{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:VTO45NSJGXBLQ4RM4NIIK5PRMV","short_pith_number":"pith:VTO45NSJ","canonical_record":{"source":{"id":"2308.01213","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2023-08-02T15:16:34Z","cross_cats_sorted":["cs.NE"],"title_canon_sha256":"ca8fdd67a65c85069c04c069e6b059dc5209531946e566b2e964c6fb0ad9ac0f","abstract_canon_sha256":"fa8e38fe37705bcf1c2030aac332fc663a32010461fd7664ec14988046ae0afd"},"schema_version":"1.0"},"canonical_sha256":"acddceb64935c2b8722ce3508575f1655a8e87ddc1d806d2d5066ef938e23c52","source":{"kind":"arxiv","id":"2308.01213","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2308.01213","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"arxiv_version","alias_value":"2308.01213v2","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2308.01213","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"pith_short_12","alias_value":"VTO45NSJGXBL","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"pith_short_16","alias_value":"VTO45NSJGXBLQ4RM","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"pith_short_8","alias_value":"VTO45NSJ","created_at":"2026-07-05T06:55:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:VTO45NSJGXBLQ4RM4NIIK5PRMV","target":"record","payload":{"canonical_record":{"source":{"id":"2308.01213","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2023-08-02T15:16:34Z","cross_cats_sorted":["cs.NE"],"title_canon_sha256":"ca8fdd67a65c85069c04c069e6b059dc5209531946e566b2e964c6fb0ad9ac0f","abstract_canon_sha256":"fa8e38fe37705bcf1c2030aac332fc663a32010461fd7664ec14988046ae0afd"},"schema_version":"1.0"},"canonical_sha256":"acddceb64935c2b8722ce3508575f1655a8e87ddc1d806d2d5066ef938e23c52","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:55:05.208454Z","signature_b64":"RjuY09nGzAi0fsXlCNsVmqMQDyHgf/oBRJsXhKAAV0mJlcV4PtRYhuGXUGmCV0AuAMIbyJ3ObMeTZ4HSdqYBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"acddceb64935c2b8722ce3508575f1655a8e87ddc1d806d2d5066ef938e23c52","last_reissued_at":"2026-07-05T06:55:05.207981Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:55:05.207981Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2308.01213","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:55:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0fubHenqr+yho/CiNhd7IBXIZCuy/EExdJxPrWPWBc1UycbxIaC67moimtuvY2xe7NgrExyfxAbHFRic5mVhAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T09:13:31.091126Z"},"content_sha256":"ae73c5612ada8c265d4e962c365a516a5f99d8c14ea7ec228c44b823bd318cce","schema_version":"1.0","event_id":"sha256:ae73c5612ada8c265d4e962c365a516a5f99d8c14ea7ec228c44b823bd318cce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:VTO45NSJGXBLQ4RM4NIIK5PRMV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Embedding Capabilities of Neural ODEs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NE"],"primary_cat":"math.DS","authors_text":"Christian Kuehn, Sara-Viola Kuntz","submitted_at":"2023-08-02T15:16:34Z","abstract_excerpt":"A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results about the exact embedding of maps in different neural ODE architectures in low and high dimension. The embedding capability of a neural ODE architecture can be increased by adding, for example, a linear layer, or augmenting the phase space. Yet, there is currently no systematic theory available and our work contributes towards this goal by developing various "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.01213","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/2308.01213/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":"2026-07-05T06:55:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f/czK6zfHppr0mc/id0NaRwk1jKfmWLI3mw2SYrKQHOT++RI0K9kXCvWzo4vfUwTLfh3rVrzdOv4EqsW+1X3Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T09:13:31.091683Z"},"content_sha256":"a3bf6e2ab0a4e62a6a7ebc251596eabad18d26e6ed67f78fabf0530a76ed304d","schema_version":"1.0","event_id":"sha256:a3bf6e2ab0a4e62a6a7ebc251596eabad18d26e6ed67f78fabf0530a76ed304d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VTO45NSJGXBLQ4RM4NIIK5PRMV/bundle.json","state_url":"https://