Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data

Numerical simulations on fluid dynamics problems primarily rely on spatially or/and temporally discretization of the governing equation using polynomials into a finite-dimensional algebraic system. Due to the multi-scale nature of the physics and sensitivity from meshing a complicated geometry, such...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Veröffentlicht in:	Computer methods in applied mechanics and engineering Jg. 361; S. 112732
Hauptverfasser:	Sun, Luning, Gao, Han, Pan, Shaowu, Wang, Jian-Xun
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Amsterdam Elsevier B.V 01.04.2020 Elsevier BV
Schlagworte:	Artificial neural networks Boundary conditions Cardiovascular flows Computational fluid dynamics Computer simulation Data points Deep learning Design optimization First principles Fluid dynamics Fluid flow Hemodynamics Internal flow Label-free Machine learning Mathematical models Meshing Navier-Stokes Navier-Stokes equations Neural networks Partial differential equations Physics Physics-informed machine learning Polynomials Simulation Training Uncertainty Uncertainty quantification Physics-informed machine learning Uncertainty quantification Navier-Stokes Cardiovascular flows Label-free Neural networks
ISSN:	0045-7825, 1879-2138
Online-Zugang:	Volltext
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!

Abstract	Numerical simulations on fluid dynamics problems primarily rely on spatially or/and temporally discretization of the governing equation using polynomials into a finite-dimensional algebraic system. Due to the multi-scale nature of the physics and sensitivity from meshing a complicated geometry, such process can be computational prohibitive for most real-time applications (e.g., clinical diagnosis and surgery planning) and many-query analyses (e.g., optimization design and uncertainty quantification). Therefore, developing a cost-effective surrogate model is of great practical significance. Deep learning (DL) has shown new promises for surrogate modeling due to its capability of handling strong nonlinearity and high dimensionality. However, the off-the-shelf DL architectures, success of which heavily relies on the large amount of training data and interpolatory nature of the problem, fail to operate when the data becomes sparse. Unfortunately, data is often insufficient in most parametric fluid dynamics problems since each data point in the parameter space requires an expensive numerical simulation based on the first principle, e.g., Navier–Stokes equations. In this paper, we provide a physics-constrained DL approach for surrogate modeling of fluid flows without relying on any simulation data. Specifically, a structured deep neural network (DNN) architecture is devised to enforce the initial and boundary conditions, and the governing partial differential equations (i.e., Navier–Stokes equations) are incorporated into the loss of the DNN to drive the training. Numerical experiments are conducted on a number of internal flows relevant to hemodynamics applications, and the forward propagation of uncertainties in fluid properties and domain geometry is studied as well. The results show excellent agreement on the flow field and forward-propagated uncertainties between the DL surrogate approximations and the first-principle numerical simulations. •Proposed a simulation-free, physics-constrained deep learning for surrogate CFD model.•Boundary-encoded neural network outperforms the one with soft boundary constraints.•Demonstrated effectiveness of the label-free learning on a few vascular flows.
