{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:ULISY7SNBHJTWHRF3WCI5IJNK2","short_pith_number":"pith:ULISY7SN","canonical_record":{"source":{"id":"2410.02151","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-10-03T02:28:17Z","cross_cats_sorted":["cs.NA","math.NA","stat.ML"],"title_canon_sha256":"5cd1b0ee96b66b6ad76a7d6603aa6314dcf04fca07056017ceebb4035c144c02","abstract_canon_sha256":"255da6be4667492985f351223d4fea0ebc53a8be5d1f9b8b9eac17431de3faa5"},"schema_version":"1.0"},"canonical_sha256":"a2d12c7e4d09d33b1e25dd848ea12d56a7b4494b00c86e345f829195ea9b7f83","source":{"kind":"arxiv","id":"2410.02151","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.02151","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"arxiv_version","alias_value":"2410.02151v1","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.02151","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"pith_short_12","alias_value":"ULISY7SNBHJT","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"pith_short_16","alias_value":"ULISY7SNBHJTWHRF","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"pith_short_8","alias_value":"ULISY7SN","created_at":"2026-07-05T09:15:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:ULISY7SNBHJTWHRF3WCI5IJNK2","target":"record","payload":{"canonical_record":{"source":{"id":"2410.02151","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-10-03T02:28:17Z","cross_cats_sorted":["cs.NA","math.NA","stat.ML"],"title_canon_sha256":"5cd1b0ee96b66b6ad76a7d6603aa6314dcf04fca07056017ceebb4035c144c02","abstract_canon_sha256":"255da6be4667492985f351223d4fea0ebc53a8be5d1f9b8b9eac17431de3faa5"},"schema_version":"1.0"},"canonical_sha256":"a2d12c7e4d09d33b1e25dd848ea12d56a7b4494b00c86e345f829195ea9b7f83","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:15:16.938924Z","signature_b64":"5f9h14e+BGJZozHFAdrCT1bikJoGkNi6jZt6j5drX6VQacV/5Wsrw2Ir40wQIW6414IzO2FO4kWZdHcTaXRVCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2d12c7e4d09d33b1e25dd848ea12d56a7b4494b00c86e345f829195ea9b7f83","last_reissued_at":"2026-07-05T09:15:16.938466Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:15:16.938466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2410.02151","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":"2026-07-05T09:15:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zV8OLpr7uL3r25hzoW1bf9VbwWCtYY8lyrhh6WxVz04OB3Z4X0syspVD4L6JY0EIi9+ca4CVA4T2oF6XUJvvBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T15:08:53.027687Z"},"content_sha256":"181a501e9cbed24454f512a9c6cfd2766b0e0e2ed1a49deab947e44118ab44c4","schema_version":"1.0","event_id":"sha256:181a501e9cbed24454f512a9c6cfd2766b0e0e2ed1a49deab947e44118ab44c4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:ULISY7SNBHJTWHRF3WCI5IJNK2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.NA","stat.ML"],"primary_cat":"cs.LG","authors_text":"Koichi Taniguchi, Satoshi Okuda, Takashi Furuya","submitted_at":"2024-10-03T02:28:17Z","abstract_excerpt":"Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the quantitative approximation theorem for solution operators of nonlinear PDEs. Our results show that neural operators can efficiently approximate these solution operators without the exponential growth in model complexity, thus strengthening the theoretical foundation of neural operators. A key insight in our proof is to transfer PDEs into the corresponding i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.02151","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/2410.02151/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":"2026-07-05T09:15:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3cXtdPvN5fl8TDkcnsjyCp8G1nJBzpb6SRpVQnJYqNjHHUbfw3F5QIjfv067437Gv7d2fgTg3gVZjXKkgnjvCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T15:08:53.028193Z"},"content_sha256":"bc4a24d1fb8e2907d35f32a76ca43ca7131b1659b1524aefcb122c06355df5b9","schema_version":"1.0","event_id":"sha256:bc4a24d1fb8e2907d35f32a76ca43ca7131b1659b1524aefcb122c06355df5b9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ULISY7SNBHJTWHRF3WCI5IJNK2/bundle.json","state_url":"https://pith.science