{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:QUZ56J53XRVXGBSUIH6XSJOFO7","short_pith_number":"pith:QUZ56J53","canonical_record":{"source":{"id":"2604.26280","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-04-29T04:21:03Z","cross_cats_sorted":["cs.NA","math.DS"],"title_canon_sha256":"02ffd29ca7939bbe52bae3aff57f3e01cf0d2ef921231c45ba00fb5f65ff2156","abstract_canon_sha256":"acdb451e64b03e8a7dde5526069af06d60aabb7eb67e0a40127f5c0a04ccd456"},"schema_version":"1.0"},"canonical_sha256":"8533df27bbbc6b73065441fd7925c577da5d62eba972b810274feb2e4411acff","source":{"kind":"arxiv","id":"2604.26280","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.26280","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"arxiv_version","alias_value":"2604.26280v2","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.26280","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"pith_short_12","alias_value":"QUZ56J53XRVX","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"pith_short_16","alias_value":"QUZ56J53XRVXGBSU","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"pith_short_8","alias_value":"QUZ56J53","created_at":"2026-06-09T01:05:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:QUZ56J53XRVXGBSUIH6XSJOFO7","target":"record","payload":{"canonical_record":{"source":{"id":"2604.26280","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-04-29T04:21:03Z","cross_cats_sorted":["cs.NA","math.DS"],"title_canon_sha256":"02ffd29ca7939bbe52bae3aff57f3e01cf0d2ef921231c45ba00fb5f65ff2156","abstract_canon_sha256":"acdb451e64b03e8a7dde5526069af06d60aabb7eb67e0a40127f5c0a04ccd456"},"schema_version":"1.0"},"canonical_sha256":"8533df27bbbc6b73065441fd7925c577da5d62eba972b810274feb2e4411acff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:18.257825Z","signature_b64":"hUnh5C3z+mFJukwy8os5gecE++vGtVqkEVhybU5XjDhNOCjiPkm+KoCFBlIfrjUOBhoJWXbv0WJqICQgzPJWCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8533df27bbbc6b73065441fd7925c577da5d62eba972b810274feb2e4411acff","last_reissued_at":"2026-06-09T01:05:18.257399Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:18.257399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.26280","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":"2026-06-09T01:05:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pn1tuRaZHHgakK9YXKcNNXhMh7nYHGebzIpYW0w5/RCe2pMIFiaRXaH/wayNijZa9abMfmMb2XtLC+GUlkabCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T23:27:43.903339Z"},"content_sha256":"0fd1ec23b0692a72a3fd2b9d7a7f37b96b07625010fa0c0101268528860ecb73","schema_version":"1.0","event_id":"sha256:0fd1ec23b0692a72a3fd2b9d7a7f37b96b07625010fa0c0101268528860ecb73"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:QUZ56J53XRVXGBSUIH6XSJOFO7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Structure-Aware Tensorial Model Reduction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Tensor factorization lets reduced bases for parameterized PDEs vary nonlinearly with parameters by encoding snapshots offline and interpolating online.","cross_cats":["cs.NA","math.DS"],"primary_cat":"math.NA","authors_text":"Anthony Gruber, Arjun Vijaywargiya, Eric C. Cyr","submitted_at":"2026-04-29T04:21:03Z","abstract_excerpt":"This work investigates a two-stage method for constructing projection-based reduced-order models (ROMs) of parameterized partial differential equations (PDEs). Based on established tensorial ROM methodology, the proposed approach reduces dimensionality offline by encoding solution snapshots using a multi-linear Tucker factorization, so that a reduced basis which varies nonlinearly with PDE parameters can be rapidly constructed online and used in a Galerkin ROM. Two novel extensions of this strategy, tailored to the cases of structured PDEs and sparse parameter sampling, are presented: the cons"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The proposed nonlinear basis ROM can effectively mitigate linear restrictions on Kolmogorov n-width while improving upon previous tensorial ROM technology, particularly in the highly nonlinear and data-limited regimes characteristic of practical use cases.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the Tucker-encoded states admit accurate RBF interpolation and that the orthonormalization with respect to a general discrete inner product preserves the error bounds sufficiently for the target PDEs when parameter sampling is sparse.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A structure-aware tensorial ROM method with orthonormalized bases and RBF interpolation of encoded states reduces dimensionality for parameterized PDEs while mitigating Kolmogorov n-width limitations in nonlinear and data-sparse regimes.