{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:L5Y6I26AQ7ERJKJHXV6P6YTXZA","short_pith_number":"pith:L5Y6I26A","canonical_record":{"source":{"id":"2402.05598","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-02-08T11:51:24Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"b28d6add82c2fa45b3e39e963e85b49c2e6c3c13c9da045ec5248c186ff11273","abstract_canon_sha256":"5f8ab734dc2368ac4ce7e95fc9a1674a1f8bfb4d3106f30f35c021c401c2f1d4"},"schema_version":"1.0"},"canonical_sha256":"5f71e46bc087c914a927bd7cff6277c80ca9fcd36060f10a07ae381bbfb0f770","source":{"kind":"arxiv","id":"2402.05598","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.05598","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"arxiv_version","alias_value":"2402.05598v1","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.05598","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"pith_short_12","alias_value":"L5Y6I26AQ7ER","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"pith_short_16","alias_value":"L5Y6I26AQ7ERJKJH","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"pith_short_8","alias_value":"L5Y6I26A","created_at":"2026-07-05T07:42:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:L5Y6I26AQ7ERJKJHXV6P6YTXZA","target":"record","payload":{"canonical_record":{"source":{"id":"2402.05598","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-02-08T11:51:24Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"b28d6add82c2fa45b3e39e963e85b49c2e6c3c13c9da045ec5248c186ff11273","abstract_canon_sha256":"5f8ab734dc2368ac4ce7e95fc9a1674a1f8bfb4d3106f30f35c021c401c2f1d4"},"schema_version":"1.0"},"canonical_sha256":"5f71e46bc087c914a927bd7cff6277c80ca9fcd36060f10a07ae381bbfb0f770","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:42:53.239935Z","signature_b64":"rpVu7z1QzSuDWEogTD3Xfqy8j+uKNA+9tvwss83AdNxUZBqTve/cZqzeHBUzYQ12HrBWtRR1ja1jBRDA20aIAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f71e46bc087c914a927bd7cff6277c80ca9fcd36060f10a07ae381bbfb0f770","last_reissued_at":"2026-07-05T07:42:53.239450Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:42:53.239450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.05598","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":"2026-07-05T07:42:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RpT6YoAyqE+lNMoIVupNlXfoQjjGNZlcEk6yHZasSdXJOSmayb9eJZeekmgFJdk9x+O4jlb1zkWarBJWibcUDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T20:50:57.355111Z"},"content_sha256":"829b94d3dbbe0b2625c063cdee6c346e78453bf72bd968a8198783002f8e1432","schema_version":"1.0","event_id":"sha256:829b94d3dbbe0b2625c063cdee6c346e78453bf72bd968a8198783002f8e1432"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:L5Y6I26AQ7ERJKJHXV6P6YTXZA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Neural operators meet conjugate gradients: The FCG-NO method for efficient PDE solving","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Alexander Rudikov, Ekaterina Muravleva, Ivan Oseledets, Vladimir Fanaskov, Yuri M. Laevsky","submitted_at":"2024-02-08T11:51:24Z","abstract_excerpt":"Deep learning solvers for partial differential equations typically have limited accuracy. We propose to overcome this problem by using them as preconditioners. More specifically, we apply discretization-invariant neural operators to learn preconditioners for the flexible conjugate gradient method (FCG). Architecture paired with novel loss function and training scheme allows for learning efficient preconditioners that can be used across different resolutions. On the theoretical side, FCG theory allows us to safely use nonlinear preconditioners that can be applied in $O(N)$ operations without co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.05598","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/2402.05598/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":"2026-07-05T07:42:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GpjzKbwc4CHSljifqY5uqXXOuMdDDpwoM1SEV+hh1IlLGmVmI3SZd82Bm2KkzrcgjhwxCLmkP9T0Z4gCD2q8Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T20:50:57.355628Z"},"content_sha256":"b4dc397244ad68eaf98cd5c25a230ca17445a31547e90fe401cb536ffc0ef747","schema_version":"1.0","event_id":"sha256:b4dc397244ad68eaf98cd5c25a230ca17445a31547e90fe401cb536ffc0ef747"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L5Y6I26AQ7ERJKJHXV6P6YTXZA/bundle.json","state_url":"https://pith.science/pith