{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:6LXLEDAAFTCVE4MU2MLFP3EWF5","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":"fdbe48a84d6b2bfedc0c053a3389596c4bbd9f1800fb3ce70f977c96b38176dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-06-06T04:03:24Z","title_canon_sha256":"e108c539875d1a1cd84efb594af714e2c4898b165f10319ad815bac9eed59d71"},"schema_version":"1.0","source":{"id":"2506.05723","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.05723","created_at":"2026-07-05T11:48:26Z"},{"alias_kind":"arxiv_version","alias_value":"2506.05723v2","created_at":"2026-07-05T11:48:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.05723","created_at":"2026-07-05T11:48:26Z"},{"alias_kind":"pith_short_12","alias_value":"6LXLEDAAFTCV","created_at":"2026-07-05T11:48:26Z"},{"alias_kind":"pith_short_16","alias_value":"6LXLEDAAFTCVE4MU","created_at":"2026-07-05T11:48:26Z"},{"alias_kind":"pith_short_8","alias_value":"6LXLEDAA","created_at":"2026-07-05T11:48:26Z"}],"graph_snapshots":[{"event_id":"sha256:60f9ca5baa91761deb83abb9af7354100d5b126202d0761b739f4717c86d9bcb","target":"graph","created_at":"2026-07-05T11:48:26Z","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/2506.05723/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Fokker-Planck (FP) equation governs the evolution of densities for stochastic dynamics of physical systems, such as the Langevin dynamics and the Lorenz system. This work simulates FP equations through a mean field control (MFC) problem. We first formulate the FP equation as a continuity equation, where the velocity field consists of the drift function and the score function, i.e., the gradient of the logarithm of the density function. Next, we design a MFC problem that matches the velocity fields in a continuity equation with the ones in the FP equation. The score functions along determin","authors_text":"Mo Zhou, Stanley Osher, Wuchen Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-06-06T04:03:24Z","title":"Simulating Fokker-Planck equations via mean field control of score-based normalizing flows"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.05723","kind":"arxiv","version":2},"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":"sha256:5e6f144aa225b918896356211df91fafa2bd65f9242e0deea0fceeda0c049647","target":"record","created_at":"2026-07-05T11:48:26Z","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":"fdbe48a84d6b2bfedc0c053a3389596c4bbd9f1800fb3ce70f977c96b38176dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-06-06T04:03:24Z","title_canon_sha256":"e108c539875d1a1cd84efb594af714e2c4898b165f10319ad815bac9eed59d71"},"schema_version":"1.0","source":{"id":"2506.05723","kind":"arxiv","version":2}},"canonical_sha256":"f2eeb20c002cc5527194d31657ec962f7547a2637c639e78b5b94b1059cc480f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f2eeb20c002cc5527194d31657ec962f7547a2637c639e78b5b94b1059cc480f","first_computed_at":"2026-07-05T11:48:26.533844Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:48:26.533844Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dLHpZJD285QqLkBC/ogqaEIoYJFAAfUbGC2xhSV6Ck3LZJeFmzdwuZcijUHaEnU1zO2GeGN55l8JUk/mLi11BA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:48:26.534417Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.05723","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e6f144aa225b918896356211df91fafa2bd65f9242e0deea0fceeda0c049647","sha256:60f9ca5baa91761deb83abb9af7354100d5b126202d0761b739f4717c86d9bcb"],"state_sha256":"8f8e65645df6d4b47c744a0434abf624ce3e11a66ec50711fa2cd519ff53dc3c"}