{"id":"https://openalex.org/W7163124393","doi":"https://doi.org/10.1007/s00366-026-02351-z","title":"Multi-patch neural solver with isogeometric mappings for partial differential equations on computer-aided design domains","display_name":"Multi-patch neural solver with isogeometric mappings for partial differential equations on computer-aided design domains","publication_year":2026,"publication_date":"2026-06-02","ids":{"openalex":"https://openalex.org/W7163124393","doi":"https://doi.org/10.1007/s00366-026-02351-z"},"language":"en","primary_location":{"id":"doi:10.1007/s00366-026-02351-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00366-026-02351-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00366-026-02351-z.pdf","source":{"id":"https://openalex.org/S89333158","display_name":"Engineering With Computers","issn_l":"0177-0667","issn":["0177-0667","1435-5663"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Engineering with Computers","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00366-026-02351-z.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057062259","display_name":"Moritz von Tresckow","orcid":"https://orcid.org/0000-0003-2416-4424"},"institutions":[{"id":"https://openalex.org/I31512782","display_name":"Technische Universit\u00e4t Darmstadt","ror":"https://ror.org/05n911h24","country_code":"DE","type":"education","lineage":["https://openalex.org/I31512782"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Moritz von Tresckow","raw_affiliation_strings":["Institute for Accelerator Science and Electromagnetic Fields, Technische Universit\u00e4t Darmstadt, Schlossgartenstr. 8, 64289, Darmstadt, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Accelerator Science and Electromagnetic Fields, Technische Universit\u00e4t Darmstadt, Schlossgartenstr. 8, 64289, Darmstadt, Germany","institution_ids":["https://openalex.org/I31512782"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059936707","display_name":"Ion Gabriel Ion","orcid":"https://orcid.org/0000-0002-2932-0396"},"institutions":[{"id":"https://openalex.org/I4210121585","display_name":"Terra Quantum (Switzerland)","ror":"https://ror.org/02y8r3w10","country_code":"CH","type":"company","lineage":["https://openalex.org/I4210121585"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Ion Gabriel Ion","raw_affiliation_strings":["Terra Quantum AG, Kornhausstr. 25, St., 9000, Gallen, Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Terra Quantum AG, Kornhausstr. 25, St., 9000, Gallen, Switzerland","institution_ids":["https://openalex.org/I4210121585"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5013517565","display_name":"Dimitrios Loukrezis","orcid":"https://orcid.org/0000-0003-1264-1182"},"institutions":[{"id":"https://openalex.org/I1341640284","display_name":"Centrum Wiskunde & Informatica","ror":"https://ror.org/00x7ekv49","country_code":"NL","type":"facility","lineage":["https://openalex.org/I1341640284","https://openalex.org/I2800991832","https://openalex.org/I4405262988"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Dimitrios Loukrezis","raw_affiliation_strings":["Scientific Computing, Centrum Wiskunde & Informatica, Science Park 123, 1098 XG, Amsterdam, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Scientific Computing, Centrum Wiskunde & Informatica, Science Park 123, 1098 XG, Amsterdam, The Netherlands","institution_ids":["https://openalex.org/I1341640284"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5013517565","https://openalex.org/A5057062259"],"corresponding_institution_ids":["https://openalex.org/I1341640284","https://openalex.org/I31512782"],"apc_list":{"value":2990,"currency":"USD","value_usd":2990},"apc_paid":{"value":2990,"currency":"USD","value_usd":2990},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.60426335,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"42","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.5795999765396118,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.5795999765396118,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.2687999904155731,"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"}},{"id":"https://openalex.org/T11115","display_name":"Topology Optimization in Engineering","score":0.037300001829862595,"subfield":{"id":"https://openalex.org/subfields/2205","display_name":"Civil and Structural Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/isogeometric-analysis","display_name":"Isogeometric analysis","score":0.8202999830245972},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.6787999868392944},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.6665999889373779},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.5458999872207642},{"id":"https://openalex.org/keywords/partial-derivative","display_name":"Partial