{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:46A4TZE4XZCOYXZ2ARFQKFGTXV","short_pith_number":"pith:46A4TZE4","canonical_record":{"source":{"id":"2507.20065","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-26T21:28:25Z","cross_cats_sorted":[],"title_canon_sha256":"7489bd97eec9e28a4799cbcf20eb67ed21b910a1445c7372dd854ca60f54567e","abstract_canon_sha256":"b96e31ebe97f97ace9e4a4098dc0684ea2f6e48be22a510e322243581936af57"},"schema_version":"1.0"},"canonical_sha256":"e781c9e49cbe44ec5f3a044b0514d3bd45756699b3704de94aa4bd72e3c23cd3","source":{"kind":"arxiv","id":"2507.20065","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.20065","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"arxiv_version","alias_value":"2507.20065v1","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.20065","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"pith_short_12","alias_value":"46A4TZE4XZCO","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"pith_short_16","alias_value":"46A4TZE4XZCOYXZ2","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"pith_short_8","alias_value":"46A4TZE4","created_at":"2026-07-05T11:44:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:46A4TZE4XZCOYXZ2ARFQKFGTXV","target":"record","payload":{"canonical_record":{"source":{"id":"2507.20065","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-26T21:28:25Z","cross_cats_sorted":[],"title_canon_sha256":"7489bd97eec9e28a4799cbcf20eb67ed21b910a1445c7372dd854ca60f54567e","abstract_canon_sha256":"b96e31ebe97f97ace9e4a4098dc0684ea2f6e48be22a510e322243581936af57"},"schema_version":"1.0"},"canonical_sha256":"e781c9e49cbe44ec5f3a044b0514d3bd45756699b3704de94aa4bd72e3c23cd3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:44:12.743197Z","signature_b64":"ybkCmLmRTBu3YR7+t1/BzPM/QR6YlK7/iyjdY3VST5JTCoNy/5j3ne9PgHA2fvqZ7mBejKE9wqDVImAt/IIxCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e781c9e49cbe44ec5f3a044b0514d3bd45756699b3704de94aa4bd72e3c23cd3","last_reissued_at":"2026-07-05T11:44:12.742782Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:44:12.742782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.20065","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":"2026-07-05T11:44:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RSIG/iZ77ZdEjTDntPDc/pWS27JffX/vNXQb/TnZrXuKoO/h3ESK3gzym7hs0YqN5+w4UoY2nogPQfe1n5DZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T06:50:05.047056Z"},"content_sha256":"a5f82fdf4941a7707724a5beeb9e96c467cc5fb56b013e77714a1c2d44c5ddfb","schema_version":"1.0","event_id":"sha256:a5f82fdf4941a7707724a5beeb9e96c467cc5fb56b013e77714a1c2d44c5ddfb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:46A4TZE4XZCOYXZ2ARFQKFGTXV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Geometric Operator Learning with Optimal Transport","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Anima Anandkumar, Nikola Kovachki, Xinyi Li, Zongyi Li","submitted_at":"2025-07-26T21:28:25Z","abstract_excerpt":"We propose integrating optimal transport (OT) into operator learning for partial differential equations (PDEs) on complex geometries. Classical geometric learning methods typically represent domains as meshes, graphs, or point clouds. Our approach generalizes discretized meshes to mesh density functions, formulating geometry embedding as an OT problem that maps these functions to a uniform density in a reference space. Compared to previous methods relying on interpolation or shared deformation, our OT-based method employs instance-dependent deformation, offering enhanced flexibility and effect"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.20065","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/2507.20065/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":"2026-07-05T11:44:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VMqQDWgnzwCMc6MSjUVYR3hIR9sIsQ/7dCAH5eEPiQKjS5tkhS85AaBYxgxmf7zR1vUF5m2Kj/h3GTdstskvCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T06:50:05.048310Z"},"content_sha256":"1f0dc2edd55bb9dadae4256e5413d9e0e13b122165e96e0fb4aaefb40c20567b","schema_version":"1.0","event_id":"sha256:1f0dc2edd55bb9dadae4256e5413d9e0e13b122165e96e0fb4aaefb40c20567b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/46A4TZE4XZCOYXZ2ARFQKFGTXV/bundle.json","state_url":"https://pith.science/pith/46A4TZE4XZCOYXZ2ARFQKFGTXV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/46A4TZE4XZCOYXZ2ARFQKFGTXV/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":"2026-08-14T06:50:05Z","links":{"resolver":"https://pith.science/pith/46A4TZE4XZCOYXZ2ARFQKFGTXV","bundle":"https://pith.science/pith/46A4TZE4XZCOYXZ2ARFQKFGTXV/bundle.json","state":"https://pith.science/pith/46A4TZE4XZCOYXZ2ARFQKFGTXV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/46A4TZE4XZCOYXZ2ARFQKFGTXV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:46A4TZE4XZCOYXZ2ARFQKFGTXV","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":"b96e31ebe97f97ace9e4a4098dc0684ea2f6e48be22a510e322243581936af57","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-26T21:28:25Z","title_canon_sha256":"7489bd97eec9e28a4799cbcf20eb67ed21b910a1445c7372dd854ca60f54567e"},"schema_version":"1.0","source":{"id":"2507.20065","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.20065","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"arxiv_version","alias_value":"2507.20065v1","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.20065","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"pith_short_12","alias_value":"46A4TZE4XZCO","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"pith_short_16","alias_value":"46A4TZE4XZCOYXZ2","created_at":"2026-07-05T11:44:12Z"},{"alias_kind":"pith_short_8","alias_value":"46A4TZE4","created_at":"2026-07-05T11:44:12Z"}],"graph_snapshots":[{"event_id":"sha256:1f0dc2edd55bb9dadae4256e5413d9e0e13b122165e96e0fb4aaefb40c20567b","target":"graph","created_at":"2026-07-05T11:44:12Z","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/2507.20065/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We propose integrating optimal transport (OT) into operator learning for partial differential equations (PDEs) on complex geometries. Classical geometric learning methods typically represent domains as meshes, graphs, or point clouds. Our approach generalizes discretized meshes to mesh density functions, formulating geometry embedding as an OT problem that maps these functions to a uniform density in a reference space. Compared to previous methods relying on interpolation or shared deformation, our OT-based method employs instance-dependent deformation, offering enhanced flexibility and effect","authors_text":"Anima Anandkumar, Nikola Kovachki, Xinyi Li, Zongyi Li","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-26T21:28:25Z","title":"Geometric Operator Learning with Optimal Transport"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.20065","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:a5f82fdf4941a7707724a5beeb9e96c467cc5fb56b013e77714a1c2d44c5ddfb","target":"record","created_at":"2026-07-05T11:44:12Z","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":"b96e31ebe97f97ace9e4a4098dc0684ea2f6e48be22a510e322243581936af57","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-26T21:28:25Z","title_canon_sha256":"7489bd97eec9e28a4799cbcf20eb67ed21b910a1445c7372dd854ca60f54567e"},"schema_version":"1.0","source":{"id":"2507.20065","kind":"arxiv","version":1}},"canonical_sha256":"e781c9e49cbe44ec5f3a044b0514d3bd45756699b3704de94aa4bd72e3c23cd3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e781c9e49cbe44ec5f3a044b0514d3bd45756699b3704de94aa4bd72e3c23cd3","first_computed_at":"2026-07-05T11:44:12.742782Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:44:12.742782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ybkCmLmRTBu3YR7+t1/BzPM/QR6YlK7/iyjdY3VST5JTCoNy/5j3ne9PgHA2fvqZ7mBejKE9wqDVImAt/IIxCA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:44:12.743197Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.20065","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a5f82fdf4941a7707724a5beeb9e96c467cc5fb56b013e77714a1c2d44c5ddfb","sha256:1f0dc2edd55bb9dadae4256e5413d9e0e13b122165e96e0fb4aaefb40c20567b"],"state_sha256":"76f0a2ec03adca21f48944e2e513c5e549331a5610822b8afe936da999c5dfe9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dY2J4PGeh33NEKQSdJqQqs6fDJwytLyjor5wKqK7vSUvb260NvMS6sRqLy7iF2O/ZBbo6Ha43U8oVPyEVb5hBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T06:50:05.054325Z","bundle_sha256":"ef927ff475d90cd1f3f0015f46767602a26e119e8fc83d586cf074e7d963a849"}}