AbstractList	Numerical simulations on fluid dynamics problems primarily rely on spatially or/and temporally discretization of the governing equation using polynomials into a finite-dimensional algebraic system. Due to the multi-scale nature of the physics and sensitivity from meshing a complicated geometry, such process can be computational prohibitive for most real-time applications (e.g., clinical diagnosis and surgery planning) and many-query analyses (e.g., optimization design and uncertainty quantification). Therefore, developing a cost-effective surrogate model is of great practical significance. Deep learning (DL) has shown new promises for surrogate modeling due to its capability of handling strong nonlinearity and high dimensionality. However, the off-the-shelf DL architectures, success of which heavily relies on the large amount of training data and interpolatory nature of the problem, fail to operate when the data becomes sparse. Unfortunately, data is often insufficient in most parametric fluid dynamics problems since each data point in the parameter space requires an expensive numerical simulation based on the first principle, e.g., Navier–Stokes equations. In this paper, we provide a physics-constrained DL approach for surrogate modeling of fluid flows without relying on any simulation data. Specifically, a structured deep neural network (DNN) architecture is devised to enforce the initial and boundary conditions, and the governing partial differential equations (i.e., Navier–Stokes equations) are incorporated into the loss of the DNN to drive the training. Numerical experiments are conducted on a number of internal flows relevant to hemodynamics applications, and the forward propagation of uncertainties in fluid properties and domain geometry is studied as well. The results show excellent agreement on the flow field and forward-propagated uncertainties between the DL surrogate approximations and the first-principle numerical simulations. •Proposed a simulation-free, physics-constrained deep learning for surrogate CFD model.•Boundary-encoded neural network outperforms the one with soft boundary constraints.•Demonstrated effectiveness of the label-free learning on a few vascular flows. Numerical simulations on fluid dynamics problems primarily rely on spatially or/and temporally discretization of the governing equation using polynomials into a finite-dimensional algebraic system. Due to the multi-scale nature of the physics and sensitivity from meshing a complicated geometry, such process can be computational prohibitive for most real-time applications (e.g., clinical diagnosis and surgery planning) and many-query analyses (e.g., optimization design and uncertainty quantification). Therefore, developing a cost-effective surrogate model is of great practical significance. Deep learning (DL) has shown new promises for surrogate modeling due to its capability of handling strong nonlinearity and high dimensionality. However, the off-the-shelf DL architectures, success of which heavily relies on the large amount of training data and interpolatory nature of the problem, fail to operate when the data becomes sparse. Unfortunately, data is often insufficient in most parametric fluid dynamics problems since each data point in the parameter space requires an expensive numerical simulation based on the first principle, e.g., Navier–Stokes equations. In this paper, we provide a physics-constrained DL approach for