derivative","score":0.5292999744415283},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5023000240325928},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.48420000076293945},{"id":"https://openalex.org/keywords/parametric-statistics","display_name":"Parametric statistics","score":0.47279998660087585},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.4562999904155731},{"id":"https://openalex.org/keywords/benchmark","display_name":"Benchmark (surveying)","score":0.42730000615119934}],"concepts":[{"id":"https://openalex.org/C2780737243","wikidata":"https://www.wikidata.org/wiki/Q17082433","display_name":"Isogeometric analysis","level":3,"score":0.8202999830245972},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.6787999868392944},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.6665999889373779},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.5458999872207642},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5333999991416931},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.5292999744415283},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5023000240325928},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.48420000076293945},{"id":"https://openalex.org/C117251300","wikidata":"https://www.wikidata.org/wiki/Q1849855","display_name":"Parametric statistics","level":2,"score":0.47279998660087585},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4562999904155731},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4546000063419342},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.43299999833106995},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.42730000615119934},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.3849000036716461},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.365200012922287},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.35359999537467957},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.32359999418258667},{"id":"https://openalex.org/C54067925","wikidata":"https://www.wikidata.org/wiki/Q3607962","display_name":"Elliptic partial differential equation","level":3,"score":0.32339999079704285},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.3140000104904175},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3086000084877014},{"id":"https://openalex.org/C29513896","wikidata":"https://www.wikidata.org/wiki/Q7489239","display_name":"Shape optimization","level":3,"score":0.295199990272522},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.2937000095844269},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.2930000126361847},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.290800005197525},{"id":"https://openalex.org/C98214594","wikidata":"https://www.wikidata.org/wiki/Q850275","display_name":"Conformal map","level":2,"score":0.28949999809265137},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.28940001130104065},{"id":"https://openalex.org/C191640071","wikidata":"https://www.wikidata.org/wiki/Q5377056","display_name":"Energy functional","level":2,"score":0.2883000075817108},{"id":"https://openalex.org/C113843644","wikidata":"https://www.wikidata.org/wiki/Q901882","display_name":"Interface (matter)","level":4,"score":0.2797999978065491},{"id":"https://openalex.org/C99636146","wikidata":"https://www.wikidata.org/wiki/Q35889","display_name":"Parametric equation","level":2,"score":0.2782999873161316},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2727000117301941},{"id":"https://openalex.org/C111615704","wikidata":"https://www.wikidata.org/wiki/Q2627459","display_name":"Hyperbolic partial differential equation","level":3,"score":0.26930001378059387},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.2639000117778778},{"id":"https://openalex.org/C80551277","wikidata":"https://www.wikidata.org/wiki/Q11210","display_name":"Coordinate system","level":2,"score":0.25870001316070557},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2574999928474426},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.25360000133514404}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s00366-026-02351-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00366-026-02351-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00366-026-02351-z.pdf","source":{"id":"https://openalex.org/S89333158","display_name":"Engineering With Computers","issn_l":"0177-0667","issn":["0177-0667","1435-5663"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Engineering with Computers","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2509.25450","is_oa":true,"landing_page_url":"https://arxiv.org/abs/2509.25450","pdf_url":"https://arxiv.org/pdf/2509.25450","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"doi:10.1007/s00366-026-02351-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00366-026-02351-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00366-026-02351-z.pdf","source":{"id":"https://openalex.org/S89333158","display_name":"Engineering With Computers","issn_l":"0177-0667","issn":["0177-0667","1435-5663"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Engineering