{"paper":{"title":"Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"CNNs approximate manifold functions and their derivatives at rates set by intrinsic dimension alone.","cross_cats":["cs.NA","math.NA"],"primary_cat":"cs.LG","authors_text":"Hanfei Zhou, Lei Shi","submitted_at":"2026-05-05T15:35:24Z","abstract_excerpt":"This paper develops convolutional neural network (CNN) methods for simultaneous Sobolev approximation and elliptic boundary value problems on compact Riemannian manifolds. We prove approximation estimates for single- and multichannel CNNs, with rates governed by the intrinsic dimension and the smoothness gap. Motivated by elliptic stability, we propose a physics-informed CNN framework with a spectral boundary loss. The boundary residual is expanded in boundary Laplace--Beltrami eigenmodes and penalized by Sobolev trace weights, matching the natural \\(\\mathcal H^{2s-1/2}(\\partial\\mathcal M^d)\\)"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish simultaneous Sobolev approximation results for single- and multichannel CNNs, showing that manifold functions and their derivatives can be approximated with rates governed by the intrinsic dimension and the smoothness gap, rather than by the ambient dimension.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the CNN architecture can be adapted to the Riemannian manifold structure such that the approximation rates transfer from Euclidean CNN theory without additional manifold-specific error terms that dominate the claimed rates.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"CNNs achieve simultaneous Sobolev approximation on manifolds with intrinsic-dimension rates and enable a PICNN for BVPs via spectral boundary loss that improves stability over standard PINNs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"CNNs approximate manifold functions and their derivatives at rates set by intrinsic dimension alone.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"82d4bd9152caa74ddcec78da37c28d6ecd791580e9c160c911269b046083f29d"},"source":{"id":"2605.04126","kind":"arxiv","version":2},"verdict":{"id":"3d2bfee3-6b15-4cee-9453-bb15dc2387b7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T18:17:53.271338Z","strongest_claim":"We establish simultaneous Sobolev approximation results for single- and multichannel CNNs, showing that manifold functions and their derivatives can be approximated with rates governed by the intrinsic dimension and the smoothness gap, rather than by the ambient dimension.","one_line_summary":"CNNs achieve simultaneous Sobolev approximation on manifolds with intrinsic-dimension rates and enable a PICNN for BVPs via spectral boundary loss that improves stability over standard PINNs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the CNN architecture can be adapted to the Riemannian manifold structure such that the approximation rates transfer from Euclidean CNN theory without additional manifold-specific error terms that dominate the claimed rates.","pith_extraction_headline":"CNNs approximate manifold functions and their derivatives at rates set by intrinsic dimension alone."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04126/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T12:42:20.649026Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T00:01:21.770953Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:58:56.578013Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e40ddb382a4e80078bcc87fc880f7650177a4667739be1d41e4619013648b2b6"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":3,"snapshot_sha256":"0b1e0cbe4b35e3337d0930356331c9092618c3b96e7caff6a5244119a22107f6"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}