{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:KOYS66QCW3RIF53PFV7S3G44G2","short_pith_number":"pith:KOYS66QC","canonical_record":{"source":{"id":"2412.07728","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-10T18:24:46Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"c9493e66b032a6a02dc899bc1180c4767dcce61f5f9ffb63a6d962d435f180e4","abstract_canon_sha256":"c72a26f2d565a0643067bbbc87bc649c36ddbe2d4634981f5180bc5466a11f4a"},"schema_version":"1.0"},"canonical_sha256":"53b12f7a02b6e282f76f2d7f2d9b9c3685bb3aafb6f22d73561dd7e75e5a02b4","source":{"kind":"arxiv","id":"2412.07728","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.07728","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"arxiv_version","alias_value":"2412.07728v1","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.07728","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"pith_short_12","alias_value":"KOYS66QCW3RI","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"pith_short_16","alias_value":"KOYS66QCW3RIF53P","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"pith_short_8","alias_value":"KOYS66QC","created_at":"2026-07-05T09:47:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:KOYS66QCW3RIF53PFV7S3G44G2","target":"record","payload":{"canonical_record":{"source":{"id":"2412.07728","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-10T18:24:46Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"c9493e66b032a6a02dc899bc1180c4767dcce61f5f9ffb63a6d962d435f180e4","abstract_canon_sha256":"c72a26f2d565a0643067bbbc87bc649c36ddbe2d4634981f5180bc5466a11f4a"},"schema_version":"1.0"},"canonical_sha256":"53b12f7a02b6e282f76f2d7f2d9b9c3685bb3aafb6f22d73561dd7e75e5a02b4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:47:14.435416Z","signature_b64":"EB/xgex5X9xkRSroBIwLVTvJS/AVV1gobyzubwjv0+tqIZ7bU43ndC/iVR1rUE6EocZKrIKdPjqmxAZvQ8yOAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53b12f7a02b6e282f76f2d7f2d9b9c3685bb3aafb6f22d73561dd7e75e5a02b4","last_reissued_at":"2026-07-05T09:47:14.434855Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:47:14.434855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.07728","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:47:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b/UICrryCR2ACCFi9bUxyFhV1ug3KU8YbSNiKTzsYH2Wd7NA+ZNswN9CA+99n0ZJXWEaxEhkDC9/nQQJtdrBDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T16:03:29.042515Z"},"content_sha256":"aa10f9ac1fd496d03bb693e25d875a8b2ee62e8b769af1b0c358169ea9ee217a","schema_version":"1.0","event_id":"sha256:aa10f9ac1fd496d03bb693e25d875a8b2ee62e8b769af1b0c358169ea9ee217a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:KOYS66QCW3RIF53PFV7S3G44G2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Solving the Poisson Equation with Dirichlet data by shallow ReLU$^\\alpha$-networks: A regularity and approximation perspective","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"math.AP","authors_text":"Malhar Vaishampayan, Stephan Wojtowytsch","submitted_at":"2024-12-10T18:24:46Z","abstract_excerpt":"For several classes of neural PDE solvers (Deep Ritz, PINNs, DeepONets), the ability to approximate the solution or solution operator to a partial differential equation (PDE) hinges on the abilitiy of a neural network to approximate the solution in the spatial variables. We analyze the capacity of neural networks to approximate solutions to an elliptic PDE assuming that the boundary condition can be approximated efficiently. Our focus is on the Laplace operator with Dirichlet boundary condition on a half space and on neural networks with a single hidden layer and an activation function that is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.07728","kind":"arxiv","version":1},"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/2412.07728/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":"2026-07-05T09:47:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/AArSpopDIpJ9ZKD9MRyHP1dEEAu9sItWITvcbr2i7vi4oK4EIu+EoGMy21YL7BiujAUMpIHoWTcPPXjjZHDBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T16:03:29.042856Z"},"content_sha256":"3f056a019c237ecf370c882def8f55fcfd828ab625fe6c9f344bff83cb4eebcd","schema_version":"1.0","event_id":"sha256:3f056a019c237ecf370c882def8f55fcfd828ab625fe6c9f344bff83cb4eebcd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KOYS66QCW3RIF53PFV7S3G44G2/bundle.json","state_url":