{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:WWZRHSLCYFCZD3W2T4FCAXUHEF","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":"7ee3982a3e69350ee03597a49995a0ec2fdfd1c7f6700fe3c0a6c4ef36368c8f","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2021-03-24T16:30:23Z","title_canon_sha256":"73a8ec9f9e4fcaf0b10084129d3872746dd28cff11e4b94d7f2fe71b4357da03"},"schema_version":"1.0","source":{"id":"2103.13330","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2103.13330","created_at":"2026-07-05T04:13:22Z"},{"alias_kind":"arxiv_version","alias_value":"2103.13330v2","created_at":"2026-07-05T04:13:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2103.13330","created_at":"2026-07-05T04:13:22Z"},{"alias_kind":"pith_short_12","alias_value":"WWZRHSLCYFCZ","created_at":"2026-07-05T04:13:22Z"},{"alias_kind":"pith_short_16","alias_value":"WWZRHSLCYFCZD3W2","created_at":"2026-07-05T04:13:22Z"},{"alias_kind":"pith_short_8","alias_value":"WWZRHSLC","created_at":"2026-07-05T04:13:22Z"}],"graph_snapshots":[{"event_id":"sha256:08532bc61c5157268a2bbab54d88cad653c3d1cd2eb69c705c1c29a890ef537c","target":"graph","created_at":"2026-07-05T04:13:22Z","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/2103.13330/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) \\cite{wan11} for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with $\\mathrm{ReLU}^2$ activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyper-parameter","authors_text":"Chenguang Duan, Xiliang Lu, Yanming Lai, Yuling Jiao, Zhijian Yang","cross_cats":["cs.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2021-03-24T16:30:23Z","title":"Convergence Rate Analysis for Deep Ritz Method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.13330","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:b1c96146c4b0d07dbb456a6da1426e2ff7df9713445fe59add83d7a85286a22e","target":"record","created_at":"2026-07-05T04:13:22Z","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":"7ee3982a3e69350ee03597a49995a0ec2fdfd1c7f6700fe3c0a6c4ef36368c8f","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2021-03-24T16:30:23Z","title_canon_sha256":"73a8ec9f9e4fcaf0b10084129d3872746dd28cff11e4b94d7f2fe71b4357da03"},"schema_version":"1.0","source":{"id":"2103.13330","kind":"arxiv","version":2}},"canonical_sha256":"b5b313c962c14591eeda9f0a205e87217fc944984a84db5303d863d06e9ac349","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b5b313c962c14591eeda9f0a205e87217fc944984a84db5303d863d06e9ac349","first_computed_at":"2026-07-05T04:13:22.694065Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:13:22.694065Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XwXp+8OBAfnfUA2OREckU8uoDcBfwau69IakM9sy2URlJgMaTAeuYqM7y/YvsUPg2rgg4rCMBpf5gc25sM96DA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:13:22.694639Z","signed_message":"canonical_sha256_bytes"},"source_id":"2103.13330","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b1c96146c4b0d07dbb456a6da1426e2ff7df9713445fe59add83d7a85286a22e","sha256:08532bc61c5157268a2bbab54d88cad653c3d1cd2eb69c705c1c29a890ef537c"],"state_sha256":"a91745d5e0fcce61db6d1400511bade1812ace14b6777fb5f9ae2b4a98e96f44"}