{"as_of":"2026-08-07T23:35:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:d8e8f06016f1e27bfd2ba8cb2f223f14cfecbb03f4bada064f438baf087c7157","coverage":[{"denominator":48,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":48,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-04T07:32:30.722586Z","state":"measured"},{"denominator":48,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":48,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-07T06:34:17.273281+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2510.25731/citation-record","integrity":"/paper/2510.25731/integrity","json":"/paper/2510.25731/citation-record.json","paper":"/paper/2510.25731"},"outbound":[{"citation":{"cited_paper":{"arxiv_id":"2311.04293","last_updated":"2023-11-07T19:07:16Z","snapshot_observed_at":"2026-08-06T00:34:49.484773Z","submitted_at":"2023-11-07T19:07:16Z","title":"Lie Point Symmetry and Physics Informed Networks","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2311.04293","snapshot_observed_at":"2026-08-04T07:32:30.350623Z","title":"Lie Point Symmetry and Physics-Informed Networks","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.350623Z"},"links":{"cited_paper":"/paper/2311.04293","citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:c96f4adb1075ac6a3e01af50b8239a0c7a34643d012ccd1d1f20b07bab41de60","observation_id":"71614209-226b-4355-8875-2285c5a8fa1a","resolution":{"observed_at":"2026-08-04T07:32:30.350623Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.361750Z","title":null,"venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.361750Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:67aa09f0a96cc99c22b25e70bd80b0adcc2d2d50805bce8a8fca7609616e62d2","observation_id":"c87031c6-7afd-414f-be4e-f63ca3e68f1b","resolution":{"observed_at":"2026-08-04T07:32:30.361750Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.369576Z","title":null,"venue":null,"work_id":null,"year":1989},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.369576Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:b20416d7baa78e1d6a043f5d190f12277fe9fbf7bcdd9b777f0b3bfeea73bb68","observation_id":"1b91d51f-0dda-44ec-aa07-a3612a747b79","resolution":{"observed_at":"2026-08-04T07:32:30.369576Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.378119Z","title":null,"venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.378119Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:5071cbea8393c4422a71cd249c68e59baef04ada3dbe5ba0a7cedb7373f92632","observation_id":"02bd3464-ddfe-4537-ae26-16fade4da92a","resolution":{"observed_at":"2026-08-04T07:32:30.378119Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.387993Z","title":"Lie Point Symmetry Data Augmentation for Neural PDE Solvers","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.387993Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:010c8feb230243684c47c9e17579a7409470e7a682a323adee06eb6506a35cb8","observation_id":"cf7f8b69-3e52-4b85-bc31-3e418856d6ad","resolution":{"observed_at":"2026-08-04T07:32:30.387993Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.395357Z","title":"A comparative study of some computer algebra packages which determine the Lie point symmetries of differential equations","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.395357Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:ee97148cf9e41970ce49f307f931d74a2e49e4823f0325351d77a6da74bda88f","observation_id":"fd54b118-0406-4d82-9415-c5963a4e42d8","resolution":{"observed_at":"2026-08-04T07:32:30.395357Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1016/0010-4655(91)90080-5","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"The computer calculation of Lie point symmetries of large systems of differential equations","venue":"Computer Physics Communications","work_id":"109d807d-d1ec-476e-b563-f8d0dcff2a05","year":1991},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.411829Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:00e373923fb6ca42bfe501cd0d6a8bec91f5b31d1ea3dd2b0f71406e201dbcb6","observation_id":"28954195-68b9-4398-8514-22d655c08c9c","resolution":{"observed_at":"2026-08-04T07:34:12.725434Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.418293Z","title":"Group equivariant convolutional networks","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.418293Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:579345b4f1acdfe1cedc7badf152b9f3170185fa861db65c7327847c06fea84f","observation_id":"0b0b2a84-8c7e-4536-804c-ebc183b35533","resolution":{"observed_at":"2026-08-04T07:32:30.418293Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.424575Z","title":"Lagrangian neural