{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:UHE7NE4JQ52347DVETXNDK6QR7","short_pith_number":"pith:UHE7NE4J","canonical_record":{"source":{"id":"2507.05164","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-07T16:18:49Z","cross_cats_sorted":["cs.LG","nlin.AO"],"title_canon_sha256":"889725354f5bf2c708a830acd7667ff110dc372154a2a5d682a20a372bb2b1ef","abstract_canon_sha256":"0e04ac6d88bcd34170b672c043fe16ac08c4d7a19794f79d04e103f2df9108f3"},"schema_version":"1.0"},"canonical_sha256":"a1c9f693898775be7c7524eed1abd08fe69b4acf6643063c047d6f108bc150ed","source":{"kind":"arxiv","id":"2507.05164","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.05164","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"arxiv_version","alias_value":"2507.05164v2","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.05164","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"pith_short_12","alias_value":"UHE7NE4JQ523","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"pith_short_16","alias_value":"UHE7NE4JQ52347DV","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"pith_short_8","alias_value":"UHE7NE4J","created_at":"2026-06-19T16:09:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:UHE7NE4JQ52347DVETXNDK6QR7","target":"record","payload":{"canonical_record":{"source":{"id":"2507.05164","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-07T16:18:49Z","cross_cats_sorted":["cs.LG","nlin.AO"],"title_canon_sha256":"889725354f5bf2c708a830acd7667ff110dc372154a2a5d682a20a372bb2b1ef","abstract_canon_sha256":"0e04ac6d88bcd34170b672c043fe16ac08c4d7a19794f79d04e103f2df9108f3"},"schema_version":"1.0"},"canonical_sha256":"a1c9f693898775be7c7524eed1abd08fe69b4acf6643063c047d6f108bc150ed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:09:49.971882Z","signature_b64":"4VQ6Ig3s4y4YQnKQ38dbzlBWOboKxGvhS0J/ctylz3aO65RZzPd6peqm0B9lf96p2DkA8KIBb+yyuilipSfBAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1c9f693898775be7c7524eed1abd08fe69b4acf6643063c047d6f108bc150ed","last_reissued_at":"2026-06-19T16:09:49.971464Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:09:49.971464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.05164","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":"2026-06-19T16:09:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2UtyBpzeXFCFfWibhFQ2mZ1T9bMH3XawkdN6Qv2cW/F98O9gA/fe1twy+ckUWxu79ajYjlMDqx2IH7zVgMvjBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T18:42:25.956059Z"},"content_sha256":"3e12a399e549be8679125b2c61710e42cc7384e3f5ed35cc6ead849dd273e6fd","schema_version":"1.0","event_id":"sha256:3e12a399e549be8679125b2c61710e42cc7384e3f5ed35cc6ead849dd273e6fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:UHE7NE4JQ52347DVETXNDK6QR7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Dynamical Systems Perspective on the Analysis of Neural Networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","nlin.AO"],"primary_cat":"math.DS","authors_text":"Christian Kuehn, Dennis Chemnitz, Maximilian Engel, Sara-Viola Kuntz","submitted_at":"2025-07-07T16:18:49Z","abstract_excerpt":"In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements. We also tackle three concrete challenges. First, we consider the process of information propagation through a neural network, i.e., we study the input-output map for different architectures. We explain the universal embedding property for augmented neural ODEs representing arbitrary functions of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.05164","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/2507.05164/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":"2026-06-19T16:09:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vW0dMWlbYH00DRWqNlXr85I5jCQYhLNZFZRYwkz6t5ME8nTV4XkxZknLR5vxMTaOe8c5mYicwb8fw1FpBChYAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T18:42:25.956456Z"},"content_sha256":"1d80bfb4efce07960714cc0692fdcf1b1422f7a5a08f5118537d7cdc7c9b1f89","schema_version":"1.0","event_id":"sha256:1d80bfb4efce07960714cc0692fdcf1b1422f7a5a08f5118537d7cdc7c9b1f89"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UHE7NE4JQ52347DVETXNDK6QR7/bundle.json","state