{"id":"https://openalex.org/W4367628371","doi":"https://doi.org/10.48550/arxiv.2304.14994","title":"A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks","display_name":"A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks","publication_year":2023,"publication_date":"2023-04-28","ids":{"openalex":"https://openalex.org/W4367628371","doi":"https://doi.org/10.48550/arxiv.2304.14994"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2304.14994","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2304.14994","pdf_url":"https://arxiv.org/pdf/2304.14994","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2304.14994","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013223417","display_name":"Marc Finzi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Finzi, Marc","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5018500293","display_name":"Andres Potapczynski","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Potapczynski, Andres","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034886733","display_name":"Matthew W. Choptuik","orcid":"https://orcid.org/0000-0001-5156-471X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Choptuik, Matthew","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5101694421","display_name":"Andrew Gordon Wilson","orcid":"https://orcid.org/0000-0002-2011-3315"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wilson, Andrew Gordon","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":true,"cited_by_count":1,"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":0.9965000152587891,"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":0.9965000152587891,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9435999989509583,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11052","display_name":"Energy Load and Power Forecasting","score":0.9433000087738037,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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/ode","display_name":"Ode","score":0.9165384769439697},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.687208890914917},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6386757493019104},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5787723064422607},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5772591829299927},{"id":"https://openalex.org/keywords/curse-of-dimensionality","display_name":"Curse of dimensionality","score":0.5603113770484924},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.5588092803955078},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.5372206568717957},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5236147046089172},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.50966477394104},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.49182772636413574},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.4457893967628479},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.35271546244621277},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.2845263183116913},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.15800315141677856},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10329616069793701}],"concepts":[{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.9165384769439697},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.687208890914917},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6386757493019104},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5787723064422607},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5772591829299927},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.5603113770484924},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.5588092803955078},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.5372206568717957},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5236147046089172},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.50966477394104},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.49182772636413574},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4457893967628479},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.35271546244621277},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.2845263183116913},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.15800315141677856},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10329616069793701},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2304.14994","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2304.14994","pdf_url":"https://arxiv.org/pdf/2304.14994","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2304.14994","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2304.14994","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,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2304.14994","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2304.14994","pdf_url":"https://arxiv.org/pdf/2304.14994","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4057891287","display_name":"RI: Small: Scalable Online Learning with Gaussian Processes","funder_award_id":"1910266","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5693414920","display_name":"NRT-HDR: FUTURE Foundations, Translation, and Responsibility for Data Science Impact","funder_award_id":"1922658","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6671297155","display_name":null,"funder_award_id":"CAREER","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7059651368","display_name":"CAREER: New Frontiers in Bayesian Deep Learning","funder_award_id":"2145492","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8543921386","display_name":"Analyzing Sequential, Multiple Assignment, Randomized Trials in the Presence of Partial Compliance","funder_award_id":"5r01da048764-04","funder_id":"https://openalex.org/F4320332161","funder_display_name":"National Institutes of Health"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320332161","display_name":"National Institutes of Health","ror":"https://ror.org/01cwqze88"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4367628371.pdf","grobid_xml":"https://content.openalex.org/works/W4367628371.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3082668976","https://openalex.org/W2517973022","https://openalex.org/W1543121148","https://openalex.org/W1580703421","https://openalex.org/W2996841952","https://openalex.org/W2618518959","https://openalex.org/W2231364979","https://openalex.org/W4295745414","https://openalex.org/W4382202915","https://openalex.org/W2059261580"],"abstract_inverted_index":{"Unlike":[0],"conventional":[1],"grid":[2],"and":[3,88,161,183],"mesh":[4],"based":[5,101,173],"methods":[6,100,135],"for":[7,50],"solving":[8],"partial":[9],"differential":[10,81],"equations":[11],"(PDEs),":[12],"neural":[13,149,203],"networks":[14],"have":[15],"the":[16,20,41,45,57,72,85,89,112,119,122,133,139,178,189,197],"potential":[17],"to":[18,27,62,126,147,155,195],"break":[19],"curse":[21],"of":[22,40,59,121,141,191,199],"dimensionality,":[23],"providing":[24],"approximate":[25],"solutions":[26],"problems":[28,65],"where":[29],"using":[30],"classical":[31],"solvers":[32],"is":[33],"difficult":[34],"or":[35],"impossible.":[36],"While":[37],"global":[38],"minimization":[39],"PDE":[42,158],"residual":[43],"over":[44],"network":[46,86,179],"parameters":[47,87],"works":[48],"well":[49],"boundary":[51],"value":[52,64],"problems,":[53],"catastrophic":[54],"forgetting":[55],"impairs":[56],"applicability":[58],"this":[60,103],"approach":[61,104],"initial":[63,159],"(IVPs).":[66],"In":[67],"an":[68,79,115,171],"alternative":[69],"local-in-time":[70],"approach,":[71],"optimization":[73],"problem":[74],"can":[75],"be":[76],"converted":[77],"into":[78],"ordinary":[80],"equation":[82],"(ODE)":[83],"on":[84,102,164],"solution":[90],"propagated":[91],"forward":[92],"in":[93,118,185,188],"time;":[94],"however,":[95],"we":[96,167],"demonstrate":[97],"that":[98],"current":[99],"suffer":[105],"from":[106,180],"two":[107],"key":[108],"issues.":[109],"First,":[110],"following":[111],"ODE":[113,134,172],"produces":[114],"uncontrolled":[116],"growth":[117],"conditioning":[120],"problem,":[123],"ultimately":[124],"leading":[125],"unacceptably":[127],"large":[128],"numerical":[129],"errors.":[130],"Second,":[131],"as":[132],"scale":[136],"cubically":[137],"with":[138,202],"number":[140,190],"model":[142],"parameters,":[143,192],"they":[144],"are":[145],"restricted":[146],"small":[148],"networks,":[150],"significantly":[151],"limiting":[152],"their":[153],"ability":[154],"represent":[156],"intricate":[157],"conditions":[160],"solutions.":[162],"Building":[163],"these":[165],"insights,":[166],"develop":[168],"Neural":[169],"IVP,":[170],"IVP":[174],"solver":[175],"which":[176],"prevents":[177],"getting":[181],"ill-conditioned":[182],"runs":[184],"time":[186],"linear":[187],"enabling":[193],"us":[194],"evolve":[196],"dynamics":[198],"challenging":[200],"PDEs":[201],"networks.":[204]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-08-18T07:49:30.821534","created_date":"2023-05-02T00:00:00"}