networks","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.424575Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:96e31308efe038b11763615c7f93089b9092112cc48ff8b330f58a7d089d3769","observation_id":"8b740931-4c56-4275-a538-c6e91e28597d","resolution":{"observed_at":"2026-08-04T07:32:30.424575Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.430019Z","title":null,"venue":null,"work_id":null,"year":1999},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.430019Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:ccaf68baf55da3a75c4b093e7a3f1b24be2985a8e3876c04128f823a4a2ba70f","observation_id":"23c79049-e10d-4c18-ab6a-127708114b36","resolution":{"observed_at":"2026-08-04T07:32:30.430019Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.436544Z","title":null,"venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.436544Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:7688c50de89fba44f0715902b382974801c244ce221fed0dab36a960aba17b17","observation_id":"39ab142c-f146-4770-9cac-103d0a48abb8","resolution":{"observed_at":"2026-08-04T07:32:30.436544Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.443942Z","title":"The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate","venue":null,"work_id":null,"year":1973},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.443942Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:9db09c0df447c47017245602a50abddfcb4cb1257102c06c8d68ac7b26754a70","observation_id":"7640a806-414f-48bc-9e12-856472531645","resolution":{"observed_at":"2026-08-04T07:32:30.443942Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.449929Z","title":"Hamiltonian neural networks","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.449929Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:d2e4cf1d08f7296098cf390960583bae6b67dda7301e82d102d7cfd0516415e8","observation_id":"a08a45ca-7fb0-43a9-b7b5-381403b726c3","resolution":{"observed_at":"2026-08-04T07:32:30.449929Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.455009Z","title":"Symbolic Software for Lie Symmetry Analysis","venue":null,"work_id":null,"year":1996},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.455009Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:48c86313f66ea872589c85ebbd0ef263946f3f2f93e40799ba00396184474fc1","observation_id":"8080db46-e4fc-4957-9235-a14e0af826e4","resolution":{"observed_at":"2026-08-04T07:32:30.455009Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.460274Z","title":"Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.460274Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:581655050b4f02fe80fde18ad62848bc0c29852db5976ef83227ed9cf90b0a4a","observation_id":"a20d57f1-e2c7-4818-8538-981dacc9ab86","resolution":{"observed_at":"2026-08-04T07:32:30.460274Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.465035Z","title":null,"venue":null,"work_id":null,"year":1978},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.465035Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:50097c1f98fea26caab36094ff003e06a7154b8fa0a6f3cfb7d2a041a380d970","observation_id":"cd16b344-08e5-406b-bd0e-26286d2c7348","resolution":{"observed_at":"2026-08-04T07:32:30.465035Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.470288Z","title":"Physics- informed machine learning","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.470288Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:75ec7e1f6ddbd0dab787ec92b992122e7fbe07c202d802cbdd852b5939cd319f","observation_id":"e0a03643-dad6-4253-813a-39c90446534d","resolution":{"observed_at":"2026-08-04T07:32:30.470288Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2109.01050","last_updated":"2021-11-11T18:47:07Z","snapshot_observed_at":"2026-07-06T11:43:47.468200Z","submitted_at":"2021-09-02T16:06:45Z","title":"Characterizing possible failure modes in physics-informed neural networks","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2109.01050","snapshot_observed_at":"2026-08-04T07:32:30.476380Z","title":"Characterizing possible failure modes in physics-informed neural networks","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.476380Z"},"links":{"cited_paper":"/paper/2109.01050","citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:b03492204c8f46d6d757f650b5e7fd674646a0d78792ac8ff1e356be1f1f24a3","observation_id":"840fd397-46ca-4d0d-b938-b5cef226346e","resolution":{"observed_at":"2026-08-04T07:32:30.476380Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"physics/9705023","last_updated":"1997-05-19T10:35:48Z","snapshot_observed_at":"2026-08-06T05:21:45.988917Z","submitted_at":"1997-05-19T10:35:48Z","title":"Artificial Neural Networks for Solving Ordinary and Partial Differential Equations","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"physics/9705023","snapshot_observed_at":"2026-08-04T07:32:30.486756Z","title":"Artificial Neural Networks for Solving Ordinary and Partial Differential Equations","venue":null,"work_id":null,"year":1998},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.486756Z"},"links":{"cited_paper":"/paper/physics/9705023","citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:2af5be405aa6b41b05cbb4d59c2113f70d882d58d5f2feba3cbaa81e4a15d2d0","observation_id":"0968042d-e724-4cc0-ae8f-9fdb929b64db","resolution":{"observed_at":"2026-08-04T07:32:30.486756Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.493525Z","title":"PINNACLE: PINN Adaptive ColLocation and Experimental Points