{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:CSY7WT3DIBUWT2SOSZT7WYXDWH","short_pith_number":"pith:CSY7WT3D","schema_version":"1.0","canonical_sha256":"14b1fb4f63406969ea4e9667fb62e3b1e2cedf3d7360cadea63abd97eee3cac9","source":{"kind":"arxiv","id":"2403.02035","version":2},"attestation_state":"computed","paper":{"title":"Exponential Expressivity of ReLU$^k$ Neural Networks on Gevrey Classes with Point Singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA"],"primary_cat":"math.NA","authors_text":"Christoph Schwab, Joost A. A. Opschoor","submitted_at":"2024-03-04T13:39:22Z","abstract_excerpt":"We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains $\\mathrm{D} \\subset \\mathbb{R}^d$, $d=2,3$. We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in $\\mathrm{D}$, comprising the countably-normed spaces of I.M. Babu\\v{s}ka and B.Q. Guo.\n  As intermediate result, we prove that continuous, piecewise polynomial high order (``$p$-version'') finite elements with e"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2403.02035","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2024-03-04T13:39:22Z","cross_cats_sorted":["cs.LG","cs.NA"],"title_canon_sha256":"d4873dc0b0887711fe5fc754adc4e2910515bd318d8ef4286903548da064f805","abstract_canon_sha256":"c99476b4386d13f8d49b92efaa7d7d5e22cd4792a9ca2204bf888ab89d0cd873"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:02:40.348695Z","signature_b64":"Zb9BLS09W8xQnlSfORzUzRiJVkoJVLFy7Bj0dRt279AfgjGgT8jASmjNXJFb23ZRg2b2kSZWyDWG91Bo0d4tCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14b1fb4f63406969ea4e9667fb62e3b1e2cedf3d7360cadea63abd97eee3cac9","last_reissued_at":"2026-07-05T09:02:40.348188Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:02:40.348188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exponential Expressivity of ReLU$^k$ Neural Networks on Gevrey Classes with Point Singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA"],"primary_cat":"math.NA","authors_text":"Christoph Schwab, Joost A. A. Opschoor","submitted_at":"2024-03-04T13:39:22Z","abstract_excerpt":"We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains $\\mathrm{D} \\subset \\mathbb{R}^d$, $d=2,3$. We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in $\\mathrm{D}$, comprising the countably-normed spaces of I.M. Babu\\v{s}ka and B.Q. Guo.\n  As intermediate result, we prove that continuous, piecewise polynomial high order (``$p$-version'') finite elements with e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.02035","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/2403.02035/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2403.02035","created_at":"2026-07-05T09:02:40.348253+00:00"},{"alias_kind":"arxiv_version","alias_value":"2403.02035v2","created_at":"2026-07-05T09:02:40.348253+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.02035","created_at":"2026-07-05T09:02:40.348253+00:00"},{"alias_kind":"pith_short_12","alias_value":"CSY7WT3DIBUW","created_at":"2026-07-05T09:02:40.348253+00:00"},{"alias_kind":"pith_short_16","alias_value":"CSY7WT3DIBUWT2SO","created_at":"2026-07-05T09:02:40.348253+00:00"},{"alias_kind":"pith_short_8","alias_value":"CSY7WT3D","created_at":"2026-07-05T09:02:40.348253+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.19104","citing_title":"On the algorithmic construction of deep ReLU networks","ref_index":17,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CSY7WT3DIBUWT2SOSZT7WYXDWH","json":"https://pith.science/pith/CSY7WT3DIBUWT2SOSZT7WYXDWH.json","graph_json":"https://pith.science/api/pith-number/CSY7WT3DIBUWT2SOSZT7WYXDWH/graph.json","events_json":"https://pith.science/api/pith-number/CSY7WT3DIBUWT2SOSZT7WYXDWH/events.json","paper":"https://pith.science/paper/CSY7WT3D"},"agent_actions":{"view_html":"https://pith.science/pith/CSY7WT3DIBUWT2SOSZT7WYXDWH","download_json":"https://pith.science/pith/CSY7WT3DIBUWT2SOSZT7WYXDWH.json","view_paper":"https://pith.science/paper/CSY7WT3D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2403.02035&json=true","fetch_graph":"https://pith.science/api/pith-number/CSY7WT3DIBUWT2SOSZT7WYXDWH/graph.json","fetch_events":"https://pith.science/api/pith-number/CSY7WT3DIBUWT2SOSZT7WYXDWH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CSY7WT3DIBUWT2SOSZT7WYXDWH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CSY7WT3DIBUWT2SOSZT7WYXDWH/action/storage_attestation","attest_author":"https://pith.science/pith/CSY7WT3DIBUWT2SOSZT7WYXDWH/action/author_attestation","sign_citation":"https://pith.science/pith/CSY7WT3DIBUWT2SOSZT7WYXDWH/action/citation_signature","submit_replication":"https://pith.science/pith/CSY7WT3DIBUWT2SOSZT7WYXDWH/action/replication_record"}},"created_at":"2026-07-05T09:02:40.348253+00:00","updated_at":"2026-07-05T09:02:40.348253+00:00"}