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Tensor factorization lets reduced bases for parameterized PDEs vary nonlinearly with parameters by encoding snapshots offline and interpolating online.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9d077cb84c2503aa3843077f46fe6efa88f9524e75b87e0981e7ec323f64d145"},"source":{"id":"2604.26280","kind":"arxiv","version":2},"verdict":{"id":"300a1f8b-bc12-435d-82fc-a6b255e1559c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T13:05:39.559896Z","strongest_claim":"The proposed nonlinear basis ROM can effectively mitigate linear restrictions on Kolmogorov n-width while improving upon previous tensorial ROM technology, particularly in the highly nonlinear and data-limited regimes characteristic of practical use cases.","one_line_summary":"A structure-aware tensorial ROM method with orthonormalized bases and RBF interpolation of encoded states reduces dimensionality for parameterized PDEs while mitigating Kolmogorov n-width limitations in nonlinear and data-sparse regimes.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the Tucker-encoded states admit accurate RBF interpolation and that the orthonormalization with respect to a general discrete inner product preserves the error bounds sufficiently for the target PDEs when parameter sampling is sparse.","pith_extraction_headline":"Tensor factorization lets reduced bases for parameterized PDEs vary nonlinearly with parameters by encoding snapshots offline and interpolating online."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.26280/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T00:39:32.806723Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:22:39.407508Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c9d11e83cd2e153624e8d922e234b21b2994a58ffd61996403ce380852b2ff04"},"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":"300a1f8b-bc12-435d-82fc-a6b255e1559c"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-09T01:05:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Rt6dLthgAPJ0BCNJzHv+w0Ev4+Y6zdsy9vEZ5I/rBo0hp2S/NsHQNJOV/Mb+KLD93xxZQ/7vUKriFSlbFomBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T23:27:43.904029Z"},"content_sha256":"5341692a833221888bf97fa82f18386a9a93f49aa547d26e60a4354967c3c51d","schema_version":"1.0","event_id":"sha256:5341692a833221888bf97fa82f18386a9a93f49aa547d26e60a4354967c3c51d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QUZ56J53XRVXGBSUIH6XSJOFO7/bundle.json","state_url":"https://pith.science/pith/QUZ56J53XRVXGBSUIH6XSJOFO7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QUZ56J53XRVXGBSUIH6XSJOFO7/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":"2026-07-03T23:27:43Z","links":{"resolver":"https://pith.science/pith/QUZ56J53XRVXGBSUIH6XSJOFO7","bundle":"https://pith.science/pith/QUZ56J53XRVXGBSUIH6XSJOFO7/bundle.json","state":"https://pith.science/pith/QUZ56J53XRVXGBSUIH6XSJOFO7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QUZ56J53XRVXGBSUIH6XSJOFO7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QUZ56J53XRVXGBSUIH6XSJOFO7","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":"acdb451e64b03e8a7dde5526069af06d60aabb7eb67e0a40127f5c0a04ccd456","cross_cats_sorted":["cs.NA","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-04-29T04:21:03Z","title_canon_sha256":"02ffd29ca7939bbe52bae3aff57f3e01cf0d2ef921231c45ba00fb5f65ff2156"},"schema_version":"1.0","source":{"id":"2604.26280","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.26280","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"arxiv_version","alias_value":"2604.26280v2","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.26280","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"pith_short_12","alias_value":"QUZ56J53XRVX","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"pith_short_16","alias_value":"QUZ56J53XRVXGBSU","created_at":"2026-06-09T01:05:18Z"},{"alias_kind":"pith_short_8","alias_value":"QUZ56J53","created_at":"2026-06-09T01:05:18Z"}],"graph_snapshots":[{"event_id":"sha256:5341692a833221888bf97fa82f18386a9a93f49aa547d26e60a4354967c3c51d","target":"graph","created_at":"2026-06-09T01:05:18Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"The proposed nonlinear basis ROM can effectively mitigate linear restrictions on Kolmogorov n-width while improving upon previous tensorial ROM technology, particularly in the highly nonlinear and data-limited regimes characteristic of practical use cases."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the Tucker-encoded states admit accurate RBF interpolation and that the orthonormalization with respect to a general discrete inner product preserves the error bounds sufficiently for the target PDEs when parameter sampling is sparse."