/pith/ULISY7SNBHJTWHRF3WCI5IJNK2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ULISY7SNBHJTWHRF3WCI5IJNK2/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":"2026-08-03T15:08:53Z","links":{"resolver":"https://pith.science/pith/ULISY7SNBHJTWHRF3WCI5IJNK2","bundle":"https://pith.science/pith/ULISY7SNBHJTWHRF3WCI5IJNK2/bundle.json","state":"https://pith.science/pith/ULISY7SNBHJTWHRF3WCI5IJNK2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ULISY7SNBHJTWHRF3WCI5IJNK2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:ULISY7SNBHJTWHRF3WCI5IJNK2","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":"255da6be4667492985f351223d4fea0ebc53a8be5d1f9b8b9eac17431de3faa5","cross_cats_sorted":["cs.NA","math.NA","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-10-03T02:28:17Z","title_canon_sha256":"5cd1b0ee96b66b6ad76a7d6603aa6314dcf04fca07056017ceebb4035c144c02"},"schema_version":"1.0","source":{"id":"2410.02151","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.02151","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"arxiv_version","alias_value":"2410.02151v1","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.02151","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"pith_short_12","alias_value":"ULISY7SNBHJT","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"pith_short_16","alias_value":"ULISY7SNBHJTWHRF","created_at":"2026-07-05T09:15:16Z"},{"alias_kind":"pith_short_8","alias_value":"ULISY7SN","created_at":"2026-07-05T09:15:16Z"}],"graph_snapshots":[{"event_id":"sha256:bc4a24d1fb8e2907d35f32a76ca43ca7131b1659b1524aefcb122c06355df5b9","target":"graph","created_at":"2026-07-05T09:15:16Z","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/2410.02151/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the quantitative approximation theorem for solution operators of nonlinear PDEs. Our results show that neural operators can efficiently approximate these solution operators without the exponential growth in model complexity, thus strengthening the theoretical foundation of neural operators. A key insight in our proof is to transfer PDEs into the corresponding i","authors_text":"Koichi Taniguchi, Satoshi Okuda, Takashi Furuya","cross_cats":["cs.NA","math.NA","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-10-03T02:28:17Z","title":"Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.02151","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":"sha256:181a501e9cbed24454f512a9c6cfd2766b0e0e2ed1a49deab947e44118ab44c4","target":"record","created_at":"2026-07-05T09:15:16Z","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":"255da6be4667492985f351223d4fea0ebc53a8be5d1f9b8b9eac17431de3faa5","cross_cats_sorted":["cs.NA","math.NA","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-10-03T02:28:17Z","title_canon_sha256":"5cd1b0ee96b66b6ad76a7d6603aa6314dcf04fca07056017ceebb4035c144c02"},"schema_version":"1.0","source":{"id":"2410.02151","kind":"arxiv","version":1}},"canonical_sha256":"a2d12c7e4d09d33b1e25dd848ea12d56a7b4494b00c86e345f829195ea9b7f83","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a2d12c7e4d09d33b1e25dd848ea12d56a7b4494b00c86e345f829195ea9b7f83","first_computed_at":"2026-07-05T09:15:16.938466Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:15:16.938466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5f9h14e+BGJZozHFAdrCT1bikJoGkNi6jZt6j5drX6VQacV/5Wsrw2Ir40wQIW6414IzO2FO4kWZdHcTaXRVCA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:15:16.938924Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.02151","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:181a501e9cbed24454f512a9c6cfd2766b0e0e2ed1a49deab947e44118ab44c4","sha256:bc4a24d1fb8e2907d35f32a76ca43ca7131b1659b1524aefcb122c06355df5b9"],"state_sha256":"e572d525bc57611d2b5b9a722b2be923de79e101cf775d44dd85db6ab60b4e84"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"byhBWqDkM1Vqsx2E7ybpIZGen2jh179m61QDLioF4gv/MC9orSSw/SCXMuZ42eAFliR7BchxE99cntnKyk0OAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T15:08:53.031934Z","bundle_sha256":"4fc1db0b989b46e87686e277cfcb89f8a8b0927a9c72d07f60ca594eeb9b3e44"}}