Selection","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.493525Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:1ce034217bf6a27651052f4509877c4c2d01543254ac67b899de86e6cee7c7a2","observation_id":"4f96d490-2977-450e-94ff-c9792e7e9e8b","resolution":{"observed_at":"2026-08-04T07:32:30.493525Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.503365Z","title":null,"venue":null,"work_id":null,"year":2007},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.503365Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:a492ef15339f49f25825f8271e2b0061c0af66111ba8c28df401967f83a6d2cb","observation_id":"c6ecedfd-4ac6-46b4-a163-fa8111b2972d","resolution":{"observed_at":"2026-08-04T07:32:30.503365Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2102.04626","last_updated":"2021-02-09T03:18:15Z","snapshot_observed_at":"2026-07-06T10:39:42.676631Z","submitted_at":"2021-02-09T03:18:15Z","title":"Physics-informed neural networks with hard constraints for inverse design","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2102.04626","snapshot_observed_at":"2026-08-04T07:32:30.510584Z","title":"Physics-informed neural networks with hard constraints for inverse design","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.510584Z"},"links":{"cited_paper":"/paper/2102.04626","citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:f34a5272807ef16de65287638eac3844381e88066d425140a8bc3258fb960fca","observation_id":"f30e6d12-7e76-45f1-a83f-56f6cf19d635","resolution":{"observed_at":"2026-08-04T07:32:30.510584Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.517931Z","title":"PPINNs: Physics-informed neural networks with prior","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.517931Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:ac8c0c69dbc005546a7a24b92bef8c34302f48fc3a2578d90bf1f66949483887","observation_id":"086fcb70-3e2d-4f92-a4ab-7b240c068751","resolution":{"observed_at":"2026-08-04T07:32:30.517931Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.523488Z","title":"Self-Supervised Learning with Lie Symmetries for Partial Differential Equations","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.523488Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:ed0444b9806a8fc46ba77fa579e97560ef86f5d26a48613fae6d02309554a93c","observation_id":"eebcaaa9-a8a8-49e5-94d2-a7de6b58c24b","resolution":{"observed_at":"2026-08-04T07:32:30.523488Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.529999Z","title":null,"venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.529999Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:d05f209c2d79aeeeadbe09ebc0aeb4a5ce74b0e78df00eb5adf9f6a6fa16da45","observation_id":"0ca61101-f1cf-4d6c-b275-10956f183707","resolution":{"observed_at":"2026-08-04T07:32:30.529999Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.537036Z","title":"Variable projection for nonlinear least squares problems","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.537036Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:739fc6e1eb849e0f729a2d1a4accd7fdce4ba4734683a4b6e8c130d604bc0292","observation_id":"09fc6ee8-768e-4634-961a-a5c7302c3f50","resolution":{"observed_at":"2026-08-04T07:32:30.537036Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.3390/sym2020658","metadata_source":"openalex","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Lie Symmetries of Differential Equations: Classical Results and Recent Contri- butions","venue":"Symmetry","work_id":"9e5de297-9a77-4207-8ee1-303ee4103794","year":2010},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.543273Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:7c6b597a046e534458169b1473a2be7899156e1ca4619badc38d6e76c4b599f3","observation_id":"3cedc9a4-e0f5-4a87-b636-a2a41fab15e5","resolution":{"observed_at":"2026-08-04T07:34:12.561422Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2105.11534","last_updated":"2021-05-07T11:22:13Z","snapshot_observed_at":"2026-07-06T11:12:21.382058Z","submitted_at":"2021-05-07T11:22:13Z","title":"ReLie: a Reduce program for Lie group analysis of differential equations","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2105.11534","snapshot_observed_at":"2026-08-04T07:32:30.552598Z","title":"ReLie: a Reduce program for Lie group analysis of differential equations","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.552598Z"},"links":{"cited_paper":"/paper/2105.11534","citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:b0d49aeb00878eeb031e867467c4a4825d647730c32d527bb05ef6b1044455dc","observation_id":"845d7a31-5e19-4439-b5ce-3f0d6c1b76f9","resolution":{"observed_at":"2026-08-04T07:32:30.552598Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.561228Z","title":null,"venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.561228Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:8f50a6ce155110c74db8da82ca1a90b904aed8de73fb15e7a88f88031a5c0a67","observation_id":"4fe470c8-9600-4be9-84dd-a76094f30d0e","resolution":{"observed_at":"2026-08-04T07:32:30.561228Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.574689Z","title":"Symmetry