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A structure-aware tensorial ROM method with orthonormalized bases and RBF interpolation of encoded states reduces dimensionality for parameterized PDEs while mitigating Kolmogorov n-width limitations in nonlinear and data-sparse regimes."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Tensor factorization lets reduced bases for parameterized PDEs vary nonlinearly with parameters by encoding snapshots offline and interpolating online."}],"snapshot_sha256":"9d077cb84c2503aa3843077f46fe6efa88f9524e75b87e0981e7ec323f64d145"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-21T00:39:32.806723Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T20:22:39.407508Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.26280/integrity.json","findings":[],"snapshot_sha256":"c9d11e83cd2e153624e8d922e234b21b2994a58ffd61996403ce380852b2ff04","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This work investigates a two-stage method for constructing projection-based reduced-order models (ROMs) of parameterized partial differential equations (PDEs). Based on established tensorial ROM methodology, the proposed approach reduces dimensionality offline by encoding solution snapshots using a multi-linear Tucker factorization, so that a reduced basis which varies nonlinearly with PDE parameters can be rapidly constructed online and used in a Galerkin ROM. Two novel extensions of this strategy, tailored to the cases of structured PDEs and sparse parameter sampling, are presented: the cons","authors_text":"Anthony Gruber, Arjun Vijaywargiya, Eric C. Cyr","cross_cats":["cs.NA","math.DS"],"headline":"Tensor factorization lets reduced bases for parameterized PDEs vary nonlinearly with parameters by encoding snapshots offline and interpolating online.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-04-29T04:21:03Z","title":"Structure-Aware Tensorial Model Reduction"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.26280","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-07T13:05:39.559896Z","id":"300a1f8b-bc12-435d-82fc-a6b255e1559c","model_set":{"reader":"grok-4.3"},"one_line_summary":"A structure-aware tensorial ROM method with orthonormalized bases and RBF interpolation of encoded states reduces dimensionality for parameterized PDEs while mitigating Kolmogorov n-width limitations in nonlinear and data-sparse regimes.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Tensor factorization lets reduced bases for parameterized PDEs vary nonlinearly with parameters by encoding snapshots offline and interpolating online.","strongest_claim":"The proposed nonlinear basis ROM can effectively mitigate linear restrictions on Kolmogorov n-width while improving upon previous tensorial ROM technology, particularly in the highly nonlinear and data-limited regimes characteristic of practical use cases.","weakest_assumption":"That the Tucker-encoded states admit accurate RBF interpolation and that the orthonormalization with respect to a general discrete inner product preserves the error bounds sufficiently for the target PDEs when parameter sampling is sparse."}},"verdict_id":"300a1f8b-bc12-435d-82fc-a6b255e1559c"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0fd1ec23b0692a72a3fd2b9d7a7f37b96b07625010fa0c0101268528860ecb73","target":"record","created_at":"2026-06-09T01:05:18Z","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":"acdb451e64b03e8a7dde5526069af06d60aabb7eb67e0a40127f5c0a04ccd456","cross_cats_sorted":["cs.NA","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-04-29T04:21:03Z","title_canon_sha256":"02ffd29ca7939bbe52bae3aff57f3e01cf0d2ef921231c45ba00fb5f65ff2156"},"schema_version":"1.0","source":{"id":"2604.26280","kind":"arxiv","version":2}},"canonical_sha256":"8533df27bbbc6b73065441fd7925c577da5d62eba972b810274feb2e4411acff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8533df27bbbc6b73065441fd7925c577da5d62eba972b810274feb2e4411acff","first_computed_at":"2026-06-09T01:05:18.257399Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T01:05:18.257399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hUnh5C3z+mFJukwy8os5gecE++vGtVqkEVhybU5XjDhNOCjiPkm+KoCFBlIfrjUOBhoJWXbv0WJqICQgzPJWCw==","signature_status":"signed_v1","signed_at":"2026-06-09T01:05:18.257825Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.26280","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0fd1ec23b0692a72a3fd2b9d7a7f37b96b07625010fa0c0101268528860ecb73","sha256:5341692a833221888bf97fa82f18386a9a93f49aa547d26e60a4354967c3c51d"],"state_sha256":"2e7f5aa4dd68c17a2f295eae013208002698918200ee257a0f61443f67156330"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+BOtpp1R+T4hgIS/Sf4hUMN+qKzOIMH9wawNYsza1GDaVitNcpLb8Xk7B4sbQUJ6rVmFOY81ksx/wZ1l7SR5Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T23:27:43.907103Z","bundle_sha256":"115525d1117b004df8ecf6cd6c72b364d9e4fac3a3b55559ed6f51618067cc2c"}}