surrogate modeling of fluid flows without relying on any simulation data. Specifically, a structured deep neural network (DNN) architecture is devised to enforce the initial and boundary conditions, and the governing partial differential equations (i.e., Navier–Stokes equations) are incorporated into the loss of the DNN to drive the training. Numerical experiments are conducted on a number of internal flows relevant to hemodynamics applications, and the forward propagation of uncertainties in fluid properties and domain geometry is studied as well. The results show excellent agreement on the flow field and forward-propagated uncertainties between the DL surrogate approximations and the first-principle numerical simulations.
ArticleNumber	112732
Author	Gao, Han Pan, Shaowu Sun, Luning Wang, Jian-Xun
Author_xml	– sequence: 1 givenname: Luning surname: Sun fullname: Sun, Luning organization: Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, United States of America – sequence: 2 givenname: Han surname: Gao fullname: Gao, Han organization: Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, United States of America – sequence: 3 givenname: Shaowu orcidid: 0000-0002-2462-362X surname: Pan fullname: Pan, Shaowu organization: Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, United States of America – sequence: 4 givenname: Jian-Xun orcidid: 0000-0002-9030-1733 surname: Wang fullname: Wang, Jian-Xun email: jwang33@nd.edu organization: Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, United States of America
BookMark	eNp9kD1PwzAQhi1UJNrCD2CLxJxgOx9OxIQqviQkBmC2HPvSOkrsYjtU_fe4lImhN_gk6x6f32eBZsYaQOia4IxgUt32mRxFRjFpMkIoy-kZmpOaNSkleT1Dc4yLMmU1LS_Qwvsex6oJnSN4n5yzaxEgGa2CQZt10lmXdMOkVTztziet8KASa5LtZu-19Km0xgcntInXCmCbDCCcOaA7HTZ2ConX4zSIoCOkRBCX6LwTg4erv75En48PH6vn9PXt6WV1_5rKIi9CGj-s2qbqoFNVqQpZU1a3FWVQtk0nWiabssYS06ZShJUyL3Kpalo0rWxbTDqWL9HN8d2ts18T-MB7OzkTV3KaM8KqijRFnCLHKems9w46vnV6FG7PCeYHm7zn0SY_2ORHm5Fh_xipw2_Ag4jhJHl3JCEG_9bguJcajASlHcjAldUn6B8UUZJt
CitedBy_id	crossref_primary_10_1016_j_chroma_2025_465831 crossref_primary_10_1017_jfm_2021_135 crossref_primary_10_1139_cgj_2023_0567 crossref_primary_10_1177_14759217231202547 crossref_primary_10_1016_j_cpc_2025_109782 crossref_primary_10_1021_acs_jpca_5c00405 crossref_primary_10_1007_s40436_025_00545_0 crossref_primary_10_1016_j_seppur_2025_132012 crossref_primary_10_1002_fld_5238 crossref_primary_10_1016_j_engappai_2024_108945 crossref_primary_10_1016_j_aei_2022_101661 crossref_primary_10_1103_d4tm_92vb crossref_primary_10_1016_j_oceaneng_2024_118341 crossref_primary_10_1109_TPEL_2025_3547390 crossref_primary_10_1016_j_tws_2024_112846 crossref_primary_10_1017_dce_2022_2 crossref_primary_10_1007_s10444_023_10065_9 crossref_primary_10_1016_j_enconman_2024_119474 crossref_primary_10_1186_s40323_024_00265_3 crossref_primary_10_1007_s12046_023_02236_7 crossref_primary_10_1016_j_autcon_2024_105790 crossref_primary_10_1016_j_cma_2024_117058 crossref_primary_10_1016_j_coastaleng_2022_104167 crossref_primary_10_1063_5_0228104 crossref_primary_10_1016_j_petrol_2022_110460 crossref_primary_10_3389_fphy_2024_1385381 crossref_primary_10_1016_j_enbuild_2024_114575 crossref_primary_10_1016_j_jocs_2025_102717 crossref_primary_10_1109_LAWP_2024_3355896 crossref_primary_10_1109_ACCESS_2024_3504962 crossref_primary_10_1016_j_engstruct_2022_115484 crossref_primary_10_1016_j_cma_2022_115668 crossref_primary_10_1016_j_combustflame_2023_113094 crossref_primary_10_1177_13694332251369102 