{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:6U37AWUW6EHU76EHQ4HMMDQT7S","short_pith_number":"pith:6U37AWUW","canonical_record":{"source":{"id":"2504.10273","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-04-14T14:40:11Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"1043439825be59dc08b4bd647dd8a77257f06c118b1b9f0f4ace095b27a8398e","abstract_canon_sha256":"df01ce08f1ff48efc39b53b05cc05503b75190218d13f803fe46244f792f64df"},"schema_version":"1.0"},"canonical_sha256":"f537f05a96f10f4ff887870ec60e13fca6442b7030278e22b3476ae4a4e54e88","source":{"kind":"arxiv","id":"2504.10273","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.10273","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"arxiv_version","alias_value":"2504.10273v2","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.10273","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"pith_short_12","alias_value":"6U37AWUW6EHU","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"pith_short_16","alias_value":"6U37AWUW6EHU76EH","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"pith_short_8","alias_value":"6U37AWUW","created_at":"2026-07-05T11:49:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:6U37AWUW6EHU76EHQ4HMMDQT7S","target":"record","payload":{"canonical_record":{"source":{"id":"2504.10273","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-04-14T14:40:11Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"1043439825be59dc08b4bd647dd8a77257f06c118b1b9f0f4ace095b27a8398e","abstract_canon_sha256":"df01ce08f1ff48efc39b53b05cc05503b75190218d13f803fe46244f792f64df"},"schema_version":"1.0"},"canonical_sha256":"f537f05a96f10f4ff887870ec60e13fca6442b7030278e22b3476ae4a4e54e88","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:49:53.336713Z","signature_b64":"DknMjcVmAD8VJEFFUlfHDw/a+Mii1bGxbNYl+EZnpRS8Risbo7hzrNqmZPzuAUDuLDjVjajjAWxKtHHWC1VADQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f537f05a96f10f4ff887870ec60e13fca6442b7030278e22b3476ae4a4e54e88","last_reissued_at":"2026-07-05T11:49:53.336210Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:49:53.336210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2504.10273","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":"2026-07-05T11:49:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KRH541T8X2iKSYQ42ETjowKYjpRQ0pYZ5rLIyvVmRuGCeyxSBVqbOy/kPrbigrU3lHp3dXViFd5QIWkLeEWMCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T01:46:49.676320Z"},"content_sha256":"fc813c54a04d94bbf2e4ef28bc48af3213feda6cb18a6a4f295225bdc5b050ed","schema_version":"1.0","event_id":"sha256:fc813c54a04d94bbf2e4ef28bc48af3213feda6cb18a6a4f295225bdc5b050ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:6U37AWUW6EHU76EHQ4HMMDQT7S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"cs.LG","authors_text":"Gaohang Chen, Lili Ju, Zhonghua Qiao","submitted_at":"2025-04-14T14:40:11Z","abstract_excerpt":"Solving partial differential equations (PDEs) with neural networks (NNs) has shown great potential in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDE formulas in the strong or weak sense, without explicitly considering some intrinsic physical properties, such as mass and momentum conservation, or energy dissipation. This limitation may result in nonphysical or unstable numerical solutions, particularly in long-term simulations. To address this issue, we propose ``Sidecar'', a novel framework that enhances the physical consis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.10273","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/2504.10273/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":"2026-07-05T11:49:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yNOSipryJkMsNkY6/N2j1RzJ3SSm3OqT9TLYdy4gm+MtDFmJEyAzupGkZ9jlXZrGXje9XO5bTyCtdobKJk9hBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T01:46:49.676969Z"},"content_sha256":"bd212aa5454b7dccc82066009bccfaa17122e918449f8800c08b5593ba8ad781","schema_version":"1.0","event_id":"sha256:bd212aa5454b7dccc82066009bccfaa17122e918449f8800c08b5593ba8ad781"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6U37AWUW6EHU76EHQ4HMMDQT7S/bundle.json","state_url":"https://pith.science/pith/6U37AWUW6EHU76EHQ4HMMDQT7S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6U37AWUW6EHU76EHQ4HMMDQT7S/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":"2026-08-16T01:46:49Z","links":{"resolver":"https://pith.science/pith/6U37AWUW6EHU76EHQ4HMMDQT7S","bundle":"https://pith.science/pith/6U37AWUW6EHU76EHQ4HMMDQT7S/bundle.json","state":"https://pith.science/pith/6U37AWUW6EHU76EHQ4HMMDQT7S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6U37AWUW6EHU76EHQ4HMMDQT7S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:6U37AWUW6EHU76EHQ4HMMDQT7S","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":"df01ce08f1ff48efc39b53b05cc05503b75190218d13f803fe46244f792f64df","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-04-14T14:40:11Z","title_canon_sha256":"1043439825be59dc08b4bd647dd8a77257f06c118b1b9f0f4ace095b27a8398e"},"schema_version":"1.0","source":{"id":"2504.10273","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.10273","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"arxiv_version","alias_value":"2504.10273v2","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.10273","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"pith_short_12","alias_value":"6U37AWUW6EHU","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"pith_short_16","alias_value":"6U37AWUW6EHU76EH","created_at":"2026-07-05T11:49:53Z"},{"alias_kind":"pith_short_8","alias_value":"6U37AWUW","created_at":"2026-07-05T11:49:53Z"}],"graph_snapshots":[{"event_id":"sha256:bd212aa5454b7dccc82066009bccfaa17122e918449f8800c08b5593ba8ad781","target":"graph","created_at":"2026-07-05T11:49:53Z","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/2504.10273/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Solving partial differential equations (PDEs) with neural networks (NNs) has shown great potential in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDE formulas in the strong or weak sense, without explicitly considering some intrinsic physical properties, such as mass and momentum conservation, or energy dissipation. This limitation may result in nonphysical or unstable numerical solutions, particularly in long-term simulations. To address this issue, we propose ``Sidecar'', a novel framework that enhances the physical consis","authors_text":"Gaohang Chen, Lili Ju, Zhonghua Qiao","cross_cats":["cs.NA","math.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-04-14T14:40:11Z","title":"A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.10273","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:fc813c54a04d94bbf2e4ef28bc48af3213feda6cb18a6a4f295225bdc5b050ed","target":"record","created_at":"2026-07-05T11:49:53Z","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":"df01ce08f1ff48efc39b53b05cc05503b75190218d13f803fe46244f792f64df","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-04-14T14:40:11Z","title_canon_sha256":"1043439825be59dc08b4bd647dd8a77257f06c118b1b9f0f4ace095b27a8398e"},"schema_version":"1.0","source":{"id":"2504.10273","kind":"arxiv","version":2}},"canonical_sha256":"f537f05a96f10f4ff887870ec60e13fca6442b7030278e22b3476ae4a4e54e88","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f537f05a96f10f4ff887870ec60e13fca6442b7030278e22b3476ae4a4e54e88","first_computed_at":"2026-07-05T11:49:53.336210Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:49:53.336210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DknMjcVmAD8VJEFFUlfHDw/a+Mii1bGxbNYl+EZnpRS8Risbo7hzrNqmZPzuAUDuLDjVjajjAWxKtHHWC1VADQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:49:53.336713Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.10273","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fc813c54a04d94bbf2e4ef28bc48af3213feda6cb18a6a4f295225bdc5b050ed","sha256:bd212aa5454b7dccc82066009bccfaa17122e918449f8800c08b5593ba8ad781"],"state_sha256":"d9c653c6dc436f09b9e03acdb398b680aca65131e3f7f26459086d5a4d9f37cb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JqsyoXBqEq/7LlDw6loHUu3L01nyD9DAaSy8Rq3gspGxgcv01XcH9RirZjVLj6gJ493NubT20GXR14MdLDDYCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T01:46:49.683097Z","bundle_sha256":"215afb8d73f0e1fc505ec52c4a50fae0bc9a8d4eaf3c9938dd0f5f4b12319974"}}