{"id":"https://openalex.org/W7212005846","doi":"https://doi.org/10.48550/arxiv.2609.08102","title":"Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators","display_name":"Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators","publication_year":2026,"publication_date":"2026-09-08","ids":{"openalex":"https://openalex.org/W7212005846","doi":"https://doi.org/10.48550/arxiv.2609.08102"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.08102","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.08102","pdf_url":"https://arxiv.org/pdf/2609.08102","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,"listed_in":[],"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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.08102","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151789654","display_name":"Han Zhang","orcid":"https://orcid.org/0000-0002-7752-8538","observed_orcids":["https://orcid.org/0000-0002-7752-8538"]},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhang, Han","raw_affiliation_strings":["Department of Mechanical Engineering and Materials Science , Duke University , Durham , NC , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering and Materials Science , Duke University , Durham , NC , USA","institution_ids":["https://openalex.org/I170897317"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032334298","display_name":"Alexander C. Ogren","orcid":"https://orcid.org/0000-0002-7277-0069","observed_orcids":["https://orcid.org/0000-0002-7277-0069"]},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ogren, Alexander","raw_affiliation_strings":["Department of Mechanical Engineering , California Institute of Technology , Pasadena , CA , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering , California Institute of Technology , Pasadena , CA , USA","institution_ids":["https://openalex.org/I122411786"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040468715","display_name":"Cynthia D Rudin","orcid":"https://orcid.org/0000-0003-4283-2780","observed_orcids":["https://orcid.org/0000-0003-4283-2780"]},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Rudin, Cynthia","raw_affiliation_strings":["Department of Electrical and Computer Engineering , Duke University , Durham , NC , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering , Duke University , Durham , NC , USA","institution_ids":["https://openalex.org/I170897317"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025395694","display_name":"Johann Guilleminot","orcid":"https://orcid.org/0000-0003-3738-4370","observed_orcids":["https://orcid.org/0000-0003-3738-4370"]},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Guilleminot, Johann","raw_affiliation_strings":["Department of Mechanical Engineering and Materials Science , Duke University , Durham , NC , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering and Materials Science , Duke University , Durham , NC , USA","institution_ids":["https://openalex.org/I170897317"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021656287","display_name":"L. Catherine Brinson","orcid":"https://orcid.org/0000-0003-2551-1563","observed_orcids":["https://orcid.org/0000-0003-2551-1563"]},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Brinson, L. Catherine","raw_affiliation_strings":["Department of Mechanical Engineering and Materials Science , Duke University , Durham , NC , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering and Materials Science , Duke University , Durham , NC , USA","institution_ids":["https://openalex.org/I170897317"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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/T11206","display_name":"Model Reduction and Neural Networks","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":1.0,"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/T10320","display_name":"Neural Networks and Applications","score":9.999999747378752e-05,"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/T11948","display_name":"Machine Learning in Materials Science","score":0.0,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/neural-operators","display_name":"neural operators","score":0.7730000019073486},{"id":"https://openalex.org/keywords/fourier-neural-operator","display_name":"Fourier neural operator","score":0.9739999771118164},{"id":"https://openalex.org/keywords/eigenmodes","display_name":"eigenmodes","score":0.6019999980926514},{"id":"https://openalex.org/keywords/eigenvalue-problem","display_name":"eigenvalue problem","score":0.9359999895095825},{"id":"https://openalex.org/keywords/elastic-wave-equation","display_name":"elastic wave equation","score":0.8389999866485596},{"id":"https://openalex.org/keywords/metamaterials","display_name":"metamaterials","score":0.9610000252723694},{"id":"https://openalex.org/keywords/surrogate-modeling","display_name":"surrogate models","score":0.6930000185966492},{"id":"https://openalex.org/keywords/sound-propagation","display_name":"acoustic wave propagation","score":0.6259999871253967},{"id":"https://openalex.org/keywords/metamaterial-design","display_name":"metamaterial design","score":0.6769999861717224},{"id":"https://openalex.org/keywords/geometric-discontinuities","display_name":"geometric discontinuities","score":0.8569999933242798}],"concepts":[{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.656000018119812},{"id":"https://openalex.org/C110367647","wikidata":"https://www.wikidata.org/wiki/Q497166","display_name":"Metamaterial","level":2,"score":0.6200000047683716},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5925999879837036},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.5185999870300293},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4749999940395355},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4325000047683716},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.4244999885559082},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4154999852180481},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.3993000090122223},{"id":"https://openalex.org/C81917197","wikidata":"https://www.wikidata.org/wiki/Q628760","display_name":"Selection (genetic algorithm)","level":2,"score":0.36000001430511475},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.33230000734329224},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.32350000739097595},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.319599986076355},{"id":"https://openalex.org/C48677424","wikidata":"https://www.wikidata.org/wiki/Q6888088","display_name":"Mode (computer interface)","level":2,"score":0.29820001125335693},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.295199990272522},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.2946000099182129},{"id":"https://openalex.org/C91748784","wikidata":"https://www.wikidata.org/wiki/Q11406","display_name":"Electromagnetism","level":2,"score":0.28679999709129333},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2851000130176544},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.27950000762939453},{"id":"https://openalex.org/C43929395","wikidata":"https://www.wikidata.org/wiki/Q1198874","display_name":"Operator theory","level":2,"score":0.2766000032424927},{"id":"https://openalex.org/C32758168","wikidata":"https://www.wikidata.org/wiki/Q900488","display_name":"Normal mode","level":3,"score":0.2667999863624573},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.26660001277923584},{"id":"https://openalex.org/C500300565","wikidata":"https://www.wikidata.org/wiki/Q925667","display_name":"Computer simulation","level":2,"score":0.25619998574256897},{"id":"https://openalex.org/C761482","wikidata":"https://www.wikidata.org/wiki/Q118093","display_name":"Transmission (telecommunications)","level":2,"score":0.25380000472068787},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.25369998812675476},{"id":"https://openalex.org/C35958075","wikidata":"https://www.wikidata.org/wiki/Q1425077","display_name":"Spectral theorem","level":3,"score":0.2522999942302704}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.08102","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.08102","pdf_url":"https://arxiv.org/pdf/2609.08102","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,"listed_in":[],"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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.08102","doi":"https://doi.org/10.48550/arxiv.2609.08102","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.08102","pdf_url":null,"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,"listed_in":[],"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.08102","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.08102","pdf_url":"https://arxiv.org/pdf/2609.08102","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,"listed_in":[],"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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G8672920617","display_name":"NRT-HDR: Harnessing AI for Understanding & Designing Materials (aiM)","funder_award_id":"2022040","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"},{"id":"https://openalex.org/F4320337367","display_name":"Division of Materials Research","ror":"https://ror.org/01pc7k308"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212005846.pdf","grobid_xml":"https://content.openalex.org/works/W7212005846.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4318828615","https://openalex.org/W4390809210","https://openalex.org/W7163043756","https://openalex.org/W4388200327","https://openalex.org/W6977454569","https://openalex.org/W7160286700","https://openalex.org/W4414427633","https://openalex.org/W4394907065","https://openalex.org/W4403499756","https://openalex.org/W4282020420"],"abstract_inverted_index":{"Machine":[0],"learning":[1,38],"surrogates":[2],"based":[3,176],"on":[4,110,153,177],"neural":[5,179],"operators":[6],"have":[7],"shown":[8],"broad":[9],"applicability":[10],"in":[11,19,171],"solving":[12],"forward":[13],"PDE":[14,58,174],"problems.":[15],"However,":[16],"eigenvalue":[17],"problems,":[18],"which":[20],"an":[21],"eigenparameter":[22],"and":[23,62,88,113,120],"one":[24],"of":[25,57,66,75,103,139,146],"several":[26],"valid":[27],"eigenmodes":[28,65],"must":[29],"be":[30],"simultaneously":[31],"solved,":[32],"remain":[33],"difficult":[34],"because":[35],"standard":[36],"operator":[37],"formulations":[39],"assume":[40],"a":[41,85,117,154],"unique":[42],"input-output":[43],"map.":[44],"This":[45],"work":[46],"demonstrates":[47],"that":[48],"Fourier":[49],"Neural":[50],"Operators":[51],"(FNOs),":[52],"combined":[53],"with":[54,126],"wavelet-based":[55],"encodings":[56,94,170],"inputs,":[59],"can":[60],"learn":[61],"predict":[63],"multiple":[64],"the":[67,99,104,132,136,140],"elastic":[68],"wave":[69],"equation,":[70],"corresponding":[71],"to":[72,98,149],"deformation":[73],"modes":[74],"acoustic":[76],"waves":[77],"propagating":[78],"through":[79],"arbitrary":[80],"metamaterial":[81,130],"geometries.":[82],"We":[83],"provide":[84],"mechanistic":[86],"explanation":[87],"experimental":[89],"evidence":[90],"for":[91,121,167],"why":[92,122],"wavelet":[93],"are":[95],"well":[96],"matched":[97],"dual":[100],"spatial-spectral":[101],"structure":[102],"FNO,":[105],"enabling":[106],"deterministic":[107],"mode":[108],"selection":[109],"both":[111],"continuous-valued":[112],"binary-valued":[114],"geometries":[115],"within":[116],"single":[118],"model,":[119],"prediction":[123],"accuracy":[124],"varies":[125],"geometric":[127],"discontinuities.":[128],"For":[129],"design,":[131],"resulting":[133],"surrogate":[134],"accelerates":[135],"simulation":[137],"stage":[138],"design":[141],"cycle":[142],"by":[143],"three":[144],"orders":[145],"magnitude":[147],"relative":[148],"finite":[150],"element":[151],"analysis":[152],"consumer-grade":[155],"CPU,":[156],"while":[157],"preserving":[158],"high":[159],"fidelity.":[160],"These":[161],"results":[162],"also":[163],"carry":[164],"broader":[165],"implications":[166],"designing":[168],"input":[169],"other":[172],"multi-mode":[173],"solvers":[175],"spectral":[178],"operators.":[180]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-10T00:00:00"}