{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:OZ2TXWJNI5TCWCG7FV4MOUE2QD","short_pith_number":"pith:OZ2TXWJN","schema_version":"1.0","canonical_sha256":"76753bd92d47662b08df2d78c7509a80e87c96b02df67ca949aaeb4525832b07","source":{"kind":"arxiv","id":"2312.14688","version":1},"attestation_state":"computed","paper":{"title":"A Mathematical Guide to Operator Learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","cs.LG","cs.NA"],"primary_cat":"math.NA","authors_text":"Alex Townsend, Nicolas Boull\\'e","submitted_at":"2023-12-22T13:43:57Z","abstract_excerpt":"Operator learning aims to discover properties of an underlying dynamical system or partial differential equation (PDE) from data. Here, we present a step-by-step guide to operator learning. We explain the types of problems and PDEs amenable to operator learning, discuss various neural network architectures, and explain how to employ numerical PDE solvers effectively. We also give advice on how to create and manage training data and conduct optimization. We offer intuition behind the various neural network architectures employed in operator learning by motivating them from the point-of-view of "},"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":"2312.14688","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2023-12-22T13:43:57Z","cross_cats_sorted":["cs.AI","cs.LG","cs.NA"],"title_canon_sha256":"e6b4ba0bb98a35a1000c8928bac951ad9ffb0a605bd4277539d45ea780a6c5ec","abstract_canon_sha256":"485d420532827b0a0ff31bbbd702195d5de7672a77f7a65a7cb80f3b06c8d177"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:55:17.839197Z","signature_b64":"v3fkqBy73WU29tECARseZz9QIvw8nuLpc7XzYXMTLABAJZFJ3JjLv1GEtEYAURcLNH8n+SNm+gA9h1MXh/J1DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76753bd92d47662b08df2d78c7509a80e87c96b02df67ca949aaeb4525832b07","last_reissued_at":"2026-07-05T10:55:17.838716Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:55:17.838716Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Mathematical Guide to Operator Learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","cs.LG","cs.NA"],"primary_cat":"math.NA","authors_text":"Alex Townsend, Nicolas Boull\\'e","submitted_at":"2023-12-22T13:43:57Z","abstract_excerpt":"Operator learning aims to discover properties of an underlying dynamical system or partial differential equation (PDE) from data. Here, we present a step-by-step guide to operator learning. We explain the types of problems and PDEs amenable to operator learning, discuss various neural network architectures, and explain how to employ numerical PDE solvers effectively. We also give advice on how to create and manage training data and conduct optimization. We offer intuition behind the various neural network architectures employed in operator learning by motivating them from the point-of-view of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.14688","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/2312.14688/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":"2312.14688","created_at":"2026-07-05T10:55:17.838767+00:00"},{"alias_kind":"arxiv_version","alias_value":"2312.14688v1","created_at":"2026-07-05T10:55:17.838767+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2312.14688","created_at":"2026-07-05T10:55:17.838767+00:00"},{"alias_kind":"pith_short_12","alias_value":"OZ2TXWJNI5TC","created_at":"2026-07-05T10:55:17.838767+00:00"},{"alias_kind":"pith_short_16","alias_value":"OZ2TXWJNI5TCWCG7","created_at":"2026-07-05T10:55:17.838767+00:00"},{"alias_kind":"pith_short_8","alias_value":"OZ2TXWJN","created_at":"2026-07-05T10:55:17.838767+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":6,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.19268","citing_title":"Patnaik-Pearson intrinsic dimension for internal representations of neural networks","ref_index":2,"is_internal_anchor":false},{"citing_arxiv_id":"2606.19268","citing_title":"Patnaik-Pearson intrinsic dimension for internal representations of neural networks","ref_index":3,"is_internal_anchor":false},{"citing_arxiv_id":"2606.08654","citing_title":"Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime","ref_index":4,"is_internal_anchor":false},{"citing_arxiv_id":"2606.00296","citing_title":"Is Zero-Shot Super-Resolution Possible in Operator Learning?","ref_index":48,"is_internal_anchor":false},{"citing_arxiv_id":"2410.06074","citing_title":"Scalable Mechanistic Neural Networks for Differential Equations and Machine Learning","ref_index":1,"is_internal_anchor":false},{"citing_arxiv_id":"2604.27052","citing_title":"Man, Machine, and Mathematics","ref_index":6,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OZ2TXWJNI5TCWCG7FV4MOUE2QD","json":"https://pith.science/pith/OZ2TXWJNI5TCWCG7FV4MOUE2QD.json","graph_json":"https://pith.science/api/pith-number/OZ2TXWJNI5TCWCG7FV4MOUE2QD/graph.json","events_json":"https://pith.science/api/pith-number/OZ2TXWJNI5TCWCG7FV4MOUE2QD/events.json","paper":"https://pith.science/paper/OZ2TXWJN"},"agent_actions":{"view_html":"https://pith.science/pith/OZ2TXWJNI5TCWCG7FV4MOUE2QD","download_json":"https://pith.science/pith/OZ2TXWJNI5TCWCG7FV4MOUE2QD.json","view_paper":"https://pith.science/paper/OZ2TXWJN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2312.14688&json=true","fetch_graph":"https://pith.science/api/pith-number/OZ2TXWJNI5TCWCG7FV4MOUE2QD/graph.json","fetch_events":"https://pith.science/api/pith-number/OZ2TXWJNI5TCWCG7FV4MOUE2QD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OZ2TXWJNI5TCWCG7FV4MOUE2QD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OZ2TXWJNI5TCWCG7FV4MOUE2QD/action/storage_attestation","attest_author":"https://pith.science/pith/OZ2TXWJNI5TCWCG7FV4MOUE2QD/action/author_attestation","sign_citation":"https://pith.science/pith/OZ2TXWJNI5TCWCG7FV4MOUE2QD/action/citation_signature","submit_replication":"https://pith.science/pith/OZ2TXWJNI5TCWCG7FV4MOUE2QD/action/replication_record"}},"created_at":"2026-07-05T10:55:17.838767+00:00","updated_at":"2026-07-05T10:55:17.838767+00:00"}