crossref_primary_10_1016_j_engappai_2023_106468 crossref_primary_10_1016_j_cma_2022_115100 crossref_primary_10_1016_j_cpc_2025_109762 crossref_primary_10_1016_j_apor_2024_104246 crossref_primary_10_1016_j_neucom_2022_05_015 crossref_primary_10_1029_2020WR029479 crossref_primary_10_1016_j_biosystemseng_2025_01_017 crossref_primary_10_1016_j_media_2025_103564 crossref_primary_10_3390_app122110986 crossref_primary_10_3390_e24081134 crossref_primary_10_1029_2024WR037736 crossref_primary_10_1038_s41598_024_84940_w crossref_primary_10_1016_j_powtec_2022_117303 crossref_primary_10_1063_5_0274006 crossref_primary_10_3390_pr13041093 crossref_primary_10_1080_19942060_2025_2535015 crossref_primary_10_1016_j_jcp_2020_109914 crossref_primary_10_1016_j_petrol_2021_110069 crossref_primary_10_1038_s41524_025_01635_0 crossref_primary_10_1016_j_applthermaleng_2024_123670 crossref_primary_10_1109_TMAG_2025_3553236 crossref_primary_10_1016_j_jmatprotec_2022_117534 crossref_primary_10_1017_jfm_2020_453 crossref_primary_10_1016_j_camwa_2024_11_009 crossref_primary_10_1016_j_cpc_2025_109582 crossref_primary_10_1016_j_cma_2022_115120 crossref_primary_10_1016_j_flowmeasinst_2023_102418 crossref_primary_10_1038_s41598_023_29822_3 crossref_primary_10_3390_en15217864 crossref_primary_10_1080_10589759_2024_2338187 crossref_primary_10_1016_j_compgeo_2024_106825 crossref_primary_10_1063_5_0056549 crossref_primary_10_1016_j_icheatmasstransfer_2023_106662 crossref_primary_10_3390_a17070279 crossref_primary_10_1016_j_engstruct_2024_119194 crossref_primary_10_1016_j_apacoust_2023_109686 crossref_primary_10_1016_j_jcp_2022_111022 crossref_primary_10_1016_j_jcp_2024_113285 crossref_primary_10_1016_j_net_2024_103353 crossref_primary_10_1007_s00366_023_01871_2 crossref_primary_10_1007_s10409_022_22302_x crossref_primary_10_1016_j_cma_2024_117130 crossref_primary_10_1016_j_ast_2024_109706 crossref_primary_10_1038_s41467_021_26434_1 crossref_primary_10_1016_j_jocs_2025_102639 crossref_primary_10_1016_j_asoc_2021_108050 crossref_primary_10_3390_s23094492 crossref_primary_10_1016_j_engappai_2023_106340 crossref_primary_10_3390_math12243892 crossref_primary_10_1016_j_ijmecsci_2024_109210 crossref_primary_10_1016_j_copbio_2021_01_002 crossref_primary_10_1016_j_engappai_2023_106425 crossref_primary_10_59717_j_xinn_energy_2025_100087 crossref_primary_10_1016_j_cma_2025_117764 crossref_primary_10_1016_j_engappai_2022_105176 crossref_primary_10_1109_ACCESS_2025_3585402 crossref_primary_10_1016_j_cma_2024_117004 crossref_primary_10_3389_fmars_2025_1547995 crossref_primary_10_1017_dce_2024_33 crossref_primary_10_1016_j_cpc_2025_109569 crossref_primary_10_1016_j_petrol_2022_110179 crossref_primary_10_1038_s42256_023_00685_7 crossref_primary_10_1016_j_asoc_2024_112632 crossref_primary_10_1007_s11831_024_10152_0 crossref_primary_10_3390_eng6050099 crossref_primary_10_1016_j_jmapro_2022_02_053 crossref_primary_10_1088_1741_4326_ae00db crossref_primary_10_1016_j_jmps_2023_105472 crossref_primary_10_32604_cmes_2021_016737 crossref_primary_10_1016_j_cma_2024_117116 crossref_primary_10_1016_j_cma_2024_117478 crossref_primary_10_1016_j_jcp_2024_113144 crossref_primary_10_1016_j_compbiomed_2025_110074 crossref_primary_10_1016_j_cma_2023_116401 crossref_primary_10_1016_j_proci_2022_07_144 crossref_primary_10_1007_s10439_022_02967_4 crossref_primary_10_1016_j_cma_2024_117355 crossref_primary_10_1137_20M1323151 crossref_primary_10_1016_j_brainresbull_2025_111318 