{"paper":{"title":"An Energy Stable Approach for Learning Derivative Operators from Noisy Data for Maxwells Equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Reduced parameterization lets SP-ADMM learn energy-conserving derivative operators for Maxwell equations directly from noisy data.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Ameh Emmanuel Sunday, Victory C. Obieke","submitted_at":"2026-01-05T08:46:15Z","abstract_excerpt":"We develop a structure-preserving ADMM method, denoted SP-ADMM, for learning energy-stable spatial derivative stencils for Maxwell equations from noisy data. Starting from the source-free Maxwell system, we focus on a one-dimensional reduction whose energy conservation depends on the skew-adjointness of the spatial derivative operator. The learned derivative is represented by a compact periodic convolution stencil. Unlike standard constrained ADMM, which learns the full stencil and imposes skew-adjointness through equality constraints, SP-ADMM enforces skew-adjointness by construction through "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"SP-ADMM achieves the smallest final-time electric-field error while preserving energy to roundoff accuracy across clean data, noisy derivative data, multiple initial conditions, different hidden skew-adjoint operators, training-set sizes, regularization parameters, constraint ablations, and long-time simulations.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That enforcing skew-adjointness solely through reduced parameterization of the positive-side stencil coefficients is sufficient to guarantee energy stability for the learned operator in the underlying Maxwell system without introducing approximation errors or limiting expressivity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"SP-ADMM learns energy-stable derivative stencils for Maxwell equations from noisy data by enforcing skew-adjointness through reduced parameterization of periodic convolution stencils.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Reduced parameterization lets SP-ADMM learn energy-conserving derivative operators for Maxwell equations directly from noisy data.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b98d243749c290d60996bb47a27870304260196dc1b77ddaf25efbb1b3337cfb"},"source":{"id":"2601.01902","kind":"arxiv","version":7},"verdict":{"id":"68a65929-177c-44f2-a45e-09c3d6bb5cd2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T18:23:27.699485Z","strongest_claim":"SP-ADMM achieves the smallest final-time electric-field error while preserving energy to roundoff accuracy across clean data, noisy derivative data, multiple initial conditions, different hidden skew-adjoint operators, training-set sizes, regularization parameters, constraint ablations, and long-time simulations.","one_line_summary":"SP-ADMM learns energy-stable derivative stencils for Maxwell equations from noisy data by enforcing skew-adjointness through reduced parameterization of periodic convolution stencils.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That enforcing skew-adjointness solely through reduced parameterization of the positive-side stencil coefficients is sufficient to guarantee energy stability for the learned operator in the underlying Maxwell system without introducing approximation errors or limiting expressivity.","pith_extraction_headline":"Reduced parameterization lets SP-ADMM learn energy-conserving derivative operators for Maxwell equations directly from noisy data."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.01902/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":2,"snapshot_sha256":"37e4b82a9fb65ac2f1b3d38953eba4aab89b22f2ff565cab52ec065e0cdbd975"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}