pith.science/pith/VTO45NSJGXBLQ4RM4NIIK5PRMV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VTO45NSJGXBLQ4RM4NIIK5PRMV/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":"2026-08-05T09:13:31Z","links":{"resolver":"https://pith.science/pith/VTO45NSJGXBLQ4RM4NIIK5PRMV","bundle":"https://pith.science/pith/VTO45NSJGXBLQ4RM4NIIK5PRMV/bundle.json","state":"https://pith.science/pith/VTO45NSJGXBLQ4RM4NIIK5PRMV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VTO45NSJGXBLQ4RM4NIIK5PRMV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:VTO45NSJGXBLQ4RM4NIIK5PRMV","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":"fa8e38fe37705bcf1c2030aac332fc663a32010461fd7664ec14988046ae0afd","cross_cats_sorted":["cs.NE"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2023-08-02T15:16:34Z","title_canon_sha256":"ca8fdd67a65c85069c04c069e6b059dc5209531946e566b2e964c6fb0ad9ac0f"},"schema_version":"1.0","source":{"id":"2308.01213","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2308.01213","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"arxiv_version","alias_value":"2308.01213v2","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2308.01213","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"pith_short_12","alias_value":"VTO45NSJGXBL","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"pith_short_16","alias_value":"VTO45NSJGXBLQ4RM","created_at":"2026-07-05T06:55:05Z"},{"alias_kind":"pith_short_8","alias_value":"VTO45NSJ","created_at":"2026-07-05T06:55:05Z"}],"graph_snapshots":[{"event_id":"sha256:a3bf6e2ab0a4e62a6a7ebc251596eabad18d26e6ed67f78fabf0530a76ed304d","target":"graph","created_at":"2026-07-05T06:55:05Z","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/2308.01213/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results about the exact embedding of maps in different neural ODE architectures in low and high dimension. The embedding capability of a neural ODE architecture can be increased by adding, for example, a linear layer, or augmenting the phase space. Yet, there is currently no systematic theory available and our work contributes towards this goal by developing various ","authors_text":"Christian Kuehn, Sara-Viola Kuntz","cross_cats":["cs.NE"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2023-08-02T15:16:34Z","title":"Embedding Capabilities of Neural ODEs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.01213","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:ae73c5612ada8c265d4e962c365a516a5f99d8c14ea7ec228c44b823bd318cce","target":"record","created_at":"2026-07-05T06:55:05Z","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":"fa8e38fe37705bcf1c2030aac332fc663a32010461fd7664ec14988046ae0afd","cross_cats_sorted":["cs.NE"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2023-08-02T15:16:34Z","title_canon_sha256":"ca8fdd67a65c85069c04c069e6b059dc5209531946e566b2e964c6fb0ad9ac0f"},"schema_version":"1.0","source":{"id":"2308.01213","kind":"arxiv","version":2}},"canonical_sha256":"acddceb64935c2b8722ce3508575f1655a8e87ddc1d806d2d5066ef938e23c52","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"acddceb64935c2b8722ce3508575f1655a8e87ddc1d806d2d5066ef938e23c52","first_computed_at":"2026-07-05T06:55:05.207981Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:55:05.207981Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RjuY09nGzAi0fsXlCNsVmqMQDyHgf/oBRJsXhKAAV0mJlcV4PtRYhuGXUGmCV0AuAMIbyJ3ObMeTZ4HSdqYBAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T06:55:05.208454Z","signed_message":"canonical_sha256_bytes"},"source_id":"2308.01213","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ae73c5612ada8c265d4e962c365a516a5f99d8c14ea7ec228c44b823bd318cce","sha256:a3bf6e2ab0a4e62a6a7ebc251596eabad18d26e6ed67f78fabf0530a76ed304d"],"state_sha256":"ece40f1a887153b5d616baad8296312b6fd754af200d91322b53d9784cd1762c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b+scZSLZH0NgMTHzgfRcvAKiu8FIfolUcmNMGdTtYgXbs78gX++y4l/qAxKJhSZb23ljKlJk1/XDC7YpGrbQDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T09:13:31.097036Z","bundle_sha256":"802baabc2240b4274f436621576ec2327b42a60c63c5c2dcf4bb7d3d6058cf1c"}}