{"paper":{"title":"Consistency of Learned Sparse Grid Quadrature Rules using NeuralODEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.PR"],"primary_cat":"math.NA","authors_text":"Emil Partow, Hanno Gottschalk, Tobias J. Riedlinger","submitted_at":"2025-07-02T09:37:16Z","abstract_excerpt":"We prove consistency of a recently proposed scheme that evaluates expected values by composing a learned transport map with Clenshaw--Curtis sparse-grid quadrature on a tractable product source. Our analysis hinges on the structural fact that composition of a $C^k_{\\mathrm{mix}}$-regular function -- which carries the fast quadrature rate $m^{-k}(\\log m)^{(d-1)(k+1)}$ -- with a $C^1$-diffeomorphism can only be guaranteed to be $C^k_{\\mathrm{mix}}$ itself, if the diffeomorphism is diagonal up to a permutation of coordinates. The fast rate is therefore available exclusively for product targets, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.01533","kind":"arxiv","version":2},"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/2507.01533/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"}