{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:2EVS4EU2ST7EORZSKJ3N4JL576","short_pith_number":"pith:2EVS4EU2","canonical_record":{"source":{"id":"2505.23002","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2025-05-29T02:22:38Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"2cf8bbf01b6a0bb88b425ced4e34f0b46c7c86d95e35d4d5991fa9ce22b11ccb","abstract_canon_sha256":"b7f1a7755e133a36384145689298be427a8158afc4a9268ab442465f3300bc07"},"schema_version":"1.0"},"canonical_sha256":"d12b2e129a94fe4747325276de257dff9b4b604383d0fb0771f985a1b0e132ba","source":{"kind":"arxiv","id":"2505.23002","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.23002","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"arxiv_version","alias_value":"2505.23002v2","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.23002","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"pith_short_12","alias_value":"2EVS4EU2ST7E","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"pith_short_16","alias_value":"2EVS4EU2ST7EORZS","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"pith_short_8","alias_value":"2EVS4EU2","created_at":"2026-07-05T12:05:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:2EVS4EU2ST7EORZSKJ3N4JL576","target":"record","payload":{"canonical_record":{"source":{"id":"2505.23002","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2025-05-29T02:22:38Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"2cf8bbf01b6a0bb88b425ced4e34f0b46c7c86d95e35d4d5991fa9ce22b11ccb","abstract_canon_sha256":"b7f1a7755e133a36384145689298be427a8158afc4a9268ab442465f3300bc07"},"schema_version":"1.0"},"canonical_sha256":"d12b2e129a94fe4747325276de257dff9b4b604383d0fb0771f985a1b0e132ba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:05:17.135065Z","signature_b64":"/tB9KnmcGeztFdh5K8MfNFTIuTQ2DTZbXWwPmqgMEE2Ej6sktpMWUElGWZfrODY/CAKqkkSYO+3MVQrs98i7CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d12b2e129a94fe4747325276de257dff9b4b604383d0fb0771f985a1b0e132ba","last_reissued_at":"2026-07-05T12:05:17.134553Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:05:17.134553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2505.23002","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":"2026-07-05T12:05:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HVA2Hu7YlkNiZ9xvWa/ztsUzfV/sG8qW5g9pD6kiKRsEwj1ocGOUFB5LMxbm2yqTTy0tserUgvdpNtOfe70sDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T11:24:27.335225Z"},"content_sha256":"60d4cafd24d3c0cdc627e193b399ec82c147f4e7fa84aa62aca8a7656414000c","schema_version":"1.0","event_id":"sha256:60d4cafd24d3c0cdc627e193b399ec82c147f4e7fa84aa62aca8a7656414000c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:2EVS4EU2ST7EORZSKJ3N4JL576","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Deep asymptotic expansion method for solving singularly perturbed time-dependent reaction-advection-diffusion equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Bangti Jin, Dmitrii Chaikovskii, Qiao Zhu, Ye Zhang","submitted_at":"2025-05-29T02:22:38Z","abstract_excerpt":"Physics-informed neural network (PINN) has shown great potential in solving partial differential equations. However, it faces challenges when dealing with problems involving steep gradients. The solutions to singularly perturbed time-dependent reaction-advection-diffusion equations exhibit internal moving transition layers with sharp gradients, and thus the standard PINN becomes ineffective. In this work, we propose a deep asymptotic expansion (DAE) method, which is inspired by asymptotic analysis and leverages deep learning to approximate the smooth part of the expansion. We first derive the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.23002","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/2505.23002/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":"2026-07-05T12:05:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c4AevLPINWguZfLIiSNfkxqYAWSVW5ch+IRUvmrDe2819qD/WsP9EbgSBjlNMWrHS8jA+NOD4ymoJJYHn/hzAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T11:24:27.335730Z"},"content_sha256":"4fd417440bd0f59ec4d9e0174377b16a0be8d4bf9d74bd3e2b0207b35b75b12e","schema_version":"1.0","event_id":"sha256:4fd417440bd0f59ec4d9e0174377b16a0be8d4bf9d74bd3e2b0207b35b75b12e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2EVS4EU2ST7EORZSKJ3N4JL576/bundle.json","state_url":"https://pith.science