Matches in SemOpenAlex for { <https://semopenalex.org/work/W3125962005> ?p ?o ?g. }
Showing items 1 to 97 of
97
with 100 items per page.
- W3125962005 abstract "Approximating the numerical solutions of partial differential equations (PDEs) using neural networks is a promising application of deep learning. The smooth architecture of a fully connected neural network is appropriate for finding the solutions of PDEs; the corresponding loss function can also be intuitively designed and guarantees the convergence for various kinds of PDEs. However, the rate of convergence has been considered as a weakness of this approach. This paper introduces a novel loss function for the training of neural networks to find the solutions of PDEs, making the training substantially efficient. Inspired by the recent studies that incorporate derivative information for the training of neural networks, we develop a loss function that guides a neural network to reduce the error in the corresponding Sobolev space. Surprisingly, a simple modification of the loss function can make the training process similar to Sobolev Training although solving PDEs with neural networks is not a fully supervised learning task. We provide several theoretical justifications for such an approach for the viscous Burgers equation and the kinetic Fokker--Planck equation. We also present several simulation results, which show that compared with the traditional L2 loss function, the proposed loss function guides the neural network to a significantly faster convergence. Moreover, we provide the empirical evidence that shows that the proposed loss function, together with the iterative sampling techniques, performs better in solving high dimensional PDEs." @default.
- W3125962005 created "2021-02-01" @default.
- W3125962005 creator A5003733262 @default.
- W3125962005 creator A5045120821 @default.
- W3125962005 creator A5046952532 @default.
- W3125962005 creator A5086339183 @default.
- W3125962005 date "2021-05-04" @default.
- W3125962005 modified "2023-10-07" @default.
- W3125962005 title "Sobolev Training for the Neural Network Solutions of PDEs" @default.
- W3125962005 cites W2008436424 @default.
- W3125962005 cites W2009566405 @default.
- W3125962005 cites W2027355161 @default.
- W3125962005 cites W2148666448 @default.
- W3125962005 cites W2749028154 @default.
- W3125962005 cites W2760972773 @default.
- W3125962005 cites W2801347329 @default.
- W3125962005 cites W2803629276 @default.
- W3125962005 cites W2899283552 @default.
- W3125962005 cites W2909891600 @default.
- W3125962005 cites W2962727772 @default.
- W3125962005 cites W2963021886 @default.
- W3125962005 cites W2963672187 @default.
- W3125962005 cites W2965391011 @default.
- W3125962005 cites W2970971581 @default.
- W3125962005 cites W3021722416 @default.
- W3125962005 cites W3035647965 @default.
- W3125962005 cites W3146803896 @default.
- W3125962005 hasPublicationYear "2021" @default.
- W3125962005 type Work @default.
- W3125962005 sameAs 3125962005 @default.
- W3125962005 citedByCount "6" @default.
- W3125962005 countsByYear W31259620052021 @default.
- W3125962005 countsByYear W31259620052022 @default.
- W3125962005 crossrefType "journal-article" @default.
- W3125962005 hasAuthorship W3125962005A5003733262 @default.
- W3125962005 hasAuthorship W3125962005A5045120821 @default.
- W3125962005 hasAuthorship W3125962005A5046952532 @default.
- W3125962005 hasAuthorship W3125962005A5086339183 @default.
- W3125962005 hasConcept C11413529 @default.
- W3125962005 hasConcept C126255220 @default.
- W3125962005 hasConcept C134306372 @default.
- W3125962005 hasConcept C14036430 @default.
- W3125962005 hasConcept C142730499 @default.
- W3125962005 hasConcept C154945302 @default.
- W3125962005 hasConcept C162324750 @default.
- W3125962005 hasConcept C2777303404 @default.
- W3125962005 hasConcept C28826006 @default.
- W3125962005 hasConcept C33923547 @default.
- W3125962005 hasConcept C41008148 @default.
- W3125962005 hasConcept C50522688 @default.
- W3125962005 hasConcept C50644808 @default.
- W3125962005 hasConcept C78458016 @default.
- W3125962005 hasConcept C86803240 @default.
- W3125962005 hasConcept C93779851 @default.
- W3125962005 hasConcept C99730327 @default.
- W3125962005 hasConceptScore W3125962005C11413529 @default.
- W3125962005 hasConceptScore W3125962005C126255220 @default.
- W3125962005 hasConceptScore W3125962005C134306372 @default.
- W3125962005 hasConceptScore W3125962005C14036430 @default.
- W3125962005 hasConceptScore W3125962005C142730499 @default.
- W3125962005 hasConceptScore W3125962005C154945302 @default.
- W3125962005 hasConceptScore W3125962005C162324750 @default.
- W3125962005 hasConceptScore W3125962005C2777303404 @default.
- W3125962005 hasConceptScore W3125962005C28826006 @default.
- W3125962005 hasConceptScore W3125962005C33923547 @default.
- W3125962005 hasConceptScore W3125962005C41008148 @default.
- W3125962005 hasConceptScore W3125962005C50522688 @default.
- W3125962005 hasConceptScore W3125962005C50644808 @default.
- W3125962005 hasConceptScore W3125962005C78458016 @default.
- W3125962005 hasConceptScore W3125962005C86803240 @default.
- W3125962005 hasConceptScore W3125962005C93779851 @default.
- W3125962005 hasConceptScore W3125962005C99730327 @default.
- W3125962005 hasOpenAccess W3125962005 @default.
- W3125962005 hasRelatedWork W1513568640 @default.
- W3125962005 hasRelatedWork W2749028154 @default.
- W3125962005 hasRelatedWork W2760972773 @default.
- W3125962005 hasRelatedWork W2988233686 @default.
- W3125962005 hasRelatedWork W3000403725 @default.
- W3125962005 hasRelatedWork W3036721570 @default.
- W3125962005 hasRelatedWork W3046173836 @default.
- W3125962005 hasRelatedWork W3084220965 @default.
- W3125962005 hasRelatedWork W3084353952 @default.
- W3125962005 hasRelatedWork W3094346332 @default.
- W3125962005 hasRelatedWork W3121217874 @default.
- W3125962005 hasRelatedWork W3126967114 @default.
- W3125962005 hasRelatedWork W3127478689 @default.
- W3125962005 hasRelatedWork W3132007915 @default.
- W3125962005 hasRelatedWork W3157506487 @default.
- W3125962005 hasRelatedWork W3164610483 @default.
- W3125962005 hasRelatedWork W3173549089 @default.
- W3125962005 hasRelatedWork W3196070124 @default.
- W3125962005 hasRelatedWork W3209680840 @default.
- W3125962005 hasRelatedWork W3128884106 @default.
- W3125962005 isParatext "false" @default.
- W3125962005 isRetracted "false" @default.
- W3125962005 magId "3125962005" @default.
- W3125962005 workType "article" @default.