{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:O7HAV35LOBIULVBG3BZSNLK5EO","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":"2d6cc3352e9fa1d331c8c4bedf438aafb99f3bd481c7566b8496d41619695ecd","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2025-09-05T15:35:04Z","title_canon_sha256":"a58de4e373e03e59219cd2cd961ad65c7e689fbd1ac07e326fb5bedbb05b91aa"},"schema_version":"1.0","source":{"id":"2509.05186","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.05186","created_at":"2026-07-05T12:06:49Z"},{"alias_kind":"arxiv_version","alias_value":"2509.05186v2","created_at":"2026-07-05T12:06:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.05186","created_at":"2026-07-05T12:06:49Z"},{"alias_kind":"pith_short_12","alias_value":"O7HAV35LOBIU","created_at":"2026-07-05T12:06:49Z"},{"alias_kind":"pith_short_16","alias_value":"O7HAV35LOBIULVBG","created_at":"2026-07-05T12:06:49Z"},{"alias_kind":"pith_short_8","alias_value":"O7HAV35L","created_at":"2026-07-05T12:06:49Z"}],"graph_snapshots":[{"event_id":"sha256:308064e108ec535cc990846951fc0cfe719078ccd1da8886b5d172e1f06f5d06","target":"graph","created_at":"2026-07-05T12:06:49Z","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/2509.05186/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In-context operator networks (ICON) are a class of operator learning methods based on the novel architectures of foundation models. Trained on a diverse set of datasets of initial and boundary conditions paired with corresponding solutions to ordinary and partial differential equations (ODEs and PDEs), ICON learns to map example condition-solution pairs of a given differential equation to an approximation of its solution operator. Here, we present a probabilistic framework that reveals ICON as implicitly performing Bayesian inference, where it computes the mean of the posterior predictive dist","authors_text":"Benjamin J. Zhang, Markos A. Katsoulakis, Siting Liu, Stanley J. Osher","cross_cats":["cs.LG","cs.NA","math.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2025-09-05T15:35:04Z","title":"Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.05186","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:b88231ddd388e0e75c7b8ee00a98b649e3e2411be265af257595efd1be8e5fa8","target":"record","created_at":"2026-07-05T12:06:49Z","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":"2d6cc3352e9fa1d331c8c4bedf438aafb99f3bd481c7566b8496d41619695ecd","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2025-09-05T15:35:04Z","title_canon_sha256":"a58de4e373e03e59219cd2cd961ad65c7e689fbd1ac07e326fb5bedbb05b91aa"},"schema_version":"1.0","source":{"id":"2509.05186","kind":"arxiv","version":2}},"canonical_sha256":"77ce0aefab705145d426d87326ad5d23a7d8a3c93cb45920ee37b4ba140219a1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"77ce0aefab705145d426d87326ad5d23a7d8a3c93cb45920ee37b4ba140219a1","first_computed_at":"2026-07-05T12:06:49.950695Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:06:49.950695Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nVymPie+OONy/iWCEed5QH9fsR4tM1UwDTt7SPQc6iZXZ5AlZIb285jFDa0981nUYDqOtMVakDknBYiHCvAEAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T12:06:49.951189Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.05186","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b88231ddd388e0e75c7b8ee00a98b649e3e2411be265af257595efd1be8e5fa8","sha256:308064e108ec535cc990846951fc0cfe719078ccd1da8886b5d172e1f06f5d06"],"state_sha256":"3d6626f99f2b3b5d8b79d423cba8245aea004939534fa81020a8359d4bd42ff0"}