2024
Learning integral operators via neural integral equations
Zappala E, Fonseca A, Caro J, Moberly A, Higley M, Cardin J, Dijk D. Learning integral operators via neural integral equations. Nature Machine Intelligence 2024, 6: 1046-1062. DOI: 10.1038/s42256-024-00886-8.Peer-Reviewed Original ResearchSelf-attentionNon-local operatorsMachine learningModeling complex systemsReal-world dataHigher-dimensional problemsComplex systemsDynamic embeddingsIntegral equationsModel capacitySpatiotemporal dependenciesIntegral operatorsSecond-kind integral equationsIntegral equation solversModeling capabilitiesNonlinear operatorsTheoretical analysisNon-local systemNumerical benchmarksMachineLearningApproximate resultsNavier-StokesEquation solverScalability
2023
Neural Integro-Differential Equations
Zappala E, de O. Fonseca A, Moberly A, Higley M, Abdallah C, Cardin J, Van Dijk D. Neural Integro-Differential Equations. Proceedings Of The AAAI Conference On Artificial Intelligence 2023, 37: 11104-11112. DOI: 10.1609/aaai.v37i9.26315.Peer-Reviewed Original ResearchIntegro-differential equationsIntegral operatorsDifferential equationsContinuous dynamical systemsNon-local dynamicsDynamical systemsInitial conditionsEquationsNeural networkTime extrapolationOperatorsIntegralsFundamental problemSuch dynamicsLatent spaceDynamicsNon-local processesBrain activity recordingsBrain dynamicsData scienceDifferential componentsIntegrandGeneralizationTheoryNetwork
2000
Challenges in Analysis
Coifman R. Challenges in Analysis. Modern Birkhäuser Classics 2000, 471-480. DOI: 10.1007/978-3-0346-0425-3_2.Peer-Reviewed Original Research
1993
Wavelet-Like Bases for the Fast Solution of Second-Kind Integral Equations
Alpert B, Beylkin G, Coifman R, Rokhlin V. Wavelet-Like Bases for the Fast Solution of Second-Kind Integral Equations. SIAM Journal On Scientific Computing 1993, 14: 159-184. DOI: 10.1137/0914010.Peer-Reviewed Original ResearchIntegral equationsSecond kind integral equationsKind integral equationsWavelet-like basisVector space basisIntegral operatorsNumerical solutionNumber of pointsFinite numberFast solutionSparse matricesNumerical resultsDiscretizationEquationsOperatorsSparse representationSingularitySolutionGeneralization
1990
Wavelets for the Fast Solution of Second-Kind Integral Equations
Alpert B, Beylkin G, Coifman R, Rokhlin V. Wavelets for the Fast Solution of Second-Kind Integral Equations. 1990 DOI: 10.21236/ada233650.Peer-Reviewed Original ResearchIntegral equationsSecond kind integral equationsKind integral equationsNon-oscillatory kernelsVector space basisIntegral operatorsNumerical solutionNumber of pointsFinite numberFast solutionSparse matricesNumerical resultsDiscretizationEquationsOperatorsSparse representationSingularitySolutionGeneralizationKernelMultiresolution Analysis in Non-Homogeneous Media
Coifman R. Multiresolution Analysis in Non-Homogeneous Media. Inverse Problems And Theoretical Imaging 1990, 259-262. DOI: 10.1007/978-3-642-75988-8_25.Peer-Reviewed Original Research
1989
Multiresolution Analysis in Non-Homogeneous Media
Coifman R. Multiresolution Analysis in Non-Homogeneous Media. Inverse Problems And Theoretical Imaging 1989, 259-262. DOI: 10.1007/978-3-642-97177-8_25.Peer-Reviewed Original Research
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