/pith/2EVS4EU2ST7EORZSKJ3N4JL576/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2EVS4EU2ST7EORZSKJ3N4JL576/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":"2026-08-08T11:24:27Z","links":{"resolver":"https://pith.science/pith/2EVS4EU2ST7EORZSKJ3N4JL576","bundle":"https://pith.science/pith/2EVS4EU2ST7EORZSKJ3N4JL576/bundle.json","state":"https://pith.science/pith/2EVS4EU2ST7EORZSKJ3N4JL576/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2EVS4EU2ST7EORZSKJ3N4JL576/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:2EVS4EU2ST7EORZSKJ3N4JL576","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":"b7f1a7755e133a36384145689298be427a8158afc4a9268ab442465f3300bc07","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2025-05-29T02:22:38Z","title_canon_sha256":"2cf8bbf01b6a0bb88b425ced4e34f0b46c7c86d95e35d4d5991fa9ce22b11ccb"},"schema_version":"1.0","source":{"id":"2505.23002","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.23002","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"arxiv_version","alias_value":"2505.23002v2","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.23002","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"pith_short_12","alias_value":"2EVS4EU2ST7E","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"pith_short_16","alias_value":"2EVS4EU2ST7EORZS","created_at":"2026-07-05T12:05:17Z"},{"alias_kind":"pith_short_8","alias_value":"2EVS4EU2","created_at":"2026-07-05T12:05:17Z"}],"graph_snapshots":[{"event_id":"sha256:4fd417440bd0f59ec4d9e0174377b16a0be8d4bf9d74bd3e2b0207b35b75b12e","target":"graph","created_at":"2026-07-05T12:05:17Z","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/2505.23002/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Physics-informed neural network (PINN) has shown great potential in solving partial differential equations. However, it faces challenges when dealing with problems involving steep gradients. The solutions to singularly perturbed time-dependent reaction-advection-diffusion equations exhibit internal moving transition layers with sharp gradients, and thus the standard PINN becomes ineffective. In this work, we propose a deep asymptotic expansion (DAE) method, which is inspired by asymptotic analysis and leverages deep learning to approximate the smooth part of the expansion. We first derive the ","authors_text":"Bangti Jin, Dmitrii Chaikovskii, Qiao Zhu, Ye Zhang","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2025-05-29T02:22:38Z","title":"Deep asymptotic expansion method for solving singularly perturbed time-dependent reaction-advection-diffusion equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.23002","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":"sha256:60d4cafd24d3c0cdc627e193b399ec82c147f4e7fa84aa62aca8a7656414000c","target":"record","created_at":"2026-07-05T12:05:17Z","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":"b7f1a7755e133a36384145689298be427a8158afc4a9268ab442465f3300bc07","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2025-05-29T02:22:38Z","title_canon_sha256":"2cf8bbf01b6a0bb88b425ced4e34f0b46c7c86d95e35d4d5991fa9ce22b11ccb"},"schema_version":"1.0","source":{"id":"2505.23002","kind":"arxiv","version":2}},"canonical_sha256":"d12b2e129a94fe4747325276de257dff9b4b604383d0fb0771f985a1b0e132ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d12b2e129a94fe4747325276de257dff9b4b604383d0fb0771f985a1b0e132ba","first_computed_at":"2026-07-05T12:05:17.134553Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:05:17.134553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/tB9KnmcGeztFdh5K8MfNFTIuTQ2DTZbXWwPmqgMEE2Ej6sktpMWUElGWZfrODY/CAKqkkSYO+3MVQrs98i7CA==","signature_status":"signed_v1","signed_at":"2026-07-05T12:05:17.135065Z","signed_message":"canonical_sha256_bytes"},"source_id":"2505.23002","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:60d4cafd24d3c0cdc627e193b399ec82c147f4e7fa84aa62aca8a7656414000c","sha256:4fd417440bd0f59ec4d9e0174377b16a0be8d4bf9d74bd3e2b0207b35b75b12e"],"state_sha256":"6c8c76aedd4b34be83276947bf959ddfe6a44c012564e3d33cabd93b64966369"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yThZsp+68RDX7jRCGARpddmP58XalbHEC4TZHF4TIJEEPNdZjctFDpSoabiiIWtCOYmrnUct2XGX7RJvBAnDDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T11:24:27.340901Z","bundle_sha256":"216745e13e852f077da746b8ff6a4e9f96094f2e0252c0b67e4312cb12086fe4"}}