crossref_primary_10_1186_s42774_021_00094_7 crossref_primary_10_1016_j_neucom_2025_129917 crossref_primary_10_1109_ACCESS_2022_3199652 crossref_primary_10_1155_2022_1781388 crossref_primary_10_1007_s40684_025_00751_y crossref_primary_10_1016_j_compchemeng_2024_108723 crossref_primary_10_1088_2632_2153_acfd09 crossref_primary_10_1016_j_jcp_2025_113798 crossref_primary_10_3390_fluids9120296 crossref_primary_10_1016_j_camwa_2022_10_003 crossref_primary_10_1016_j_jhydrol_2023_130048 crossref_primary_10_3390_make7030086 crossref_primary_10_1016_j_artmed_2024_102995 crossref_primary_10_3390_batteries9060301 crossref_primary_10_1016_j_cma_2022_114871 crossref_primary_10_1016_j_cma_2025_118279 crossref_primary_10_1088_2631_7990_adea23 crossref_primary_10_1016_j_ast_2024_109518 crossref_primary_10_1016_j_cma_2022_115839 crossref_primary_10_1016_j_cpc_2024_109428 crossref_primary_10_1016_j_engappai_2025_111554 crossref_primary_10_1016_j_cnsns_2024_108242 crossref_primary_10_1080_20550340_2025_2547335 crossref_primary_10_1109_JSEN_2022_3199154 crossref_primary_10_1016_j_cma_2022_115831 crossref_primary_10_1016_j_cpc_2024_109422 crossref_primary_10_1016_j_jii_2025_100899 crossref_primary_10_1016_j_engappai_2025_110347 crossref_primary_10_1016_j_engstruct_2025_120018 crossref_primary_10_1038_s41524_022_00712_y crossref_primary_10_1016_j_eswa_2024_125079 crossref_primary_10_1007_s00521_024_09883_9 crossref_primary_10_1145_3604283 crossref_primary_10_1016_j_cma_2022_114740 crossref_primary_10_1016_j_engappai_2021_104232 crossref_primary_10_1109_TBME_2022_3163428 crossref_primary_10_1017_jfm_2025_10448 crossref_primary_10_1016_j_cma_2025_118284 crossref_primary_10_1016_j_cma_2025_118280 crossref_primary_10_1016_j_enbuild_2024_115023 crossref_primary_10_1016_j_jcp_2023_111919 crossref_primary_10_1016_j_jcp_2023_111912 crossref_primary_10_1016_j_compfluid_2024_106421 crossref_primary_10_1016_j_engappai_2023_106660 crossref_primary_10_1016_j_enbuild_2024_114295 crossref_primary_10_1016_j_cpc_2023_109010 crossref_primary_10_1016_j_jcp_2022_111731 crossref_primary_10_1016_j_eswa_2021_115409 crossref_primary_10_1080_10618562_2022_2154758 crossref_primary_10_1016_j_cma_2021_114211 crossref_primary_10_1016_j_cma_2025_118015 crossref_primary_10_1016_j_commatsci_2024_113502 crossref_primary_10_1016_j_procs_2025_03_106 crossref_primary_10_1016_j_compfluid_2024_106302 crossref_primary_10_1016_j_oceaneng_2024_120239 crossref_primary_10_1016_j_cnsns_2024_108229 crossref_primary_10_1177_30494826251336314 crossref_primary_10_1016_j_advwatres_2024_104870 crossref_primary_10_1016_j_oceaneng_2021_109406 crossref_primary_10_1063_5_0216609 crossref_primary_10_1016_j_cma_2024_116907 crossref_primary_10_1016_j_cma_2022_115852 crossref_primary_10_1016_j_tws_2024_111693 crossref_primary_10_1007_s00521_022_07294_2 crossref_primary_10_1016_j_compgeo_2023_105511 crossref_primary_10_1016_j_engappai_2025_111458 crossref_primary_10_1016_j_jcp_2020_110079 crossref_primary_10_1016_j_mfglet_2023_08_074 crossref_primary_10_1080_17455030_2022_2083264 crossref_primary_10_1016_j_ast_2024_109648 crossref_primary_10_1016_j_compstruct_2022_116354 crossref_primary_10_1016_j_biosystemseng_2023_04_012 crossref_primary_10_1016_j_enganabound_2025_106207 crossref_primary_10_1016_j_jcp_2021_110364 crossref_primary_10_1016_j_neucom_2023_126890 crossref_primary_10_1109_ACCESS_2022_3208103 crossref_primary_10_1016_j_cma_2023_115902 crossref_primary_10_1016_j_compbiomed_2025_110265 