{"paper":{"title":"Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Augmenting low-regularity integrators with scaled neural corrections yields global error C(ε_net + δ) τ^γ ln(1/τ) for nonlinear dispersive equations.","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Huanhuan Gao, Zhangyong Liang","submitted_at":"2026-05-06T12:50:36Z","abstract_excerpt":"We propose HIN-LRI, a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error. A base low-regularity integrator provides a consistent first-order approximation to nonlinear dispersive PDEs, while a lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close. An explicit time-step scaling on the neural correction ensures that its Lipschitz contribution remains $\\mathcal{O}(\\tau)$, yielding a Gronwall stability factor bounded unifo"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Under stated assumptions, the global error satisfies C(ε_net + δ) τ^γ ln(1/τ); experiments on three dispersive benchmarks with rough data show improved accuracy over analytical integrators, splitting methods, and neural PDE surrogates, with stable spatial refinement and out-of-distribution transfer.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The neural correction's Lipschitz contribution remains O(τ) after explicit time-step scaling, and the network approximation quality ε_net plus training shortfall δ are small enough for the Gronwall factor to stay bounded independently of spatial resolution.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A hybrid solver-neural framework achieves global error O(τ^γ ln(1/τ)) for nonlinear dispersive equations by training a lightweight network on the residual defect inside the solver loop while preserving uniform stability.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Augmenting low-regularity integrators with scaled neural corrections yields global error C(ε_net + δ) τ^γ ln(1/τ) for nonlinear dispersive equations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2f76cdec3e75919645a8cd43a40bb1c8953bb668bad1dc7ef33003c090ddcc3d"},"source":{"id":"2605.04853","kind":"arxiv","version":2},"verdict":{"id":"24d9c73a-71b4-40cf-bc64-3cd75f30d2ad","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T17:01:33.311741Z","strongest_claim":"Under stated assumptions, the global error satisfies C(ε_net + δ) τ^γ ln(1/τ); experiments on three dispersive benchmarks with rough data show improved accuracy over analytical integrators, splitting methods, and neural PDE surrogates, with stable spatial refinement and out-of-distribution transfer.","one_line_summary":"A hybrid solver-neural framework achieves global error O(τ^γ ln(1/τ)) for nonlinear dispersive equations by training a lightweight network on the residual defect inside the solver loop while preserving uniform stability.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The neural correction's Lipschitz contribution remains O(τ) after explicit time-step scaling, and the network approximation quality ε_net plus training shortfall δ are small enough for the Gronwall factor to stay bounded independently of spatial resolution.","pith_extraction_headline":"Augmenting low-regularity integrators with scaled neural corrections yields global error C(ε_net + δ) τ^γ ln(1/τ) for nonlinear dispersive equations."},"integrity":{"clean":false,"summary":{"advisory":0,"critical":1,"by_detector":{"doi_compliance":{"total":1,"advisory":0,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2605.04853/integrity.json","findings":[{"note":"Identifier '10.1007/s10543-021-00895-8.bit' is syntactically valid but the DOI registry (doi.org) returned 404, and Crossref / OpenAlex / internal corpus also have no record. The cited work could not be located through any authoritative source.","detector":"doi_compliance","severity":"critical","ref_index":20,"audited_at":"2026-05-19T14:06:30.273886Z","detected_doi":"10.1007/s10543-021-00895-8.bit","finding_type":"unresolvable_identifier","verdict_class":"cross_source","detected_arxiv_id":null}],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T10:42:52.640775Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T22:01:28.070467Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:06:30.273886Z","status":"completed","version":"1.0.0","findings_count":1}],"snapshot_sha256":"6a8283dd9f80616f6278d150114bcde2ac860e1fb5f1587a9aa5c2ab69d9253b"},"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"}