_url":"https://pith.science/pith/UHE7NE4JQ52347DVETXNDK6QR7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UHE7NE4JQ52347DVETXNDK6QR7/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":"2026-06-28T18:42:25Z","links":{"resolver":"https://pith.science/pith/UHE7NE4JQ52347DVETXNDK6QR7","bundle":"https://pith.science/pith/UHE7NE4JQ52347DVETXNDK6QR7/bundle.json","state":"https://pith.science/pith/UHE7NE4JQ52347DVETXNDK6QR7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UHE7NE4JQ52347DVETXNDK6QR7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:UHE7NE4JQ52347DVETXNDK6QR7","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":"0e04ac6d88bcd34170b672c043fe16ac08c4d7a19794f79d04e103f2df9108f3","cross_cats_sorted":["cs.LG","nlin.AO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-07T16:18:49Z","title_canon_sha256":"889725354f5bf2c708a830acd7667ff110dc372154a2a5d682a20a372bb2b1ef"},"schema_version":"1.0","source":{"id":"2507.05164","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.05164","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"arxiv_version","alias_value":"2507.05164v2","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.05164","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"pith_short_12","alias_value":"UHE7NE4JQ523","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"pith_short_16","alias_value":"UHE7NE4JQ52347DV","created_at":"2026-06-19T16:09:49Z"},{"alias_kind":"pith_short_8","alias_value":"UHE7NE4J","created_at":"2026-06-19T16:09:49Z"}],"graph_snapshots":[{"event_id":"sha256:1d80bfb4efce07960714cc0692fdcf1b1422f7a5a08f5118537d7cdc7c9b1f89","target":"graph","created_at":"2026-06-19T16:09:49Z","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/2507.05164/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements. We also tackle three concrete challenges. First, we consider the process of information propagation through a neural network, i.e., we study the input-output map for different architectures. We explain the universal embedding property for augmented neural ODEs representing arbitrary functions of ","authors_text":"Christian Kuehn, Dennis Chemnitz, Maximilian Engel, Sara-Viola Kuntz","cross_cats":["cs.LG","nlin.AO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-07T16:18:49Z","title":"A Dynamical Systems Perspective on the Analysis of Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.05164","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":"sha256:3e12a399e549be8679125b2c61710e42cc7384e3f5ed35cc6ead849dd273e6fd","target":"record","created_at":"2026-06-19T16:09:49Z","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":"0e04ac6d88bcd34170b672c043fe16ac08c4d7a19794f79d04e103f2df9108f3","cross_cats_sorted":["cs.LG","nlin.AO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-07T16:18:49Z","title_canon_sha256":"889725354f5bf2c708a830acd7667ff110dc372154a2a5d682a20a372bb2b1ef"},"schema_version":"1.0","source":{"id":"2507.05164","kind":"arxiv","version":2}},"canonical_sha256":"a1c9f693898775be7c7524eed1abd08fe69b4acf6643063c047d6f108bc150ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a1c9f693898775be7c7524eed1abd08fe69b4acf6643063c047d6f108bc150ed","first_computed_at":"2026-06-19T16:09:49.971464Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:09:49.971464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4VQ6Ig3s4y4YQnKQ38dbzlBWOboKxGvhS0J/ctylz3aO65RZzPd6peqm0B9lf96p2DkA8KIBb+yyuilipSfBAA==","signature_status":"signed_v1","signed_at":"2026-06-19T16:09:49.971882Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.05164","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e12a399e549be8679125b2c61710e42cc7384e3f5ed35cc6ead849dd273e6fd","sha256:1d80bfb4efce07960714cc0692fdcf1b1422f7a5a08f5118537d7cdc7c9b1f89"],"state_sha256":"d0f1e773f52e98e52dc6526f558e5c56cca50ac8ea9ab3636bb1f86354bddd56"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"idu4ykoicbi+I7gDnp7D7FEVA7muUXxZEfA6K0Z4AEQJDiwPmS6wJ1g6eWOjVaPPqKW2tWb4Wdzf8dQsQ7+VBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T18:42:25.958516Z","bundle_sha256":"641b6d29b61dd95e0dd84d93d632b4ccbe1211fa5baad2a40ccd0af13a7dfbdb"}}