{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:6LXLEDAAFTCVE4MU2MLFP3EWF5","short_pith_number":"pith:6LXLEDAA","schema_version":"1.0","canonical_sha256":"f2eeb20c002cc5527194d31657ec962f7547a2637c639e78b5b94b1059cc480f","source":{"kind":"arxiv","id":"2506.05723","version":2},"attestation_state":"computed","paper":{"title":"Simulating Fokker-Planck equations via mean field control of score-based normalizing flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Mo Zhou, Stanley Osher, Wuchen Li","submitted_at":"2025-06-06T04:03:24Z","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.05723","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-06-06T04:03:24Z","cross_cats_sorted":[],"title_canon_sha256":"e108c539875d1a1cd84efb594af714e2c4898b165f10319ad815bac9eed59d71","abstract_canon_sha256":"fdbe48a84d6b2bfedc0c053a3389596c4bbd9f1800fb3ce70f977c96b38176dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:48:26.534417Z","signature_b64":"dLHpZJD285QqLkBC/ogqaEIoYJFAAfUbGC2xhSV6Ck3LZJeFmzdwuZcijUHaEnU1zO2GeGN55l8JUk/mLi11BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2eeb20c002cc5527194d31657ec962f7547a2637c639e78b5b94b1059cc480f","last_reissued_at":"2026-07-05T11:48:26.533844Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:48:26.533844Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Simulating Fokker-Planck equations via mean field control of score-based normalizing flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Mo Zhou, Stanley Osher, Wuchen Li","submitted_at":"2025-06-06T04:03:24Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.05723","kind":"arxiv","version":2},"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/2506.05723/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.05723","created_at":"2026-07-05T11:48:26.533922+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.05723v2","created_at":"2026-07-05T11:48:26.533922+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.05723","created_at":"2026-07-05T11:48:26.533922+00:00"},{"alias_kind":"pith_short_12","alias_value":"6LXLEDAAFTCV","created_at":"2026-07-05T11:48:26.533922+00:00"},{"alias_kind":"pith_short_16","alias_value":"6LXLEDAAFTCVE4MU","created_at":"2026-07-05T11:48:26.533922+00:00"},{"alias_kind":"pith_short_8","alias_value":"6LXLEDAA","created_at":"2026-07-05T11:48:26.533922+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6LXLEDAAFTCVE4MU2MLFP3EWF5","json":"https://pith.science/pith/6LXLEDAAFTCVE4MU2MLFP3EWF5.json","graph_json":"https://pith.science/api/pith-number/6LXLEDAAFTCVE4MU2MLFP3EWF5/graph.json","events_json":"https://pith.science/api/pith-number/6LXLEDAAFTCVE4MU2MLFP3EWF5/events.json","paper":"https://pith.science/paper/6LXLEDAA"},"agent_actions":{"view_html":"https://pith.science/pith/6LXLEDAAFTCVE4MU2MLFP3EWF5","download_json":"https://pith.science/pith/6LXLEDAAFTCVE4MU2MLFP3EWF5.json","view_paper":"https://pith.science/paper/6LXLEDAA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.05723&json=true","fetch_graph":"https://pith.science/api/pith-number/6LXLEDAAFTCVE4MU2MLFP3EWF5/graph.json","fetch_events":"https://pith.science/api/pith-number/6LXLEDAAFTCVE4MU2MLFP3EWF5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6LXLEDAAFTCVE4MU2MLFP3EWF5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6LXLEDAAFTCVE4MU2MLFP3EWF5/action/storage_attestation","attest_author":"https://pith.science/pith/6LXLEDAAFTCVE4MU2MLFP3EWF5/action/author_attestation","sign_citation":"https://pith.science/pith/6LXLEDAAFTCVE4MU2MLFP3EWF5/action/citation_signature","submit_replication":"https://pith.science/pith/6LXLEDAAFTCVE4MU2MLFP3EWF5/action/replication_record"}},"created_at":"2026-07-05T11:48:26.533922+00:00","updated_at":"2026-07-05T11:48:26.533922+00:00"}