066c20a0fec99182805fea870fb6c1edb5241d66a46ea"}}