/L5Y6I26AQ7ERJKJHXV6P6YTXZA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L5Y6I26AQ7ERJKJHXV6P6YTXZA/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":"2026-08-23T20:50:57Z","links":{"resolver":"https://pith.science/pith/L5Y6I26AQ7ERJKJHXV6P6YTXZA","bundle":"https://pith.science/pith/L5Y6I26AQ7ERJKJHXV6P6YTXZA/bundle.json","state":"https://pith.science/pith/L5Y6I26AQ7ERJKJHXV6P6YTXZA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L5Y6I26AQ7ERJKJHXV6P6YTXZA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:L5Y6I26AQ7ERJKJHXV6P6YTXZA","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":"5f8ab734dc2368ac4ce7e95fc9a1674a1f8bfb4d3106f30f35c021c401c2f1d4","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-02-08T11:51:24Z","title_canon_sha256":"b28d6add82c2fa45b3e39e963e85b49c2e6c3c13c9da045ec5248c186ff11273"},"schema_version":"1.0","source":{"id":"2402.05598","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.05598","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"arxiv_version","alias_value":"2402.05598v1","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.05598","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"pith_short_12","alias_value":"L5Y6I26AQ7ER","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"pith_short_16","alias_value":"L5Y6I26AQ7ERJKJH","created_at":"2026-07-05T07:42:53Z"},{"alias_kind":"pith_short_8","alias_value":"L5Y6I26A","created_at":"2026-07-05T07:42:53Z"}],"graph_snapshots":[{"event_id":"sha256:b4dc397244ad68eaf98cd5c25a230ca17445a31547e90fe401cb536ffc0ef747","target":"graph","created_at":"2026-07-05T07:42:53Z","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/2402.05598/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Deep learning solvers for partial differential equations typically have limited accuracy. We propose to overcome this problem by using them as preconditioners. More specifically, we apply discretization-invariant neural operators to learn preconditioners for the flexible conjugate gradient method (FCG). Architecture paired with novel loss function and training scheme allows for learning efficient preconditioners that can be used across different resolutions. On the theoretical side, FCG theory allows us to safely use nonlinear preconditioners that can be applied in $O(N)$ operations without co","authors_text":"Alexander Rudikov, Ekaterina Muravleva, Ivan Oseledets, Vladimir Fanaskov, Yuri M. Laevsky","cross_cats":["cs.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-02-08T11:51:24Z","title":"Neural operators meet conjugate gradients: The FCG-NO method for efficient PDE solving"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.05598","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":"sha256:829b94d3dbbe0b2625c063cdee6c346e78453bf72bd968a8198783002f8e1432","target":"record","created_at":"2026-07-05T07:42:53Z","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":"5f8ab734dc2368ac4ce7e95fc9a1674a1f8bfb4d3106f30f35c021c401c2f1d4","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2024-02-08T11:51:24Z","title_canon_sha256":"b28d6add82c2fa45b3e39e963e85b49c2e6c3c13c9da045ec5248c186ff11273"},"schema_version":"1.0","source":{"id":"2402.05598","kind":"arxiv","version":1}},"canonical_sha256":"5f71e46bc087c914a927bd7cff6277c80ca9fcd36060f10a07ae381bbfb0f770","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f71e46bc087c914a927bd7cff6277c80ca9fcd36060f10a07ae381bbfb0f770","first_computed_at":"2026-07-05T07:42:53.239450Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:42:53.239450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rpVu7z1QzSuDWEogTD3Xfqy8j+uKNA+9tvwss83AdNxUZBqTve/cZqzeHBUzYQ12HrBWtRR1ja1jBRDA20aIAg==","signature_status":"signed_v1","signed_at":"2026-07-05T07:42:53.239935Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.05598","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:829b94d3dbbe0b2625c063cdee6c346e78453bf72bd968a8198783002f8e1432","sha256:b4dc397244ad68eaf98cd5c25a230ca17445a31547e90fe401cb536ffc0ef747"],"state_sha256":"e70e97861f55e3a3351af1c5c558ad45104f6b436c9cba45dc72f226005c5793"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QIP55oA3ZhfsxmwbYrRGAPOd20ZQxHoSLC/EHa4D6FoESInsLTmYpU+URdRnGbRKz8VkMJqKu4ipKnUiVCFtAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T20:50:57.359395Z","bundle_sha256":"ffb5f361bbed55e2bcc1ad181bf1bdbb47412f5048976bf7c03c7d3412e1e18d"}}