crossref_primary_10_1016_j_engappai_2023_106724 crossref_primary_10_1016_j_cma_2022_114790 crossref_primary_10_1186_s40645_024_00654_7 crossref_primary_10_1615_IntJMultCompEng_2025059444 crossref_primary_10_1109_ACCESS_2024_3481671 crossref_primary_10_1016_j_neucom_2020_09_006 crossref_primary_10_3390_app14020859 crossref_primary_10_1186_s42774_024_00197_x crossref_primary_10_1016_j_jmsy_2022_04_004 crossref_primary_10_1016_j_taml_2020_01_039 crossref_primary_10_1061_JGGEFK_GTENG_13267 crossref_primary_10_1016_j_compgeo_2025_107110 crossref_primary_10_1016_j_knosys_2024_111831 crossref_primary_10_1016_j_jcp_2021_110296 crossref_primary_10_1029_2021WR030595 crossref_primary_10_1016_j_pnucene_2022_104339 crossref_primary_10_1016_j_cma_2022_115757 crossref_primary_10_1002_tal_70002 crossref_primary_10_1016_j_cageo_2023_105360 crossref_primary_10_3390_math12213430 crossref_primary_10_1016_j_jcp_2021_110192 crossref_primary_10_1016_j_cma_2022_114784 crossref_primary_10_1016_j_jobe_2024_109231 crossref_primary_10_1007_s00158_024_03935_3 crossref_primary_10_1007_s44379_025_00015_1 crossref_primary_10_1007_s12206_024_0624_9 crossref_primary_10_1016_j_jcp_2022_111419 crossref_primary_10_1061_JCCEE5_CPENG_6151 crossref_primary_10_1016_j_geoen_2023_212516 crossref_primary_10_1016_j_ress_2022_108900 crossref_primary_10_1038_s41467_024_46246_3 crossref_primary_10_1016_j_jmps_2024_105714 crossref_primary_10_1016_j_advwatres_2022_104243 crossref_primary_10_1016_j_cma_2021_114399 crossref_primary_10_1016_j_cma_2021_114037 crossref_primary_10_1016_j_anucene_2023_110181 crossref_primary_10_1016_j_compfluid_2023_106164 crossref_primary_10_1016_j_jcp_2025_114027 crossref_primary_10_1016_j_earscirev_2025_105276 crossref_primary_10_1038_s41598_023_44541_5 crossref_primary_10_1016_j_cageo_2023_105477 crossref_primary_10_1016_j_ijheatmasstransfer_2024_125480 crossref_primary_10_1049_itr2_12551 crossref_primary_10_2118_203997_PA crossref_primary_10_1016_j_egyai_2025_100485 crossref_primary_10_1016_j_jcp_2025_114370 crossref_primary_10_3390_mca28020062 crossref_primary_10_1016_j_engappai_2022_105686 crossref_primary_10_1016_j_cma_2022_115771 crossref_primary_10_1016_j_icheatmasstransfer_2024_107897 crossref_primary_10_1016_j_cma_2024_117075 crossref_primary_10_3390_sym17081334 crossref_primary_10_1016_j_cma_2024_117071 crossref_primary_10_1016_j_ast_2024_109207 crossref_primary_10_3390_app13074539 crossref_primary_10_1061_JENMDT_EMENG_6643 crossref_primary_10_1016_j_taml_2025_100585 crossref_primary_10_3390_pr13061834 crossref_primary_10_2118_208602_PA crossref_primary_10_1016_j_compstruc_2022_106863 crossref_primary_10_3847_1538_4357_ade43e crossref_primary_10_1063_5_0281538 crossref_primary_10_1016_j_petrol_2021_109046 crossref_primary_10_3390_mca28020052 crossref_primary_10_3390_rs15205075 crossref_primary_10_1016_j_asoc_2025_113394 crossref_primary_10_1063_5_0132433 crossref_primary_10_1007_s10483_023_2993_9 crossref_primary_10_1016_j_anucene_2023_109840 crossref_primary_10_1016_j_eswa_2021_115146 crossref_primary_10_1016_j_cageo_2024_105611 crossref_primary_10_1016_j_camwa_2024_08_035 crossref_primary_10_1177_14759217221117238 crossref_primary_10_3390_en16124558 crossref_primary_10_1016_j_cma_2021_114507 crossref_primary_10_1016_j_euromechflu_2025_204352 crossref_primary_10_1038_s41598_022_16463_1 crossref_primary_10_1016_j_advwatres_2021_104051 crossref_primary_10_1016_j_ress_2022_108835 crossref_primary_10_1007_s10915_022_01980_y