{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:G4VKWWZ2Q5TUSDPU7S2T2UP26Z","short_pith_number":"pith:G4VKWWZ2","canonical_record":{"source":{"id":"2202.06137","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2022-02-12T20:37:04Z","cross_cats_sorted":["physics.comp-ph"],"title_canon_sha256":"d7e10a775bc6c0713b85c66ed9718aac50da348840b13bb74ada30024968ff2f","abstract_canon_sha256":"a7d83a703a72b22f483bf7d2931fdfa8eb3e19ca9d37817067fabbeb511122cb"},"schema_version":"1.0"},"canonical_sha256":"372aab5b3a8767490df4fcb53d51faf64e6179118eee5f725632c4ba7290f8f7","source":{"kind":"arxiv","id":"2202.06137","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2202.06137","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"arxiv_version","alias_value":"2202.06137v1","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.06137","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"pith_short_12","alias_value":"G4VKWWZ2Q5TU","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"pith_short_16","alias_value":"G4VKWWZ2Q5TUSDPU","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"pith_short_8","alias_value":"G4VKWWZ2","created_at":"2026-07-05T03:56:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:G4VKWWZ2Q5TUSDPU7S2T2UP26Z","target":"record","payload":{"canonical_record":{"source":{"id":"2202.06137","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2022-02-12T20:37:04Z","cross_cats_sorted":["physics.comp-ph"],"title_canon_sha256":"d7e10a775bc6c0713b85c66ed9718aac50da348840b13bb74ada30024968ff2f","abstract_canon_sha256":"a7d83a703a72b22f483bf7d2931fdfa8eb3e19ca9d37817067fabbeb511122cb"},"schema_version":"1.0"},"canonical_sha256":"372aab5b3a8767490df4fcb53d51faf64e6179118eee5f725632c4ba7290f8f7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:56:21.056629Z","signature_b64":"U9sJkxnY4c8WuyIm7/NlZEEk3c0AQNDXBBzTHLr0yplsFvS+WuGKFt8cv8UbTsPnUZepDlrg78117YjHWo0qDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"372aab5b3a8767490df4fcb53d51faf64e6179118eee5f725632c4ba7290f8f7","last_reissued_at":"2026-07-05T03:56:21.056213Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:56:21.056213Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2202.06137","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":"2026-07-05T03:56:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jZACYTgLL+SToM8wP7VU//iLAnSWSFXIRHDQaFp/1/6HQ0Z6EdUJV7ZCgPg5mb6qDj68qbsxphppj88ptUoKCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T11:02:29.948869Z"},"content_sha256":"f698d9f25f0a5a17ed3b4dad2bf9796111683e7092c5a893c30a8d35c8a1647c","schema_version":"1.0","event_id":"sha256:f698d9f25f0a5a17ed3b4dad2bf9796111683e7092c5a893c30a8d35c8a1647c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:G4VKWWZ2Q5TUSDPU7S2T2UP26Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"MIONet: Learning multiple-input operators via tensor product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["physics.comp-ph"],"primary_cat":"cs.LG","authors_text":"Lu Lu, Pengzhan Jin, Shuai Meng","submitted_at":"2022-02-12T20:37:04Z","abstract_excerpt":"As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all the existing neural operators are only designed to learn operators defined on a single Banach space, i.e., the input of the operator is a single function. Here, for the first time, we study the operator regression via neural networks for multiple-input operators defined on the product of Banach spaces. We first prove a universal approximation theorem of con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.06137","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/2202.06137/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":"2026-07-05T03:56:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3SvMtNp+WUX6tvEkZHLKEWd6kuL0WQEDGQrTsCoOcu/bHHHJSYC7V6Kc0RgKsfsNlpUlQtgZy5DDgaNdhRgjAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T11:02:29.949364Z"},"content_sha256":"11e4ba72f8ec86b97f45dab5314378539667a474c6dbe225a7605e6edd03a3ec","schema_version":"1.0","event_id":"sha256:11e4ba72f8ec86b97f45dab5314378539667a474c6dbe225a7605e6edd03a3ec"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G4VKWWZ2Q5TUSDPU7S2T2UP26Z/bundle.json","state