Pesquisadores de matemática aclamam avanço nas aplicações de inteligaªncia artificial
Pela primeira vez, cientistas da computaa§a£o e matema¡ticos usaram inteligaªncia artificial para ajudar a provar ou sugerir novos teoremas matema¡ticos nos campos complexos da teoria do na³ e da teoria da representaa§a£o.
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)
O professor Geordie Williamson FRS éum matema¡tico lider no campo da teoria da representação e diretor do Instituto de Pesquisa Matema¡tica da Universidade de Sydney. Crédito: Louise Cooper / University of Sydney
Pela primeira vez, cientistas da computação e matema¡ticos usaram inteligaªncia artificial para ajudar a provar ou sugerir novos teoremas matema¡ticos nos campos complexos da teoria do na³ e da teoria da representação.
Os resultados surpreendentes foram publicados hoje na revista cientafica proeminente, Nature .
O professor Geordie Williamson édiretor do Instituto de Pesquisa Matema¡tica da Universidade de Sydney e um dos matema¡ticos mais importantes do mundo. Como coautor do artigo, ele aplicou o poder dos processos de IA do Deep Mind para explorar conjecturas em seu campo de especialidade, a teoria da representação.
Seus coautores eram da DeepMind - a equipe de cientistas da computação por trás do AlphaGo , o primeiro programa de computador a derrotar com sucesso um campea£o mundial no jogo Go em 2016.
O professor Williamson disse: "Os problemas de matemática são amplamente considerados alguns dos problemas mais desafiadores intelectualmente.
"Embora os matema¡ticos tenham usado o aprendizado de ma¡quina para auxiliar na análise de conjuntos de dados complexos, esta éa primeira vez que usamos computadores para nos ajudar a formular conjecturas ou sugerir possaveis linhas de ataque para ideias não comprovadas em matemática."
Provando conjecturas matemáticas
O professor Williamson éum lider mundialmente reconhecido na teoria da representação, o ramo da matemática que explora o espaço dimensional superior usando a¡lgebra linear .
Em 2018, ele foi eleito o mais jovem membro vivo da Royal Society de Londres, a associação cientafica mais antiga e provavelmente de maior prestígio do mundo.
"Trabalhar para provar ou refutar conjecturas de longa data em meu campo envolve a consideração de, a s vezes, espaço infinito e conjuntos extremamente complexos de equações em maºltiplasDimensões ", disse o professor Williamson.
Embora os computadores tenham sido usados ​​ha¡ muito tempo para gerar dados para a matemática experimental, a tarefa de identificar padraµes interessantes se baseou principalmente na intuição dos pra³prios matema¡ticos.
Isso agora mudou.
O professor Williamson usou a IA do DeepMind para aproxima¡-lo de provar uma velha conjectura sobre os polina´mios de Kazhdan-Lusztig, que não foi resolvida por 40 anos. As conjecturas dizem respeito a simetria profunda em a¡lgebra de dimensão superior.
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Os co-autores, o professor Marc Lackeby e o professor Andra¡s Juha¡sz, da University of Oxford, deram um passo adiante no processo. Eles descobriram uma conexão surpreendente entre invariantes alganãbricos e geomanãtricos de nós, estabelecendo um teorema completamente novo em matemática.
Na teoria dos nós, os invariantes são usados ​​para resolver o problema de distinguir os nosuns dos outros. Eles também ajudam os matema¡ticos a entender as propriedades dos nose como isso se relaciona com outros ramos da matemática.
Embora seja de profundo interesse por si são, a teoria do na³ também tem inaºmeras aplicações nas ciências físicas, desde a compreensão dos filamentos do DNA, dina¢mica dos fluidos e a interação de forças na coroa solar.
O professor Juha¡sz disse: “Os matema¡ticos puros trabalham formulando conjecturas e provando-as, resultando em teoremas. Mas de onde vão as conjecturas?
"Demonstramos que, quando guiado pela intuição matemática, o aprendizado de ma¡quina fornece uma estrutura poderosa que pode revelar conjecturas interessantes e prova¡veis ​​em áreas onde uma grande quantidade de dados estãodisponavel ou onde os objetos são muito grandes para estudar com manãtodos cla¡ssicos."
O professor Lackeby disse: "Tem sido fascinante usar o aprendizado de ma¡quina para descobrir conexões novas e inesperadas entre diferentes áreas da matemática. Acredito que o trabalho que fizemos em Oxford e em Sydney em colaboração com a DeepMind demonstra que o aprendizado de ma¡quina pode ser um ferramenta genuinamente útil na pesquisa matemática. "
O autor principal da DeepMind, Dr. Alex Davies, disse: "Achamos que as técnicas de IA já estãosuficientemente avana§adas para ter um impacto na aceleração do progresso cientafico em muitas disciplinas diferentes. A matemática pura éum exemplo e esperamos que este artigo da Nature possa inspirar outros pesquisadores considerar o potencial da IA ​​como uma ferramenta útil no campo. "
O professor Williamson disse: "IA éuma ferramenta extraordina¡ria. Este trabalho éuma das primeiras vezes em que demonstrou sua utilidade para matema¡ticos puros, como eu."
"A intuição pode nos levar muito longe, mas a IA pode nos ajudar a encontrar conexões que a mente humana nem sempre pode detectar facilmente."
Os autores esperam que este trabalho possa servir de modelo para aprofundar a colaboração entre os campos da matemática e da inteligaªncia artificial para alcana§ar resultados surpreendentes, alavancando os respectivos pontos fortes da matemática. e do aprendizado de ma¡quina.
"Para mim, essas descobertas nos lembram que a inteligaªncia não éuma única varia¡vel, como um número de QI. A inteligaªncia émais bem pensada como um espaço multidimensional com maºltiplos eixos: inteligaªncia acadaªmica, inteligaªncia emocional, inteligaªncia social", disse o professor Williamson.
"Minha esperana§a éque a IA possa fornecer outro eixo de inteligaªncia para trabalharmos, e que esse novo eixo aprofunde nossa compreensão do mundo matema¡tico."