crossref_primary_10_1080_24725579_2021_1879322 crossref_primary_10_1109_TCAD_2024_3506867 crossref_primary_10_1007_s00707_023_03691_3 crossref_primary_10_1016_j_cma_2022_114909 crossref_primary_10_1002_er_7879 crossref_primary_10_1063_5_0291125 crossref_primary_10_1016_j_tafmec_2023_103925 crossref_primary_10_1016_j_cma_2021_114502 crossref_primary_10_1016_j_jcp_2023_112278 crossref_primary_10_1038_s44172_024_00303_3 crossref_primary_10_1007_s10915_022_01939_z crossref_primary_10_1002_nme_7228 crossref_primary_10_3390_biomimetics9020072 crossref_primary_10_1016_j_cma_2020_113492 crossref_primary_10_1016_j_jcp_2023_112291 crossref_primary_10_32604_cmes_2023_031093 crossref_primary_10_3390_en18112888 crossref_primary_10_1016_j_jhydrol_2024_131263 crossref_primary_10_1016_j_buildenv_2024_111175 crossref_primary_10_1016_j_ifacol_2022_09_070 crossref_primary_10_1007_s00466_020_01952_9 crossref_primary_10_1016_j_ifacol_2022_07_481 crossref_primary_10_1016_j_cma_2023_116139 crossref_primary_10_1016_j_seppur_2023_124631 crossref_primary_10_1016_j_jclepro_2023_139039 crossref_primary_10_1016_j_jcp_2021_110666 crossref_primary_10_1007_s00466_022_02252_0 crossref_primary_10_1109_TGRS_2022_3144636 crossref_primary_10_1016_j_camwa_2024_01_021 crossref_primary_10_1016_j_jpse_2025_100257 crossref_primary_10_1016_j_oceaneng_2023_114597 crossref_primary_10_1016_j_eswa_2024_123758 crossref_primary_10_1016_j_ijheatmasstransfer_2023_124593 crossref_primary_10_1016_j_ymssp_2022_109039 crossref_primary_10_3390_w16071041 crossref_primary_10_1016_j_geoen_2024_212788 crossref_primary_10_1007_s10462_025_11322_7 crossref_primary_10_1016_j_jcp_2021_110754 crossref_primary_10_1016_j_cma_2024_116983 crossref_primary_10_1016_j_compositesa_2025_108857 crossref_primary_10_1016_j_jmsy_2024_07_002 crossref_primary_10_1002_zamm_202300712 crossref_primary_10_1109_ACCESS_2024_3354058 crossref_primary_10_3390_math9161939 crossref_primary_10_3390_s23020663 crossref_primary_10_1016_j_advwatres_2023_104609 crossref_primary_10_1088_2632_2153_acffa4 crossref_primary_10_1016_j_ymssp_2022_109277 crossref_primary_10_1016_j_anucene_2022_109234 crossref_primary_10_1038_s42005_024_01521_z crossref_primary_10_1016_j_inffus_2023_102041 crossref_primary_10_1016_j_cma_2020_113226 crossref_primary_10_1016_j_jcp_2021_110768 crossref_primary_10_1137_20M1318043 crossref_primary_10_1002_nme_7135 crossref_primary_10_1016_j_cma_2024_117709 crossref_primary_10_1016_j_petrol_2021_109880 crossref_primary_10_2118_214288_PA crossref_primary_10_1016_j_jcp_2021_110521 crossref_primary_10_1109_MIC_2022_3212085 crossref_primary_10_1061_JENMDT_EMENG_7463 crossref_primary_10_1140_epjp_s13360_025_06220_4 crossref_primary_10_1016_j_engappai_2022_105516 crossref_primary_10_1016_j_cpc_2023_108775 crossref_primary_10_1016_j_cma_2021_114424 crossref_primary_10_1016_j_matdes_2020_109180 crossref_primary_10_1016_j_cma_2024_116965 crossref_primary_10_1007_s12551_020_00776_4 crossref_primary_10_1016_j_cma_2024_116847 crossref_primary_10_1016_j_renene_2022_10_013 crossref_primary_10_1038_s41524_025_01718_y crossref_primary_10_1016_j_ynexs_2024_100016 crossref_primary_10_1162_neco_a_01647 crossref_primary_10_1007_s00466_023_02434_4 crossref_primary_10_1016_j_cej_2021_131220 crossref_primary_10_1109_ACCESS_2022_3162827 crossref_primary_10_1016_j_oceaneng_2023_115393 crossref_primary_10_1016_j_oceaneng_2025_121026 crossref_primary_10_1063_5_0284621 crossref_primary_10_3390_atmos14040759 