with Computers","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Affordable and clean energy","score":0.71974116563797,"id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320323384","display_name":"Technische Universit\u00e4t Darmstadt","ror":"https://ror.org/05n911h24"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7163124393.pdf","grobid_xml":"https://content.openalex.org/works/W7163124393.grobid-xml"},"referenced_works_count":69,"referenced_works":["https://openalex.org/W1940709639","https://openalex.org/W1982343173","https://openalex.org/W1998777442","https://openalex.org/W2000425991","https://openalex.org/W2005390260","https://openalex.org/W2037832437","https://openalex.org/W2107979247","https://openalex.org/W2117127065","https://openalex.org/W2128095813","https://openalex.org/W2141671098","https://openalex.org/W2194775991","https://openalex.org/W2483194617","https://openalex.org/W2749028154","https://openalex.org/W2760972773","https://openalex.org/W2809918478","https://openalex.org/W2899283552","https://openalex.org/W2963066470","https://openalex.org/W2997814214","https://openalex.org/W2998847955","https://openalex.org/W3010849941","https://openalex.org/W3011806874","https://openalex.org/W3015865829","https://openalex.org/W3098546160","https://openalex.org/W3137392741","https://openalex.org/W3153200540","https://openalex.org/W3185095713","https://openalex.org/W3208953217","https://openalex.org/W3213310542","https://openalex.org/W3216718982","https://openalex.org/W3217139884","https://openalex.org/W4206261456","https://openalex.org/W4220717841","https://openalex.org/W4223971218","https://openalex.org/W4237572979","https://openalex.org/W4237713727","https://openalex.org/W4285226320","https://openalex.org/W4293210563","https://openalex.org/W4299968561","https://openalex.org/W4306920631","https://openalex.org/W4309265129","https://openalex.org/W4310153797","https://openalex.org/W4313334586","https://openalex.org/W4322731021","https://openalex.org/W4376638025","https://openalex.org/W4382051831","https://openalex.org/W4382652314","https://openalex.org/W4383225215","https://openalex.org/W4385495295","https://openalex.org/W4386387954","https://openalex.org/W4387317432","https://openalex.org/W4391780410","https://openalex.org/W4391848578","https://openalex.org/W4394989066","https://openalex.org/W4395475030","https://openalex.org/W4398238760","https://openalex.org/W4399840416","https://openalex.org/W4400293893","https://openalex.org/W4400763213","https://openalex.org/W4401284683","https://openalex.org/W4401346328","https://openalex.org/W4402484076","https://openalex.org/W4402774937","https://openalex.org/W4403069620","https://openalex.org/W4403936200","https://openalex.org/W4404103034","https://openalex.org/W4404350998","https://openalex.org/W4404874492","https://openalex.org/W4407350629","https://openalex.org/W4407430485"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"This":[1],"work":[2],"develops":[3],"a":[4,109],"computational":[5],"framework":[6,84],"that":[7,32],"combines":[8],"physics-informed":[9],"neural":[10,30,76,140],"networks":[11,31],"with":[12,128],"multi-patch":[13],"isogeometric":[14,48],"mappings":[15],"to":[16,124,142],"solve":[17],"partial":[18,97],"differential":[19,98],"equations":[20],"on":[21,34,108,113],"complex":[22,144],"computer-aided":[23,150],"design":[24,151],"geometries.":[25],"The":[26,100,119],"method":[27,105],"utilizes":[28],"patch-local":[29],"operate":[33],"the":[35,42,46,55,82,87,92,96,103,135,138,148],"parametric":[36],"space":[37],"of":[38,45,58,95,102,137],"each":[39],"patch,":[40],"i.e.":[41],"reference":[43,125],"domain":[44],"local":[47],"map.":[49],"A":[50],"custom":[51],"output":[52],"layer":[53],"enables":[54],"strong":[56],"imposition":[57],"Dirichlet":[59],"boundary":[60],"conditions.":[61],"Solution":[62],"conformity":[63],"across":[64],"interfaces":[65],"between":[66],"non-uniform":[67],"rational":[68],"B-spline":[69],"patches":[70],"is":[71,79,106],"enforced":[72],"using":[73,81],"dedicated":[74],"interface":[75],"networks.":[77],"Training":[78],"performed":[80],"variational":[83],"by":[85],"minimizing":[86],"energy":[88],"functional":[89],"derived":[90],"after":[91],"weak":[93],"form":[94],"equation.":[99],"effectiveness":[101],"suggested":[104,139],"demonstrated":[107],"benchmark":[110],"problem":[111],"and":[112],"two":[114],"highly":[115],"non-trivial":[116],"engineering":[117,145],"applications.":[118],"results":[120],"show":[121],"excellent":[122],"agreement":[123],"solutions":[126],"obtained":[127],"high-fidelity":[129],"finite":[130],"element":[131],"solvers,":[132],"thus":[133],"highlighting":[134],"potential":[136],"solver":[141],"tackle":[143],"problems":[146],"given":[147],"corresponding":[149],"models.":[152]},"counts_by_year":[],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2026-06-03T00:00:00"}