{"paper":{"title":"Neural Preconditioned Born Series: A Metric-Matched Framework for Learning-based Preconditioners","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Neural Preconditioned Born Series replaces the scalar Born correction with a learned map in residual coordinates induced by a constant-coefficient reference operator.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Jiwei Jia, Juntao Wang, Xinliang Liu","submitted_at":"2026-03-19T06:17:20Z","abstract_excerpt":"High-frequency Helmholtz problems in heterogeneous media remain challenging for both classical iterative methods and end-to-end neural PDE solvers. We propose Neural Preconditioned Born Series (NPBS), a learned iterative preconditioning framework that operates in preconditioned residual coordinates induced by the Convergent Born Series (CBS). Existing learned Born-series methods primarily use Born-style unrolling for forward wavefield prediction, while learned Helmholtz preconditioners are usually formulated in physical residual coordinates. NPBS fills this gap by recasting Born-series iterati"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Numerical results on heterogeneous Helmholtz benchmarks show that the metric-matched formulation consistently reduces iteration counts relative to direct residual learning and classical CBS, with stronger benefits in more ill-conditioned regimes.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The equivalence between Born-series residuals and shifted-Laplacian left preconditioning holds for the chosen constant-coefficient references, and the learned residual-to-correction map generalizes from the training distribution to unseen heterogeneous media without degrading the iteration.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"NPBS learns a residual-to-correction map inside Born-series coordinates with a metric-matched objective, reducing iterations versus direct residual learning and classical CBS on heterogeneous Helmholtz benchmarks.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Neural Preconditioned Born Series replaces the scalar Born correction with a learned map in residual coordinates induced by a constant-coefficient reference operator.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b786e90e4e58d295cb36646c0b3f03b935e048bebfa4cc51c35bd9723f838ea8"},"source":{"id":"2603.18527","kind":"arxiv","version":4},"verdict":{"id":"ffe02589-553e-41ba-8d73-94f9f5e75c5f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T09:05:17.697803Z","strongest_claim":"Numerical results on heterogeneous Helmholtz benchmarks show that the metric-matched formulation consistently reduces iteration counts relative to direct residual learning and classical CBS, with stronger benefits in more ill-conditioned regimes.","one_line_summary":"NPBS learns a residual-to-correction map inside Born-series coordinates with a metric-matched objective, reducing iterations versus direct residual learning and classical CBS on heterogeneous Helmholtz benchmarks.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The equivalence between Born-series residuals and shifted-Laplacian left preconditioning holds for the chosen constant-coefficient references, and the learned residual-to-correction map generalizes from the training distribution to unseen heterogeneous media without degrading the iteration.","pith_extraction_headline":"Neural Preconditioned Born Series replaces the scalar Born correction with a learned map in residual coordinates induced by a constant-coefficient reference operator."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.18527/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"}