{"paper":{"title":"Neural Operator: Graph Kernel Network for Partial Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A single set of network parameters can describe mappings between infinite-dimensional spaces and their finite approximations using graph kernel networks.","cross_cats":["cs.NA","math.NA","stat.ML"],"primary_cat":"cs.LG","authors_text":"Andrew Stuart, Anima Anandkumar, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Nikola Kovachki, Zongyi Li","submitted_at":"2020-03-07T01:56:20Z","abstract_excerpt":"The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators). The key innovation in our work is that a single set of network parameters, within a carefully designed network architecture, may be used to describe mappings between infinite-dimensional spaces and between different finite-dimensional approximations of those spaces"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A single set of network parameters, within a carefully designed network architecture, may be used to describe mappings between infinite-dimensional spaces and between different finite-dimensional approximations of those spaces.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That message passing on graphs can faithfully approximate the required integral operators for the target PDE mappings without introducing discretization-dependent artifacts that break generalization.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Graph Kernel Networks learn PDE solution operators that generalize across discretization methods and grid resolutions using graph-based kernel integration.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A single set of network parameters can describe mappings between infinite-dimensional spaces and their finite approximations using graph kernel networks.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e5e6833e714884f08fc95423b8a07bb3df63a1f21c0311209370258b8fc6ed80"},"source":{"id":"2003.03485","kind":"arxiv","version":1},"verdict":{"id":"186e514a-8a15-49cb-8c39-1e283d135728","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T22:13:20.539096Z","strongest_claim":"A single set of network parameters, within a carefully designed network architecture, may be used to describe mappings between infinite-dimensional spaces and between different finite-dimensional approximations of those spaces.","one_line_summary":"Graph Kernel Networks learn PDE solution operators that generalize across discretization methods and grid resolutions using graph-based kernel integration.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That message passing on graphs can faithfully approximate the required integral operators for the target PDE mappings without introducing discretization-dependent artifacts that break generalization.","pith_extraction_headline":"A single set of network parameters can describe mappings between infinite-dimensional spaces and their finite approximations using graph kernel networks."},"references":{"count":141,"sample":[{"doi":"10.1137/07070111x","year":2009,"title":"Kolda and Brett W","work_id":"00b6411a-bd17-4725-9d1c-60a55cd201b5","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Hsu and Sham M","work_id":"0bca2428-01fc-484e-8b9b-57e0ae50249b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/090752286","year":2011,"title":"Tensor-train decomposition","work_id":"880f1684-5487-419a-885a-823cf8b643c4","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Fast adaptive interpolation of multi-dimensional arrays in tensor train format , isbn =","work_id":"c3ef5637-48d5-4929-a1d6-233d247e9e96","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Gorodetsky and Sertac Karaman and Youssef M","work_id":"e4c45c80-38b9-445a-83ba-205f1910fbc7","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":141,"snapshot_sha256":"f7789c609d414f0db2016e8242d8b48e843f3951c6aa40f44745ab3d709ebc3e","internal_anchors":16},"formal_canon":{"evidence_count":2,"snapshot_sha256":"fb2252c5ab474b3359f68b4c90dac6b7e167f0a9e5dcb53de459f27eb7ebdcdf"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}