{"id":"https://openalex.org/W7164012504","doi":"https://doi.org/10.48550/arxiv.2606.09278","title":"Internalizing Geometric Law: Learning from Solver Residuals for Precision-Critical Generation","display_name":"Internalizing Geometric Law: Learning from Solver Residuals for Precision-Critical Generation","publication_year":2026,"publication_date":"2026-06-08","ids":{"openalex":"https://openalex.org/W7164012504","doi":"https://doi.org/10.48550/arxiv.2606.09278"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.09278","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.09278","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.09278","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5138257002","display_name":"Rafael Cabral","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cabral, Rafael","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138248288","display_name":"Pang Zixi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zixi, Pang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5076867415","display_name":"Ziyi Shou","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Shou, Ziyi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5138267341","display_name":"Shen Xin","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xin, Shen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11948","display_name":"Machine Learning in Materials Science","score":0.17190000414848328,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11948","display_name":"Machine Learning in Materials Science","score":0.17190000414848328,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11273","display_name":"Advanced Graph Neural Networks","score":0.1005999967455864,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.09030000120401382,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/outlier","display_name":"Outlier","score":0.613099992275238},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.5534999966621399},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.46860000491142273},{"id":"https://openalex.org/keywords/verifiable-secret-sharing","display_name":"Verifiable secret sharing","score":0.429500013589859},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.39480000734329224},{"id":"https://openalex.org/keywords/observability","display_name":"Observability","score":0.3880999982357025},{"id":"https://openalex.org/keywords/spark","display_name":"SPARK (programming language)","score":0.38670000433921814},{"id":"https://openalex.org/keywords/overhead","display_name":"Overhead (engineering)","score":0.36570000648498535},{"id":"https://openalex.org/keywords/scheme","display_name":"Scheme (mathematics)","score":0.34610000252723694},{"id":"https://openalex.org/keywords/satisfiability-modulo-theories","display_name":"Satisfiability modulo theories","score":0.34540000557899475}],"concepts":[{"id":"https://openalex.org/C79337645","wikidata":"https://www.wikidata.org/wiki/Q779824","display_name":"Outlier","level":2,"score":0.613099992275238},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5856000185012817},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.5534999966621399},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.46860000491142273},{"id":"https://openalex.org/C85847156","wikidata":"https://www.wikidata.org/wiki/Q59015987","display_name":"Verifiable secret sharing","level":3,"score":0.429500013589859},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.39480000734329224},{"id":"https://openalex.org/C36299963","wikidata":"https://www.wikidata.org/wiki/Q1369844","display_name":"Observability","level":2,"score":0.3880999982357025},{"id":"https://openalex.org/C2781215313","wikidata":"https://www.wikidata.org/wiki/Q3493345","display_name":"SPARK (programming language)","level":2,"score":0.38670000433921814},{"id":"https://openalex.org/C2779960059","wikidata":"https://www.wikidata.org/wiki/Q7113681","display_name":"Overhead (engineering)","level":2,"score":0.36570000648498535},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3628999888896942},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36149999499320984},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.34610000252723694},{"id":"https://openalex.org/C164155591","wikidata":"https://www.wikidata.org/wiki/Q2067766","display_name":"Satisfiability modulo theories","level":2,"score":0.34540000557899475},{"id":"https://openalex.org/C12725497","wikidata":"https://www.wikidata.org/wiki/Q810247","display_name":"Baseline (sea)","level":2,"score":0.3418999910354614},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3418999910354614},{"id":"https://openalex.org/C79581498","wikidata":"https://www.wikidata.org/wiki/Q1367530","display_name":"Suite","level":2,"score":0.34130001068115234},{"id":"https://openalex.org/C20729856","wikidata":"https://www.wikidata.org/wiki/Q2078279","display_name":"Geometric