{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:WTL4IGZOUERGVTPZKJIUZPZTL3","short_pith_number":"pith:WTL4IGZO","schema_version":"1.0","canonical_sha256":"b4d7c41b2ea1226acdf952514cbf335ef7f95c119a005c4a2294fe45a3c7ff3c","source":{"kind":"arxiv","id":"2005.03180","version":2},"attestation_state":"computed","paper":{"title":"Model Reduction and Neural Networks for Parametric PDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","stat.ML"],"primary_cat":"math.NA","authors_text":"Andrew M. Stuart, Bamdad Hosseini, Kaushik Bhattacharya, Nikola B. Kovachki","submitted_at":"2020-05-07T00:09:27Z","abstract_excerpt":"We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, "},"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":"2005.03180","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-05-07T00:09:27Z","cross_cats_sorted":["cs.LG","cs.NA","stat.ML"],"title_canon_sha256":"43a080843e08d92bc670172c8849dcad9a6c3f3d0877a9ff2755d02898151fd1","abstract_canon_sha256":"82f95841c6f73b31a1f8909a9ad64a1603d8c934ba1530f1cf4c9156c69c151f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:50:19.399505Z","signature_b64":"XWns8RKNlPh/FP99M8P/ZdPdREPKusvrrfctnwLHEb32aj7MX7IL6MrcvZmKTHk/K6JOfAVJ23dPkzdsUSuSDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4d7c41b2ea1226acdf952514cbf335ef7f95c119a005c4a2294fe45a3c7ff3c","last_reissued_at":"2026-07-05T02:50:19.399007Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:50:19.399007Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Model Reduction and Neural Networks for Parametric PDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","stat.ML"],"primary_cat":"math.NA","authors_text":"Andrew M. Stuart, Bamdad Hosseini, Kaushik Bhattacharya, Nikola B. Kovachki","submitted_at":"2020-05-07T00:09:27Z","abstract_excerpt":"We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.03180","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/2005.03180/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":"2005.03180","created_at":"2026-07-05T02:50:19.399070+00:00"},{"alias_kind":"arxiv_version","alias_value":"2005.03180v2","created_at":"2026-07-05T02:50:19.399070+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.03180","created_at":"2026-07-05T02:50:19.399070+00:00"},{"alias_kind":"pith_short_12","alias_value":"WTL4IGZOUERG","created_at":"2026-07-05T02:50:19.399070+00:00"},{"alias_kind":"pith_short_16","alias_value":"WTL4IGZOUERGVTPZ","created_at":"2026-07-05T02:50:19.399070+00:00"},{"alias_kind":"pith_short_8","alias_value":"WTL4IGZO","created_at":"2026-07-05T02:50:19.399070+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.07738","citing_title":"Physics-Informed Reduced-Order Operator Learning for Hyperelasticity in Continuum Micromechanics","ref_index":9,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WTL4IGZOUERGVTPZKJIUZPZTL3","json":"https://pith.science/pith/WTL4IGZOUERGVTPZKJIUZPZTL3.json","graph_json":"https://pith.science/api/pith-number/WTL4IGZOUERGVTPZKJIUZPZTL3/graph.json","events_json":"https://pith.science/api/pith-number/WTL4IGZOUERGVTPZKJIUZPZTL3/events.json","paper":"https://pith.science/paper/WTL4IGZO"},"agent_actions":{"view_html":"https://pith.science/pith/WTL4IGZOUERGVTPZKJIUZPZTL3","download_json":"https://pith.science/pith/WTL4IGZOUERGVTPZKJIUZPZTL3.json","view_paper":"https://pith.science/paper/WTL4IGZO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2005.03180&json=true","fetch_graph":"https://pith.science/api/pith-number/WTL4IGZOUERGVTPZKJIUZPZTL3/graph.json","fetch_events":"https://pith.science/api/pith-number/WTL4IGZOUERGVTPZKJIUZPZTL3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WTL4IGZOUERGVTPZKJIUZPZTL3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WTL4IGZOUERGVTPZKJIUZPZTL3/action/storage_attestation","attest_author":"https://pith.science/pith/WTL4IGZOUERGVTPZKJIUZPZTL3/action/author_attestation","sign_citation":"https://pith.science/pith/WTL4IGZOUERGVTPZKJIUZPZTL3/action/citation_signature","submit_replication":"https://pith.science/pith/WTL4IGZOUERGVTPZKJIUZPZTL3/action/replication_record"}},"created_at":"2026-07-05T02:50:19.399070+00:00","updated_at":"2026-07-05T02:50:19.399070+00:00"}