{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:XSZALDYJQFQMNGF53S3FZQ27BW","short_pith_number":"pith:XSZALDYJ","schema_version":"1.0","canonical_sha256":"bcb2058f098160c698bddcb65cc35f0d9503b1eb24a5b767d8af2afa73893952","source":{"kind":"arxiv","id":"2606.12735","version":1},"attestation_state":"computed","paper":{"title":"Physics-Informed Neural Networks and Radial Basis Functions for PDEs with Dirac Delta Sources","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Alexandre Tartakovsky, Manuel Reyna","submitted_at":"2026-06-10T22:53:43Z","abstract_excerpt":"Physics-Informed Neural Networks (PINNs) are a machine learning method for solving forward and inverse Partial Differential Equations (PDEs). When applied to PDEs with Dirac delta functions in the forcing terms, boundary conditions, or initial conditions, PINNs require approximating them with smooth surrogate functions, a practice that can introduce significant modeling errors. In this work, we exploit the interpretation of PINNs as Residual Least Squares (RLS) methods and show that this perspective enables direct treatment of Dirac delta terms by integrating the weak-form equation. Among RLS "},"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":"2606.12735","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-10T22:53:43Z","cross_cats_sorted":[],"title_canon_sha256":"3af2af3e5ac6e3723fd0e478a979fe7317b8dae1fa2daeef2fe9f56dafc70242","abstract_canon_sha256":"d034597bde2286b40946dd45d87e873c6153c11dd12c93f41c31bf47833f813f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:08:48.111473Z","signature_b64":"80PFVm7q8ib5w5hscji5QauHfs/cbSsUmPJp6xdnKXDiSLABFpdns1oA8IupyF6E9M4+cw4mfrTkAdFLTjR8BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bcb2058f098160c698bddcb65cc35f0d9503b1eb24a5b767d8af2afa73893952","last_reissued_at":"2026-06-12T01:08:48.110588Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:08:48.110588Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Physics-Informed Neural Networks and Radial Basis Functions for PDEs with Dirac Delta Sources","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Alexandre Tartakovsky, Manuel Reyna","submitted_at":"2026-06-10T22:53:43Z","abstract_excerpt":"Physics-Informed Neural Networks (PINNs) are a machine learning method for solving forward and inverse Partial Differential Equations (PDEs). When applied to PDEs with Dirac delta functions in the forcing terms, boundary conditions, or initial conditions, PINNs require approximating them with smooth surrogate functions, a practice that can introduce significant modeling errors. In this work, we exploit the interpretation of PINNs as Residual Least Squares (RLS) methods and show that this perspective enables direct treatment of Dirac delta terms by integrating the weak-form equation. Among RLS "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12735","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/2606.12735/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":"2606.12735","created_at":"2026-06-12T01:08:48.110729+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.12735v1","created_at":"2026-06-12T01:08:48.110729+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.12735","created_at":"2026-06-12T01:08:48.110729+00:00"},{"alias_kind":"pith_short_12","alias_value":"XSZALDYJQFQM","created_at":"2026-06-12T01:08:48.110729+00:00"},{"alias_kind":"pith_short_16","alias_value":"XSZALDYJQFQMNGF5","created_at":"2026-06-12T01:08:48.110729+00:00"},{"alias_kind":"pith_short_8","alias_value":"XSZALDYJ","created_at":"2026-06-12T01:08:48.110729+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XSZALDYJQFQMNGF53S3FZQ27BW","json":"https://pith.science/pith/XSZALDYJQFQMNGF53S3FZQ27BW.json","graph_json":"https://pith.science/api/pith-number/XSZALDYJQFQMNGF53S3FZQ27BW/graph.json","events_json":"https://pith.science/api/pith-number/XSZALDYJQFQMNGF53S3FZQ27BW/events.json","paper":"https://pith.science/paper/XSZALDYJ"},"agent_actions":{"view_html":"https://pith.science/pith/XSZALDYJQFQMNGF53S3FZQ27BW","download_json":"https://pith.science/pith/XSZALDYJQFQMNGF53S3FZQ27BW.json","view_paper":"https://pith.science/paper/XSZALDYJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.12735&json=true","fetch_graph":"https://pith.science/api/pith-number/XSZALDYJQFQMNGF53S3FZQ27BW/graph.json","fetch_events":"https://pith.science/api/pith-number/XSZALDYJQFQMNGF53S3FZQ27BW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XSZALDYJQFQMNGF53S3FZQ27BW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XSZALDYJQFQMNGF53S3FZQ27BW/action/storage_attestation","attest_author":"https://pith.science/pith/XSZALDYJQFQMNGF53S3FZQ27BW/action/author_attestation","sign_citation":"https://pith.science/pith/XSZALDYJQFQMNGF53S3FZQ27BW/action/citation_signature","submit_replication":"https://pith.science/pith/XSZALDYJQFQMNGF53S3FZQ27BW/action/replication_record"}},"created_at":"2026-06-12T01:08:48.110729+00:00","updated_at":"2026-06-12T01:08:48.110729+00:00"}