programming","level":2,"score":0.3325999975204468},{"id":"https://openalex.org/C4069607","wikidata":"https://www.wikidata.org/wiki/Q868732","display_name":"Aliasing","level":3,"score":0.3246000111103058},{"id":"https://openalex.org/C48677424","wikidata":"https://www.wikidata.org/wiki/Q6888088","display_name":"Mode (computer interface)","level":2,"score":0.3228999972343445},{"id":"https://openalex.org/C97256817","wikidata":"https://www.wikidata.org/wiki/Q1462316","display_name":"Spurious relationship","level":2,"score":0.3147999942302704},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.31130000948905945},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.30149999260902405},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.30090001225471497},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.2978000044822693},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.2962000072002411},{"id":"https://openalex.org/C136520226","wikidata":"https://www.wikidata.org/wiki/Q302814","display_name":"Geometric data analysis","level":2,"score":0.2904999852180481},{"id":"https://openalex.org/C160234255","wikidata":"https://www.wikidata.org/wiki/Q812535","display_name":"Bayesian inference","level":3,"score":0.28130000829696655},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.271699994802475},{"id":"https://openalex.org/C5274069","wikidata":"https://www.wikidata.org/wiki/Q2285707","display_name":"Categorical variable","level":2,"score":0.2685999870300293},{"id":"https://openalex.org/C177212765","wikidata":"https://www.wikidata.org/wiki/Q627335","display_name":"Workflow","level":2,"score":0.2669000029563904},{"id":"https://openalex.org/C7305733","wikidata":"https://www.wikidata.org/wiki/Q207961","display_name":"Geometric shape","level":2,"score":0.2653999924659729},{"id":"https://openalex.org/C2779843651","wikidata":"https://www.wikidata.org/wiki/Q7390335","display_name":"SIGNAL (programming language)","level":2,"score":0.2635999917984009},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.2533999979496002}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.09278","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.09278","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.09278","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.09278","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","score":0.7614616751670837,"id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Large":[0],"Language":[1],"Models":[2],"frequently":[3],"hallucinate":[4],"in":[5],"precision-critical":[6],"domains":[7],"such":[8],"as":[9,78],"technical":[10],"diagramming":[11],"and":[12,64,138,163,183],"mechanical":[13],"design,":[14],"where":[15],"outputs":[16],"must":[17,38],"satisfy":[18,40],"strict":[19],"geometric":[20,25,54],"constraints.":[21,44],"We":[22,178],"study":[23],"open-ended":[24],"synthesis":[26],"from":[27],"natural":[28,150],"language:":[29],"translating":[30],"free-form":[31],"descriptions":[32],"into":[33,60,131],"precise":[34],"constructions":[35],"whose":[36],"entities":[37],"simultaneously":[39],"dozens":[41],"of":[42,69],"interacting":[43],"To":[45,118],"make":[46],"this":[47,176],"tractable,":[48],"we":[49,81,86,121],"release":[50,179],"PyGeoX,":[51],"a":[52,61,66,79,83,100,106],"programmable":[53],"DSL":[55],"that":[56,96],"compiles":[57],"declarative":[58],"constraints":[59],"differentiable":[62],"loss,":[63],"PyGeoX-Bench,":[65],"stratified":[67],"suite":[68],"300":[70],"problems":[71],"with":[72,170],"per-constraint":[73,133],"verifiable":[74],"rewards.":[75],"Using":[76],"PyGeoX":[77],"verifier,":[80],"identify":[82],"failure":[84],"mode":[85],"call":[87],"Outlier":[88],"Gradient":[89],"Masking:":[90],"under":[91,143],"global-norm":[92],"rewards":[93],"(any":[94],"scheme":[95],"aggregates":[97],"residuals":[98],"through":[99],"single":[101,107],"norm,":[102],"for":[103,152],"example,":[104],"$\\exp(-\\mathrm{MSE})$),":[105],"outlier":[108],"constraint":[109],"can":[110],"nullify":[111],"the":[112,129,149,157,164,180],"learning":[113],"signal":[114],"across":[115],"all":[116],"others.":[117],"address":[119],"this,":[120],"propose":[122],"Saturating":[123],"Additive":[124],"Rewards":[125],"(SAR),":[126],"which":[127],"decompose":[128],"reward":[130],"bounded":[132],"terms,":[134],"preserving":[135],"partial":[136],"progress":[137],"ensuring":[139],"consistent":[140],"gradients":[141],"even":[142],"severe":[144],"violations.":[145],"Against":[146],"MSE-based":[147],"rewards,":[148],"baseline":[151],"geometry":[153],"solvers,":[154],"SAR":[155],"improves":[156],"hard-tier":[158],"solving":[159],"rate":[160],"by":[161],"$2.3\\times$,":[162],"resulting":[165],"8B":[166],"model":[167],"is":[168],"competitive":[169],"much":[171],"larger":[172],"frontier":[173],"systems":[174],"on":[175],"benchmark.":[177],"engine,":[181],"benchmark,":[182],"data":[184],"at":[185],"https://github.com/Huawei-AI4Math/PyGeoX.":[186]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-06-10T00:00:00"}
