{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:4AS2ZKMOYSXQNPV7YB34JPDMRG","short_pith_number":"pith:4AS2ZKMO","canonical_record":{"source":{"id":"2607.07637","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T16:55:16Z","cross_cats_sorted":["cs.NA","math.NA","math.OC"],"title_canon_sha256":"43a70e7b1ff9e03a87ce005bad5e46b7e643f5010e0f4877847c9af8e15feed7","abstract_canon_sha256":"35119e717249990fe4052de2b8bf831c40d3fa67fc7ec1a859530abc6c3d7d9d"},"schema_version":"1.0"},"canonical_sha256":"e025aca98ec4af06bebfc077c4bc6c898facda784a249858c50a7ed67961ffe3","source":{"kind":"arxiv","id":"2607.07637","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.07637","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"arxiv_version","alias_value":"2607.07637v1","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.07637","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"pith_short_12","alias_value":"4AS2ZKMOYSXQ","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"pith_short_16","alias_value":"4AS2ZKMOYSXQNPV7","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"pith_short_8","alias_value":"4AS2ZKMO","created_at":"2026-07-09T01:20:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:4AS2ZKMOYSXQNPV7YB34JPDMRG","target":"record","payload":{"canonical_record":{"source":{"id":"2607.07637","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T16:55:16Z","cross_cats_sorted":["cs.NA","math.NA","math.OC"],"title_canon_sha256":"43a70e7b1ff9e03a87ce005bad5e46b7e643f5010e0f4877847c9af8e15feed7","abstract_canon_sha256":"35119e717249990fe4052de2b8bf831c40d3fa67fc7ec1a859530abc6c3d7d9d"},"schema_version":"1.0"},"canonical_sha256":"e025aca98ec4af06bebfc077c4bc6c898facda784a249858c50a7ed67961ffe3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-09T01:20:37.632446Z","signature_b64":"IceCxUUbi1U1XOZyxQ+SAsI++6xAJxyHDlHALUVzMT6LpS3jgf3FX+NLoFnfYBCvZ9fhte12V6rXQ5spDf1fAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e025aca98ec4af06bebfc077c4bc6c898facda784a249858c50a7ed67961ffe3","last_reissued_at":"2026-07-09T01:20:37.632001Z","signature_status":"signed_v1","first_computed_at":"2026-07-09T01:20:37.632001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.07637","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":"2026-07-09T01:20:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x3hxmOAwGi8poOqnpWtWRtCi+GV4gXbI1aPha3suRy/roiMoBUK0LblJhdquZ4d0x/hfZJO6Rq29lsYjKR/JDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T18:43:43.855161Z"},"content_sha256":"34a76e8a2d3a17309797cc77185a3611d88d753ecb4bb356d2ab15c993cabe88","schema_version":"1.0","event_id":"sha256:34a76e8a2d3a17309797cc77185a3611d88d753ecb4bb356d2ab15c993cabe88"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:4AS2ZKMOYSXQNPV7YB34JPDMRG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An optimal control approach for neural network architecture adaptation with a posteriori error estimation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.NA","math.OC"],"primary_cat":"cs.LG","authors_text":"C G Krishnanunni, Tan Bui-Thanh, Thomas Scott","submitted_at":"2026-07-08T16:55:16Z","abstract_excerpt":"This work presents a novel approach for adapting neural network architecture along the depth based on a posteriori error estimation. By formulating neural network training as a continuous-time optimal control problem, we derive rigorous error estimates that quantify how approximation error distributes across network layers. This error decomposition enables a principled depth adaptation strategy: new layers are inserted at locations of maximum estimated error, allowing the network to efficiently capture complex, nonlinear variations in the underlying problem. Our framework introduces a novel ne"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07637","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/2607.07637/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":"2026-07-09T01:20:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pKUF1KKhCHD2xwuIGoZisrw2eMq0ypuO47XIK+J+Mq/d784JM4lPNQSdBp/Xu1ZBXChgLkKeyw91N93yeXQSDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T18:43:43.855687Z"},"content_sha256":"5ef0becae012294c5bc7159cc027f34e3cf22bea3926863ccd51b0d6abe9d98c","schema_version":"1.0","event_id":"sha256:5ef0becae012294c5bc7159cc027f34e3cf22bea3926863ccd51b0d6abe9d98c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4AS2ZKMOYSXQNPV7YB34JPDMRG/bundle.json","state_url":"https://pith.science/pith/4AS2ZKMOYSXQNPV7YB34JPDMRG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4AS2ZKMOYSXQNPV7YB34JPDMRG/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":"2026-08-10T18:43:43Z","links":{"resolver":"https://pith.science/pith/4AS2ZKMOYSXQNPV7YB34JPDMRG","bundle":"https://pith.science/pith/4AS2ZKMOYSXQNPV7YB34JPDMRG/bundle.json","state":"https://pith.science/pith/4AS2ZKMOYSXQNPV7YB34JPDMRG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4AS2ZKMOYSXQNPV7YB34JPDMRG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4AS2ZKMOYSXQNPV7YB34JPDMRG","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":"35119e717249990fe4052de2b8bf831c40d3fa67fc7ec1a859530abc6c3d7d9d","cross_cats_sorted":["cs.NA","math.NA","math.OC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T16:55:16Z","title_canon_sha256":"43a70e7b1ff9e03a87ce005bad5e46b7e643f5010e0f4877847c9af8e15feed7"},"schema_version":"1.0","source":{"id":"2607.07637","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.07637","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"arxiv_version","alias_value":"2607.07637v1","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.07637","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"pith_short_12","alias_value":"4AS2ZKMOYSXQ","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"pith_short_16","alias_value":"4AS2ZKMOYSXQNPV7","created_at":"2026-07-09T01:20:37Z"},{"alias_kind":"pith_short_8","alias_value":"4AS2ZKMO","created_at":"2026-07-09T01:20:37Z"}],"graph_snapshots":[{"event_id":"sha256:5ef0becae012294c5bc7159cc027f34e3cf22bea3926863ccd51b0d6abe9d98c","target":"graph","created_at":"2026-07-09T01:20:37Z","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/2607.07637/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This work presents a novel approach for adapting neural network architecture along the depth based on a posteriori error estimation. By formulating neural network training as a continuous-time optimal control problem, we derive rigorous error estimates that quantify how approximation error distributes across network layers. This error decomposition enables a principled depth adaptation strategy: new layers are inserted at locations of maximum estimated error, allowing the network to efficiently capture complex, nonlinear variations in the underlying problem. Our framework introduces a novel ne","authors_text":"C G Krishnanunni, Tan Bui-Thanh, Thomas Scott","cross_cats":["cs.NA","math.NA","math.OC"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T16:55:16Z","title":"An optimal control approach for neural network architecture adaptation with a posteriori error estimation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07637","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":"sha256:34a76e8a2d3a17309797cc77185a3611d88d753ecb4bb356d2ab15c993cabe88","target":"record","created_at":"2026-07-09T01:20:37Z","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":"35119e717249990fe4052de2b8bf831c40d3fa67fc7ec1a859530abc6c3d7d9d","cross_cats_sorted":["cs.NA","math.NA","math.OC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T16:55:16Z","title_canon_sha256":"43a70e7b1ff9e03a87ce005bad5e46b7e643f5010e0f4877847c9af8e15feed7"},"schema_version":"1.0","source":{"id":"2607.07637","kind":"arxiv","version":1}},"canonical_sha256":"e025aca98ec4af06bebfc077c4bc6c898facda784a249858c50a7ed67961ffe3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e025aca98ec4af06bebfc077c4bc6c898facda784a249858c50a7ed67961ffe3","first_computed_at":"2026-07-09T01:20:37.632001Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-09T01:20:37.632001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IceCxUUbi1U1XOZyxQ+SAsI++6xAJxyHDlHALUVzMT6LpS3jgf3FX+NLoFnfYBCvZ9fhte12V6rXQ5spDf1fAw==","signature_status":"signed_v1","signed_at":"2026-07-09T01:20:37.632446Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.07637","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:34a76e8a2d3a17309797cc77185a3611d88d753ecb4bb356d2ab15c993cabe88","sha256:5ef0becae012294c5bc7159cc027f34e3cf22bea3926863ccd51b0d6abe9d98c"],"state_sha256":"aaee7bb8800bb9ce1f5a79d86ed2f6eed6c7f22366177a46915defbaa5f9bc44"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"26U25h1NwDXczNGIDK8EiQSbaR0Y0hKYcplyiXGeRIFEkTdsjP0sOTQOFlZBQx8gGTALHcy8yBlpId9LAryKBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T18:43:43.860267Z","bundle_sha256":"07e0c2fae11dc6b4791bb9250d03eeed89ffff1c61fdd7da707775e4d5a992ca"}}