and explicit solutions of partial differential equations","venue":null,"work_id":null,"year":1992},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.574689Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:0a0646e2ef3cf3195fd0e626d0099613588c7c35e8c19404165dc8c92f742e8c","observation_id":"e74ef88d-16be-489c-a6cb-0919a67ac672","resolution":{"observed_at":"2026-08-04T07:32:30.574689Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.582749Z","title":null,"venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.582749Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:f41bd5d5b0d2f225724ba80858de5891bb7b11416b8d6f7b4461e25d6ec777ef","observation_id":"deab01f0-41cf-46c5-a5c4-c27c16c57283","resolution":{"observed_at":"2026-08-04T07:32:30.582749Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.589043Z","title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial dif- ferential equations","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.589043Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:f2504c337b0bc9a4094980a3d5cdd924ee13ced5e7df834d60d834bccca09979","observation_id":"d61fcbef-15d6-4f16-ab5e-b64d121e9365","resolution":{"observed_at":"2026-08-04T07:32:30.589043Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.599998Z","title":"Challenges in Training PINNs: A Loss Landscape Perspective","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.599998Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:302703c6bc17c76528d6445d4ec7ace021584ca016542636e316111bb7cdc20d","observation_id":"1aa65342-a797-47e5-9f63-81dc41cd8e64","resolution":{"observed_at":"2026-08-04T07:32:30.599998Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.608461Z","title":null,"venue":null,"work_id":null,"year":1991},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.608461Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:64eb3e7a47dce9c5f3d201bcd4be1d2cdba2bdaa627a59034da9632a6c9f02ba","observation_id":"952b260a-c04a-4c9a-83e6-e691d2a8ab9b","resolution":{"observed_at":"2026-08-04T07:32:30.608461Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1708.07469","last_updated":"2018-09-05T19:39:17Z","snapshot_observed_at":"2026-07-06T05:56:35.070217Z","submitted_at":"2017-08-24T15:50:24Z","title":"DGM: A deep learning algorithm for solving partial differential equations","version":5},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1708.07469","snapshot_observed_at":"2026-08-04T07:32:30.619525Z","title":"DGM: A deep learning algorithm for solving partial dif- ferential equations","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.619525Z"},"links":{"cited_paper":"/paper/1708.07469","citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:3d9a70dd8b5cb3c99bc5e26d687bf8222bb3e3d2d92e6d39294adbe775f510d7","observation_id":"0638c4e1-cb13-465c-90e8-575fdf97c1b3","resolution":{"observed_at":"2026-08-04T07:32:30.619525Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1409.8364","last_updated":"2014-09-30T02:18:30Z","snapshot_observed_at":"2026-07-06T03:55:57.965897Z","submitted_at":"2014-09-30T02:18:30Z","title":"Practical Guide to the Symbolic Computation of Symmetries of Differential Equations","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1409.8364","snapshot_observed_at":"2026-08-04T07:32:30.629362Z","title":"Practical Guide to the Symbolic Computation of Symmetries of Differential Equations","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.629362Z"},"links":{"cited_paper":"/paper/1409.8364","citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:ae38386f6ea068efa56fafb503317ce0d4ee8079280ae48abf7c25de89e1a9b9","observation_id":"c139d7d5-2271-41d5-b2a6-0fee18d06f25","resolution":{"observed_at":"2026-08-04T07:32:30.629362Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.636863Z","title":"Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.636863Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:aa6b5e2877602effa40fd802d28381ffb7cd0fe17225d32d8a660f592d071461","observation_id":"b6748a8c-a81b-46a4-bcae-a8c66d8218db","resolution":{"observed_at":"2026-08-04T07:32:30.636863Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.644623Z","title":"Solving Differential Equations with Genetic Programming","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.644623Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:8783e324672b5442924e2b55a45dcb9bd5348196e29d7b41a6359c48032c98d4","observation_id":"8b5fba76-9a4b-4e30-bb1a-bd4ebeaf1fc5","resolution":{"observed_at":"2026-08-04T07:32:30.644623Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.652382Z","title":"Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.652382Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:6ad73d6788578f8eadad9ee9a9235df7d4a6fa8119f8b8f19f909291804e5c79","observation_id":"724d0a6a-9144-4f6a-9146-e2ad521e8029","resolution":{"observed_at":"2026-08-04T07:32:30.652382Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2308.08468","last_updated":"2023-08-16T16:19:25Z","snapshot_observed_at":"2026-07-06T16:06:54.634531Z","submitted_at":"2023-08-16T16:19:25Z","title":"An