_url":"https://pith.science/pith/G4VKWWZ2Q5TUSDPU7S2T2UP26Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G4VKWWZ2Q5TUSDPU7S2T2UP26Z/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":"2026-08-09T11:02:29Z","links":{"resolver":"https://pith.science/pith/G4VKWWZ2Q5TUSDPU7S2T2UP26Z","bundle":"https://pith.science/pith/G4VKWWZ2Q5TUSDPU7S2T2UP26Z/bundle.json","state":"https://pith.science/pith/G4VKWWZ2Q5TUSDPU7S2T2UP26Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G4VKWWZ2Q5TUSDPU7S2T2UP26Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:G4VKWWZ2Q5TUSDPU7S2T2UP26Z","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":"a7d83a703a72b22f483bf7d2931fdfa8eb3e19ca9d37817067fabbeb511122cb","cross_cats_sorted":["physics.comp-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2022-02-12T20:37:04Z","title_canon_sha256":"d7e10a775bc6c0713b85c66ed9718aac50da348840b13bb74ada30024968ff2f"},"schema_version":"1.0","source":{"id":"2202.06137","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2202.06137","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"arxiv_version","alias_value":"2202.06137v1","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.06137","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"pith_short_12","alias_value":"G4VKWWZ2Q5TU","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"pith_short_16","alias_value":"G4VKWWZ2Q5TUSDPU","created_at":"2026-07-05T03:56:21Z"},{"alias_kind":"pith_short_8","alias_value":"G4VKWWZ2","created_at":"2026-07-05T03:56:21Z"}],"graph_snapshots":[{"event_id":"sha256:11e4ba72f8ec86b97f45dab5314378539667a474c6dbe225a7605e6edd03a3ec","target":"graph","created_at":"2026-07-05T03:56:21Z","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/2202.06137/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all the existing neural operators are only designed to learn operators defined on a single Banach space, i.e., the input of the operator is a single function. Here, for the first time, we study the operator regression via neural networks for multiple-input operators defined on the product of Banach spaces. We first prove a universal approximation theorem of con","authors_text":"Lu Lu, Pengzhan Jin, Shuai Meng","cross_cats":["physics.comp-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2022-02-12T20:37:04Z","title":"MIONet: Learning multiple-input operators via tensor product"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.06137","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:f698d9f25f0a5a17ed3b4dad2bf9796111683e7092c5a893c30a8d35c8a1647c","target":"record","created_at":"2026-07-05T03:56:21Z","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":"a7d83a703a72b22f483bf7d2931fdfa8eb3e19ca9d37817067fabbeb511122cb","cross_cats_sorted":["physics.comp-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2022-02-12T20:37:04Z","title_canon_sha256":"d7e10a775bc6c0713b85c66ed9718aac50da348840b13bb74ada30024968ff2f"},"schema_version":"1.0","source":{"id":"2202.06137","kind":"arxiv","version":1}},"canonical_sha256":"372aab5b3a8767490df4fcb53d51faf64e6179118eee5f725632c4ba7290f8f7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"372aab5b3a8767490df4fcb53d51faf64e6179118eee5f725632c4ba7290f8f7","first_computed_at":"2026-07-05T03:56:21.056213Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:56:21.056213Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U9sJkxnY4c8WuyIm7/NlZEEk3c0AQNDXBBzTHLr0yplsFvS+WuGKFt8cv8UbTsPnUZepDlrg78117YjHWo0qDg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:56:21.056629Z","signed_message":"canonical_sha256_bytes"},"source_id":"2202.06137","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f698d9f25f0a5a17ed3b4dad2bf9796111683e7092c5a893c30a8d35c8a1647c","sha256:11e4ba72f8ec86b97f45dab5314378539667a474c6dbe225a7605e6edd03a3ec"],"state_sha256":"69ac2af771e7a078b24fd266de0197953c4be6e73114fc068199cb6ffc6f7ad0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o58x/ooBGdAMvK6/KfIcSFlSRwlRP5sWEmRiH/28ioviOfP+mtIxX3QLGzTqkhZ6mGsrV57B+2MMYgKDXq80Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T11:02:29.953584Z","bundle_sha256":"2a48d6b97d54329e47e406661dab3bc7a4592bcbaa459b5fdcec20e34b847b51"}}