{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TFVEFDJ6XIAE6ZZW6T2S6JG3E3","short_pith_number":"pith:TFVEFDJ6","canonical_record":{"source":{"id":"2606.22075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-20T14:50:40Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"f360f1c078e5de537df412600b62b3b00dcaf63208c804fc2bf6c7ff4205fbd1","abstract_canon_sha256":"8950bdb715605db0a331e9d0b16bb91c4b12fa4f9e4aa5b4cce69a1a264f840d"},"schema_version":"1.0"},"canonical_sha256":"996a428d3eba004f6736f4f52f24db26d58caf5c8a6290c49a2f2137d81826ab","source":{"kind":"arxiv","id":"2606.22075","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.22075","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"arxiv_version","alias_value":"2606.22075v1","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22075","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"pith_short_12","alias_value":"TFVEFDJ6XIAE","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"pith_short_16","alias_value":"TFVEFDJ6XIAE6ZZW","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"pith_short_8","alias_value":"TFVEFDJ6","created_at":"2026-06-23T02:13:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TFVEFDJ6XIAE6ZZW6T2S6JG3E3","target":"record","payload":{"canonical_record":{"source":{"id":"2606.22075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-20T14:50:40Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"f360f1c078e5de537df412600b62b3b00dcaf63208c804fc2bf6c7ff4205fbd1","abstract_canon_sha256":"8950bdb715605db0a331e9d0b16bb91c4b12fa4f9e4aa5b4cce69a1a264f840d"},"schema_version":"1.0"},"canonical_sha256":"996a428d3eba004f6736f4f52f24db26d58caf5c8a6290c49a2f2137d81826ab","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T02:13:27.492248Z","signature_b64":"Z5+yutYe5+MLa8y+240PYxfrJZJRbX67hf5fmGjOqdnfxMUp0rJvpCjqiiQ2ZtpPTaJI0E5wInT9P7GMbxhBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"996a428d3eba004f6736f4f52f24db26d58caf5c8a6290c49a2f2137d81826ab","last_reissued_at":"2026-06-23T02:13:27.491780Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T02:13:27.491780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.22075","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":"2026-06-23T02:13:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3FgYhs6qXhpgMNgsCGinkzR0HHChK52eJkVWcUyAsyYa+GSHCCHByrHEgpGBH/9C3sc9kzP/DJIaBeeS4rDcAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T21:51:42.671724Z"},"content_sha256":"d6a0c2b48b006c15cfd57eff0888bb51a587bf0c3e6a95a07461cd9315d3f450","schema_version":"1.0","event_id":"sha256:d6a0c2b48b006c15cfd57eff0888bb51a587bf0c3e6a95a07461cd9315d3f450"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TFVEFDJ6XIAE6ZZW6T2S6JG3E3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Frequency-Domain Neural ODEs for Modeling Non-Linear Dynamical Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"cs.LG","authors_text":"Ayman A. El-Badawy, Mohammed Ashraf","submitted_at":"2026-06-20T14:50:40Z","abstract_excerpt":"Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better gener"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22075","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/2606.22075/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":"2026-06-23T02:13:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"51UFyXA24Fe4OHs90MFU0fgp4PzeEfAgddEx90+yrsoCJKk54ZGOphgEVmSap6qeEGaclGNc1lxph7JgkPI3AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T21:51:42.672099Z"},"content_sha256":"db1c05348acbabea8a67ed543c8093f1793148591041754b53f1719889f1e25e","schema_version":"1.0","event_id":"sha256:db1c05348acbabea8a67ed543c8093f1793148591041754b53f1719889f1e25e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TFVEFDJ6XIAE6ZZW6T2S6JG3E3/bundle.json","state_url":"https://pith.science/pith/TFVEFDJ6XIAE6ZZW6T2S6JG3E3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TFVEFDJ6XIAE6ZZW6T2S6JG3E3/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":"2026-07-02T21:51:42Z","links":{"resolver":"https://pith.science/pith/TFVEFDJ6XIAE6ZZW6T2S6JG3E3","bundle":"https://pith.science/pith/TFVEFDJ6XIAE6ZZW6T2S6JG3E3/bundle.json","state":"https://pith.science/pith/TFVEFDJ6XIAE6ZZW6T2S6JG3E3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TFVEFDJ6XIAE6ZZW6T2S6JG3E3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TFVEFDJ6XIAE6ZZW6T2S6JG3E3","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":"8950bdb715605db0a331e9d0b16bb91c4b12fa4f9e4aa5b4cce69a1a264f840d","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-20T14:50:40Z","title_canon_sha256":"f360f1c078e5de537df412600b62b3b00dcaf63208c804fc2bf6c7ff4205fbd1"},"schema_version":"1.0","source":{"id":"2606.22075","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.22075","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"arxiv_version","alias_value":"2606.22075v1","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22075","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"pith_short_12","alias_value":"TFVEFDJ6XIAE","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"pith_short_16","alias_value":"TFVEFDJ6XIAE6ZZW","created_at":"2026-06-23T02:13:27Z"},{"alias_kind":"pith_short_8","alias_value":"TFVEFDJ6","created_at":"2026-06-23T02:13:27Z"}],"graph_snapshots":[{"event_id":"sha256:db1c05348acbabea8a67ed543c8093f1793148591041754b53f1719889f1e25e","target":"graph","created_at":"2026-06-23T02:13:27Z","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/2606.22075/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better gener","authors_text":"Ayman A. El-Badawy, Mohammed Ashraf","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-20T14:50:40Z","title":"Frequency-Domain Neural ODEs for Modeling Non-Linear Dynamical Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22075","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:d6a0c2b48b006c15cfd57eff0888bb51a587bf0c3e6a95a07461cd9315d3f450","target":"record","created_at":"2026-06-23T02:13:27Z","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":"8950bdb715605db0a331e9d0b16bb91c4b12fa4f9e4aa5b4cce69a1a264f840d","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-20T14:50:40Z","title_canon_sha256":"f360f1c078e5de537df412600b62b3b00dcaf63208c804fc2bf6c7ff4205fbd1"},"schema_version":"1.0","source":{"id":"2606.22075","kind":"arxiv","version":1}},"canonical_sha256":"996a428d3eba004f6736f4f52f24db26d58caf5c8a6290c49a2f2137d81826ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"996a428d3eba004f6736f4f52f24db26d58caf5c8a6290c49a2f2137d81826ab","first_computed_at":"2026-06-23T02:13:27.491780Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T02:13:27.491780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z5+yutYe5+MLa8y+240PYxfrJZJRbX67hf5fmGjOqdnfxMUp0rJvpCjqiiQ2ZtpPTaJI0E5wInT9P7GMbxhBAQ==","signature_status":"signed_v1","signed_at":"2026-06-23T02:13:27.492248Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.22075","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d6a0c2b48b006c15cfd57eff0888bb51a587bf0c3e6a95a07461cd9315d3f450","sha256:db1c05348acbabea8a67ed543c8093f1793148591041754b53f1719889f1e25e"],"state_sha256":"f2134ebb648ce02e9f388d21854f4e1cdde1e0bb0a6432348f4714936eb8c50e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z/EID7dSJ94xOpkv+iZsKCS/+MgVR+ytEK86lHSteuXSx0kdqp4VTKMC+n/OF6uCV1k8GSupJYdwONFTi4wZBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T21:51:42.674058Z","bundle_sha256":"8d28ecfa2c023684e527801eec5677528a1f1fcb6529fc62dcbd010e881b52ac"}}