Expert's Guide to Training Physics-informed Neural Networks","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2308.08468","snapshot_observed_at":"2026-08-04T07:32:30.658305Z","title":"An Expert’s Guide to Training Physics- informed Neural Networks","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.658305Z"},"links":{"cited_paper":"/paper/2308.08468","citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:fd04f252802fb33456ca53ffb2a12736424ed3fe930f1c7f6f82d6d8dd06753a","observation_id":"8a877869-91ed-4362-8ba5-6438aa6b0235","resolution":{"observed_at":"2026-08-04T07:32:30.658305Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.666328Z","title":"Understanding and Mitigating Gradient Flow Patholo- gies in Physics-Informed Neural Networks","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.666328Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:8f5a7535f2330ceb3eb3ae2b318793697241f22168251b292b441f544cb1b6e1","observation_id":"59853b10-f777-4e0b-95ff-f6dcc4208899","resolution":{"observed_at":"2026-08-04T07:32:30.666328Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.673538Z","title":"When and why PINNs fail to train: A neural tangent kernel perspective","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.673538Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:39d666ee907d2f9436f5b8ca1497893093e38f44111aaa4df6bb07eddb5066bc","observation_id":"e4c82f8c-4d11-4607-9d5e-6195ff49192b","resolution":{"observed_at":"2026-08-04T07:32:30.673538Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.683089Z","title":null,"venue":null,"work_id":null,"year":1967},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.683089Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:360224bee36504bf45f133abc98bb4755070419a82834b0c5843860353a89a93","observation_id":"52164e0b-f8bf-49c5-a9d4-5b87ad7bf93d","resolution":{"observed_at":"2026-08-04T07:32:30.683089Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.692841Z","title":"A Comprehensive Study of Non-Adaptive and Residual-Based Adaptive Sampling for Physics-Informed Neural Networks","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.692841Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:1ab390db3633abe6cdf0574af0d44a2fbbb85cf6b1b879e0c8a5aab4e0309416","observation_id":"4eec0d56-6a4a-411f-a5fe-fa52c1f0ed5b","resolution":{"observed_at":"2026-08-04T07:32:30.692841Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.702433Z","title":"Latent Space Symmetry Discovery","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.702433Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:64fc4da43598c751ef5d192df69f756672d72b08b8b20f0ad0914136d9eff04c","observation_id":"e0ac995d-7365-4fe8-8685-3c9882200523","resolution":{"observed_at":"2026-08-04T07:32:30.702433Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.709327Z","title":"Symmetry-Informed Governing Equa- tion Discovery","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.709327Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:b1f5b519dac09af1395dfd0f3cab29b7d7f6b6c42898e3429217ac5b8fc98c45","observation_id":"36b6e1ba-8146-4a53-b5f3-413dc43152dc","resolution":{"observed_at":"2026-08-04T07:32:30.709327Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.716933Z","title":"A Symmetry Group Based Supervised Learning Method for Solving Partial Differential Equations","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.716933Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:c2888a4131474f6d8a0156a8fda3f049d8eb01e13996a5aaa35c1d823c49f28d","observation_id":"f9d82f1c-49de-465f-bbf1-2155dba210ab","resolution":{"observed_at":"2026-08-04T07:32:30.716933Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T07:32:30.722586Z","title":"Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries","version":2},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-04T07:32:30.722586Z"},"links":{"citing_paper":"/paper/2510.25731"},"observation_digest":"sha256:e0553fc1c49ec1154491b6ac619333c4f505e4821c91d3c6672d132854aeb36a","observation_id":"01c64f81-0d87-427a-9505-8f5f507fee0f","resolution":{"observed_at":"2026-08-04T07:32:30.722586Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2510.25731","last_updated":"2026-06-25T18:41:25Z","latest_version":2,"primary_category":"cs.LG","snapshot_observed_at":"2026-08-07T13:33:21.617919Z","submitted_at":"2025-10-29T17:37:27Z","title":"LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries"},"reference_resolution":{"displayed":48,"state_counts":{"malformed_identifier":3,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":43,"verified_exact":2,"verified_fuzzy":0},"total_outbound_references":48},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-07T06:34:17.273281+00:00","source":"crossref"},{"observed_at":"2026-08-07T06:34:11.927384+00:00","source":"retraction_watch"}],"thesis":"As of 7 August 2026, this Paper Citation Record lists 48 of 48 outbound references and 0 inbound Pith citation observations for arXiv:2510.25731."}