{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:6HXROIZI76OMQWKGATJEF2H44P","short_pith_number":"pith:6HXROIZI","canonical_record":{"source":{"id":"2406.14808","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2024-06-21T01:13:18Z","cross_cats_sorted":["cs.LG","stat.ME","stat.ML","stat.TH"],"title_canon_sha256":"d456270de209334f204b92630e2e6d7553398165ca3f235a59e0ece994cabe00","abstract_canon_sha256":"30ac2de9cf8738a9094212340f1325319ba8ecf6243af408b24f71efd906d82b"},"schema_version":"1.0"},"canonical_sha256":"f1ef172328ff9cc8594604d242e8fce3e34e84442f181930ad5bd06f4d4f90e5","source":{"kind":"arxiv","id":"2406.14808","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.14808","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"arxiv_version","alias_value":"2406.14808v1","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.14808","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"pith_short_12","alias_value":"6HXROIZI76OM","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"pith_short_16","alias_value":"6HXROIZI76OMQWKG","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"pith_short_8","alias_value":"6HXROIZI","created_at":"2026-07-05T08:35:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:6HXROIZI76OMQWKGATJEF2H44P","target":"record","payload":{"canonical_record":{"source":{"id":"2406.14808","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2024-06-21T01:13:18Z","cross_cats_sorted":["cs.LG","stat.ME","stat.ML","stat.TH"],"title_canon_sha256":"d456270de209334f204b92630e2e6d7553398165ca3f235a59e0ece994cabe00","abstract_canon_sha256":"30ac2de9cf8738a9094212340f1325319ba8ecf6243af408b24f71efd906d82b"},"schema_version":"1.0"},"canonical_sha256":"f1ef172328ff9cc8594604d242e8fce3e34e84442f181930ad5bd06f4d4f90e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:35:08.696883Z","signature_b64":"38kB4c6Y/jZYoDoS5KcjFkrJkJ8aD+u5Mo5/qNRftisvQ9f1+GjP+y52jMI/LL4h9zzTLoBaZvv9Xl3WVuKxBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f1ef172328ff9cc8594604d242e8fce3e34e84442f181930ad5bd06f4d4f90e5","last_reissued_at":"2026-07-05T08:35:08.696457Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:35:08.696457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2406.14808","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:35:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lahIPRMBkvs3D9nbUBAxFPcJ+ceJMlmA9EkXJxVIl5Z5kXs7VwJBRRED/jRgJVis+38HQRi1uozf+Sdg+8FUCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T16:06:03.850772Z"},"content_sha256":"67981b872560e958ccc0b956f70ff023c9c106f061403a084e62dbf199e8fa0a","schema_version":"1.0","event_id":"sha256:67981b872560e958ccc0b956f70ff023c9c106f061403a084e62dbf199e8fa0a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:6HXROIZI76OMQWKGATJEF2H44P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the estimation rate of Bayesian PINN for inverse problems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","stat.ME","stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"Debarghya Mukherjee, Yi Sun, Yves Atchade","submitted_at":"2024-06-21T01:13:18Z","abstract_excerpt":"Solving partial differential equations (PDEs) and their inverse problems using Physics-informed neural networks (PINNs) is a rapidly growing approach in the physics and machine learning community. Although several architectures exist for PINNs that work remarkably in practice, our theoretical understanding of their performances is somewhat limited. In this work, we study the behavior of a Bayesian PINN estimator of the solution of a PDE from $n$ independent noisy measurement of the solution. We focus on a class of equations that are linear in their parameters (with unknown coefficients $\\theta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.14808","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/2406.14808/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":"2026-07-05T08:35:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MVMTmQ6lqLu0QuYfsI6V05LbVV1boYUBeF/rd0TW60Uf16xIkVTk5MJ4LtXAG7xW9mbowfEtFvuvsNhNtQCwCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T16:06:03.851266Z"},"content_sha256":"7fbbdf80814455d260d5074e950a2c08e43198a0f7c737c6482f8cd6669dbc50","schema_version":"1.0","event_id":"sha256:7fbbdf80814455d260d5074e950a2c08e43198a0f7c737c6482f8cd6669dbc50"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6HXROIZI76OMQWKGATJEF2H44P/bundle.json","state_url":"https://pith.science/pith/6HXROIZI76OMQWKGATJEF2H44P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6HXROIZI76OMQWKGATJEF2H44P/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":"2026-08-06T16:06:03Z","links":{"resolver":"https://pith.science/pith/6HXROIZI76OMQWKGATJEF2H44P","bundle":"https://pith.science/pith/6HXROIZI76OMQWKGATJEF2H44P/bundle.json","state":"https://pith.science/pith/6HXROIZI76OMQWKGATJEF2H44P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6HXROIZI76OMQWKGATJEF2H44P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:6HXROIZI76OMQWKGATJEF2H44P","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":"30ac2de9cf8738a9094212340f1325319ba8ecf6243af408b24f71efd906d82b","cross_cats_sorted":["cs.LG","stat.ME","stat.ML","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2024-06-21T01:13:18Z","title_canon_sha256":"d456270de209334f204b92630e2e6d7553398165ca3f235a59e0ece994cabe00"},"schema_version":"1.0","source":{"id":"2406.14808","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.14808","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"arxiv_version","alias_value":"2406.14808v1","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.14808","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"pith_short_12","alias_value":"6HXROIZI76OM","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"pith_short_16","alias_value":"6HXROIZI76OMQWKG","created_at":"2026-07-05T08:35:08Z"},{"alias_kind":"pith_short_8","alias_value":"6HXROIZI","created_at":"2026-07-05T08:35:08Z"}],"graph_snapshots":[{"event_id":"sha256:7fbbdf80814455d260d5074e950a2c08e43198a0f7c737c6482f8cd6669dbc50","target":"graph","created_at":"2026-07-05T08:35:08Z","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/2406.14808/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Solving partial differential equations (PDEs) and their inverse problems using Physics-informed neural networks (PINNs) is a rapidly growing approach in the physics and machine learning community. Although several architectures exist for PINNs that work remarkably in practice, our theoretical understanding of their performances is somewhat limited. In this work, we study the behavior of a Bayesian PINN estimator of the solution of a PDE from $n$ independent noisy measurement of the solution. We focus on a class of equations that are linear in their parameters (with unknown coefficients $\\theta","authors_text":"Debarghya Mukherjee, Yi Sun, Yves Atchade","cross_cats":["cs.LG","stat.ME","stat.ML","stat.TH"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2024-06-21T01:13:18Z","title":"On the estimation rate of Bayesian PINN for inverse problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.14808","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":"sha256:67981b872560e958ccc0b956f70ff023c9c106f061403a084e62dbf199e8fa0a","target":"record","created_at":"2026-07-05T08:35:08Z","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":"30ac2de9cf8738a9094212340f1325319ba8ecf6243af408b24f71efd906d82b","cross_cats_sorted":["cs.LG","stat.ME","stat.ML","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2024-06-21T01:13:18Z","title_canon_sha256":"d456270de209334f204b92630e2e6d7553398165ca3f235a59e0ece994cabe00"},"schema_version":"1.0","source":{"id":"2406.14808","kind":"arxiv","version":1}},"canonical_sha256":"f1ef172328ff9cc8594604d242e8fce3e34e84442f181930ad5bd06f4d4f90e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f1ef172328ff9cc8594604d242e8fce3e34e84442f181930ad5bd06f4d4f90e5","first_computed_at":"2026-07-05T08:35:08.696457Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:35:08.696457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"38kB4c6Y/jZYoDoS5KcjFkrJkJ8aD+u5Mo5/qNRftisvQ9f1+GjP+y52jMI/LL4h9zzTLoBaZvv9Xl3WVuKxBA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:35:08.696883Z","signed_message":"canonical_sha256_bytes"},"source_id":"2406.14808","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:67981b872560e958ccc0b956f70ff023c9c106f061403a084e62dbf199e8fa0a","sha256:7fbbdf80814455d260d5074e950a2c08e43198a0f7c737c6482f8cd6669dbc50"],"state_sha256":"0bf67efa503ee149650775f4f6fbbf237ae188913ad45f84113bef6f88827b0d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KdbUnHvL4ylizM2bb5H0BEfZ2Hke0Fo1SRckGi5VelLDTKCyxWiAykS4PWdGsNWbOgrPPozIrbfQg07HE3FaAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T16:06:03.854801Z","bundle_sha256":"02b49e59451d8a1e71b83ff24a25c3e5e052583491459ea144401093e5a6b82d"}}