{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:AX336QV67MYPRHQ4WFMZHEI7VX","short_pith_number":"pith:AX336QV6","canonical_record":{"source":{"id":"2607.02203","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-02T14:11:20Z","cross_cats_sorted":["physics.flu-dyn"],"title_canon_sha256":"8a6d282ef55102c4ef6a99592b45fbd0b6c378392573a161d51ce5f6ec626139","abstract_canon_sha256":"641881df16ce9ba28b0231558d7be3ecf00061c42468deca16a56539c8ac7556"},"schema_version":"1.0"},"canonical_sha256":"05f7bf42befb30f89e1cb15993911faddb4a3fadaeedce4dfbd352f9f6bae0e7","source":{"kind":"arxiv","id":"2607.02203","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02203","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02203v1","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02203","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"pith_short_12","alias_value":"AX336QV67MYP","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"pith_short_16","alias_value":"AX336QV67MYPRHQ4","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"pith_short_8","alias_value":"AX336QV6","created_at":"2026-07-03T01:17:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:AX336QV67MYPRHQ4WFMZHEI7VX","target":"record","payload":{"canonical_record":{"source":{"id":"2607.02203","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-02T14:11:20Z","cross_cats_sorted":["physics.flu-dyn"],"title_canon_sha256":"8a6d282ef55102c4ef6a99592b45fbd0b6c378392573a161d51ce5f6ec626139","abstract_canon_sha256":"641881df16ce9ba28b0231558d7be3ecf00061c42468deca16a56539c8ac7556"},"schema_version":"1.0"},"canonical_sha256":"05f7bf42befb30f89e1cb15993911faddb4a3fadaeedce4dfbd352f9f6bae0e7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:45.032270Z","signature_b64":"bKpBlnFHwUURfnz9q2GvHjbpm/0kEOf0iUtrNx952AAZTRpx2dlDq4hJ74Wzt+I0yzEnfxsEVtbZQwlcQ3X4AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05f7bf42befb30f89e1cb15993911faddb4a3fadaeedce4dfbd352f9f6bae0e7","last_reissued_at":"2026-07-03T01:17:45.031875Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:45.031875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.02203","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":"2026-07-03T01:17:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tUlem/fdAix2HVhlp0CasmnyFscG0KTU1Ug2UtZAPMTnRd+VLC2YRQSDbZgHePwo3xmxuWquI1oV6VL3E550DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T20:47:51.128959Z"},"content_sha256":"4b9f2877186728cc8ff86b17b9cb5a755a26b610d039d15e98f9d94a12f102b5","schema_version":"1.0","event_id":"sha256:4b9f2877186728cc8ff86b17b9cb5a755a26b610d039d15e98f9d94a12f102b5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:AX336QV67MYPRHQ4WFMZHEI7VX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["physics.flu-dyn"],"primary_cat":"cs.LG","authors_text":"Amirhossein Arzani, Mojgan Alishiri","submitted_at":"2026-07-02T14:11:20Z","abstract_excerpt":"Operator learning has emerged as a powerful tool for modeling complex physical systems in functional spaces. However, their neural network-based architectures make them opaque models, obscuring the reasoning behind their predictions. In this work, we introduce a self-explainable operator learning framework that overcomes this challenge by reformulating operator learning as a linear combination of generalized functional linear models expressed through integral equations. Exploiting the additive decomposability of these integral equations, we divide the input domain into subdomains and compute l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02203","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/2607.02203/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":"2026-07-03T01:17:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G9uy84kwLMPB97c6tIhJ2RCMubl5hSy9haCXrfC87d3NTOwaSTfa1hfTo+t7CcD9a5+K7icxRB6AXB4C0D6/Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T20:47:51.129350Z"},"content_sha256":"cd4a5093e141ed1014c9b411cd7d9eb3878f97f39b3c1570b38723a81a90a64f","schema_version":"1.0","event_id":"sha256:cd4a5093e141ed1014c9b411cd7d9eb3878f97f39b3c1570b38723a81a90a64f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AX336QV67MYPRHQ4WFMZHEI7VX/bundle.json","state_url":"https://pith.science/pith/AX336QV67MYPRHQ4WFMZHEI7VX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AX336QV67MYPRHQ4WFMZHEI7VX/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":"2026-07-03T20:47:51Z","links":{"resolver":"https://pith.science/pith/AX336QV67MYPRHQ4WFMZHEI7VX","bundle":"https://pith.science/pith/AX336QV67MYPRHQ4WFMZHEI7VX/bundle.json","state":"https://pith.science/pith/AX336QV67MYPRHQ4WFMZHEI7VX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AX336QV67MYPRHQ4WFMZHEI7VX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:AX336QV67MYPRHQ4WFMZHEI7VX","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":"641881df16ce9ba28b0231558d7be3ecf00061c42468deca16a56539c8ac7556","cross_cats_sorted":["physics.flu-dyn"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-02T14:11:20Z","title_canon_sha256":"8a6d282ef55102c4ef6a99592b45fbd0b6c378392573a161d51ce5f6ec626139"},"schema_version":"1.0","source":{"id":"2607.02203","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02203","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02203v1","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02203","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"pith_short_12","alias_value":"AX336QV67MYP","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"pith_short_16","alias_value":"AX336QV67MYPRHQ4","created_at":"2026-07-03T01:17:45Z"},{"alias_kind":"pith_short_8","alias_value":"AX336QV6","created_at":"2026-07-03T01:17:45Z"}],"graph_snapshots":[{"event_id":"sha256:cd4a5093e141ed1014c9b411cd7d9eb3878f97f39b3c1570b38723a81a90a64f","target":"graph","created_at":"2026-07-03T01:17:45Z","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/2607.02203/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Operator learning has emerged as a powerful tool for modeling complex physical systems in functional spaces. However, their neural network-based architectures make them opaque models, obscuring the reasoning behind their predictions. In this work, we introduce a self-explainable operator learning framework that overcomes this challenge by reformulating operator learning as a linear combination of generalized functional linear models expressed through integral equations. Exploiting the additive decomposability of these integral equations, we divide the input domain into subdomains and compute l","authors_text":"Amirhossein Arzani, Mojgan Alishiri","cross_cats":["physics.flu-dyn"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-02T14:11:20Z","title":"Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02203","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":"sha256:4b9f2877186728cc8ff86b17b9cb5a755a26b610d039d15e98f9d94a12f102b5","target":"record","created_at":"2026-07-03T01:17:45Z","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":"641881df16ce9ba28b0231558d7be3ecf00061c42468deca16a56539c8ac7556","cross_cats_sorted":["physics.flu-dyn"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-02T14:11:20Z","title_canon_sha256":"8a6d282ef55102c4ef6a99592b45fbd0b6c378392573a161d51ce5f6ec626139"},"schema_version":"1.0","source":{"id":"2607.02203","kind":"arxiv","version":1}},"canonical_sha256":"05f7bf42befb30f89e1cb15993911faddb4a3fadaeedce4dfbd352f9f6bae0e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"05f7bf42befb30f89e1cb15993911faddb4a3fadaeedce4dfbd352f9f6bae0e7","first_computed_at":"2026-07-03T01:17:45.031875Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:45.031875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bKpBlnFHwUURfnz9q2GvHjbpm/0kEOf0iUtrNx952AAZTRpx2dlDq4hJ74Wzt+I0yzEnfxsEVtbZQwlcQ3X4AA==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:45.032270Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.02203","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b9f2877186728cc8ff86b17b9cb5a755a26b610d039d15e98f9d94a12f102b5","sha256:cd4a5093e141ed1014c9b411cd7d9eb3878f97f39b3c1570b38723a81a90a64f"],"state_sha256":"c683aabbcf2acbd3cdcbbebf9b8906a7d22fc360fa4d9fb712a8580929eacc87"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0ucWe7D59EZGKMgof7be6VwFsbnbtpO3VqJLaNBP1tgLWlz810Zgvf40Abz2INRrw1n8PtUu9CP7flGcskWhDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T20:47:51.131360Z","bundle_sha256":"8e928d5056f854a36bed7c101e707b25928debd4d4edd1fb1efdb473b431be1a"}}