"https://pith.science/pith/KOYS66QCW3RIF53PFV7S3G44G2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KOYS66QCW3RIF53PFV7S3G44G2/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":"2026-08-06T16:03:29Z","links":{"resolver":"https://pith.science/pith/KOYS66QCW3RIF53PFV7S3G44G2","bundle":"https://pith.science/pith/KOYS66QCW3RIF53PFV7S3G44G2/bundle.json","state":"https://pith.science/pith/KOYS66QCW3RIF53PFV7S3G44G2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KOYS66QCW3RIF53PFV7S3G44G2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:KOYS66QCW3RIF53PFV7S3G44G2","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":"c72a26f2d565a0643067bbbc87bc649c36ddbe2d4634981f5180bc5466a11f4a","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-10T18:24:46Z","title_canon_sha256":"c9493e66b032a6a02dc899bc1180c4767dcce61f5f9ffb63a6d962d435f180e4"},"schema_version":"1.0","source":{"id":"2412.07728","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.07728","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"arxiv_version","alias_value":"2412.07728v1","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.07728","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"pith_short_12","alias_value":"KOYS66QCW3RI","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"pith_short_16","alias_value":"KOYS66QCW3RIF53P","created_at":"2026-07-05T09:47:14Z"},{"alias_kind":"pith_short_8","alias_value":"KOYS66QC","created_at":"2026-07-05T09:47:14Z"}],"graph_snapshots":[{"event_id":"sha256:3f056a019c237ecf370c882def8f55fcfd828ab625fe6c9f344bff83cb4eebcd","target":"graph","created_at":"2026-07-05T09:47:14Z","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/2412.07728/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For several classes of neural PDE solvers (Deep Ritz, PINNs, DeepONets), the ability to approximate the solution or solution operator to a partial differential equation (PDE) hinges on the abilitiy of a neural network to approximate the solution in the spatial variables. We analyze the capacity of neural networks to approximate solutions to an elliptic PDE assuming that the boundary condition can be approximated efficiently. Our focus is on the Laplace operator with Dirichlet boundary condition on a half space and on neural networks with a single hidden layer and an activation function that is","authors_text":"Malhar Vaishampayan, Stephan Wojtowytsch","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-10T18:24:46Z","title":"Solving the Poisson Equation with Dirichlet data by shallow ReLU$^\\alpha$-networks: A regularity and approximation perspective"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.07728","kind":"arxiv","version":1},"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:aa10f9ac1fd496d03bb693e25d875a8b2ee62e8b769af1b0c358169ea9ee217a","target":"record","created_at":"2026-07-05T09:47:14Z","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":"c72a26f2d565a0643067bbbc87bc649c36ddbe2d4634981f5180bc5466a11f4a","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-12-10T18:24:46Z","title_canon_sha256":"c9493e66b032a6a02dc899bc1180c4767dcce61f5f9ffb63a6d962d435f180e4"},"schema_version":"1.0","source":{"id":"2412.07728","kind":"arxiv","version":1}},"canonical_sha256":"53b12f7a02b6e282f76f2d7f2d9b9c3685bb3aafb6f22d73561dd7e75e5a02b4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"53b12f7a02b6e282f76f2d7f2d9b9c3685bb3aafb6f22d73561dd7e75e5a02b4","first_computed_at":"2026-07-05T09:47:14.434855Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:47:14.434855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EB/xgex5X9xkRSroBIwLVTvJS/AVV1gobyzubwjv0+tqIZ7bU43ndC/iVR1rUE6EocZKrIKdPjqmxAZvQ8yOAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:47:14.435416Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.07728","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa10f9ac1fd496d03bb693e25d875a8b2ee62e8b769af1b0c358169ea9ee217a","sha256:3f056a019c237ecf370c882def8f55fcfd828ab625fe6c9f344bff83cb4eebcd"],"state_sha256":"3bf6d8d41fe31bb48634ccd32b7db56177a84aad28c73270dfec9e90022614b4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jjE1Qfn/AXyHXsTJRsVY4Su68drb4fovgSFvTckEDGFEyf4bt5Z3q6zGmkdeaNEW19VVLGGNeBPE9Sm6I0i4BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T16:03:29.046796Z","bundle_sha256":"a7b8463783592ed933de2cee1547d9c4aefb7c7456d35b95fe9d5b307584b188"}}