crossref_primary_10_1016_j_cma_2020_113047 crossref_primary_10_1016_j_taml_2025_100609 crossref_primary_10_1016_j_apenergy_2023_120855 crossref_primary_10_1109_TCAD_2023_3269385 crossref_primary_10_1016_j_geoen_2023_212474 crossref_primary_10_1109_LGRS_2022_3176867 crossref_primary_10_1016_j_cpc_2022_108382 crossref_primary_10_3390_axioms14050385 crossref_primary_10_1145_3549076 crossref_primary_10_1088_1572_9494_ada3ce crossref_primary_10_1137_22M1518189 crossref_primary_10_1063_5_0272317 crossref_primary_10_1016_j_compfluid_2025_106700 crossref_primary_10_1190_geo2022_0479_1 crossref_primary_10_1007_s10596_023_10232_3 crossref_primary_10_3390_w13040423 crossref_primary_10_1017_dsj_2023_25 crossref_primary_10_1002_fld_4888 crossref_primary_10_1103_PhysRevLett_126_098302 crossref_primary_10_1016_j_cnsns_2024_108544 crossref_primary_10_1016_j_jfoodeng_2022_111137 crossref_primary_10_1007_s10915_023_02328_w crossref_primary_10_1038_s41598_023_41039_y crossref_primary_10_3390_en16093820 crossref_primary_10_1002_eer2_70005 crossref_primary_10_1007_s10483_023_3050_9 crossref_primary_10_1145_3648506 crossref_primary_10_1038_s41524_024_01307_5 crossref_primary_10_1088_2632_2153_ac3712 crossref_primary_10_1016_j_jcp_2021_110551 crossref_primary_10_1016_j_petrol_2021_109694 crossref_primary_10_1016_j_cma_2021_114524 crossref_primary_10_5194_hess_26_4469_2022 crossref_primary_10_1139_cgj_2025_0118 crossref_primary_10_1016_j_energy_2024_131676 crossref_primary_10_1016_j_strusafe_2022_102256 crossref_primary_10_1016_j_aiig_2025_100148 crossref_primary_10_1016_j_scs_2024_105750 crossref_primary_10_1007_s00521_022_07092_w crossref_primary_10_3390_s23052792 crossref_primary_10_3390_s22103697 crossref_primary_10_1088_1361_6560_ae02dd crossref_primary_10_1016_j_jcp_2024_113012 crossref_primary_10_1016_j_rineng_2024_101931 crossref_primary_10_1016_j_jcp_2024_113494 crossref_primary_10_1016_j_cma_2024_117102 crossref_primary_10_1007_s00521_022_07257_7 crossref_primary_10_1016_j_ijheatfluidflow_2025_110011 crossref_primary_10_1016_j_ress_2024_110083 crossref_primary_10_1177_03611981241266837 crossref_primary_10_1016_j_compbiomed_2024_108706 crossref_primary_10_1016_j_physd_2024_134399 crossref_primary_10_1029_2022WR033168 crossref_primary_10_1016_j_cma_2021_113938 crossref_primary_10_1016_j_icheatmasstransfer_2024_108362 crossref_primary_10_1155_2022_9873112 crossref_primary_10_1016_j_jcp_2023_112435 crossref_primary_10_1016_j_cma_2024_117458 crossref_primary_10_1016_j_cma_2024_117211 crossref_primary_10_1016_j_triboint_2023_108871 crossref_primary_10_1103_PhysRevFluids_8_064603 crossref_primary_10_1108_HFF_09_2023_0568 crossref_primary_10_1007_s12273_025_1263_5 crossref_primary_10_3390_ai5030074 crossref_primary_10_3390_app132413312 crossref_primary_10_1016_j_jwpe_2025_108749 crossref_primary_10_1016_j_cma_2024_117691 crossref_primary_10_1016_j_apor_2025_104699 crossref_primary_10_1002_cnm_3905 crossref_primary_10_1063_5_0207372 crossref_primary_10_1109_TGRS_2025_3594153 crossref_primary_10_1063_5_0284425 crossref_primary_10_3390_app12105232 crossref_primary_10_1038_s41524_023_01165_7 crossref_primary_10_1038_s41598_024_65664_3 crossref_primary_10_3389_fphy_2022_1004417 crossref_primary_10_3390_e26080649 crossref_primary_10_1016_j_matdes_2023_112034 crossref_primary_10_1016_j_jcp_2023_112462 crossref_primary_10_1016_j_cma_2023_116671 crossref_primary_10_1016_j_jcp_2023_112342 crossref_primary_10_1016_j_jcp_2023_112584 crossref_primary_10_1016_j_jcp_2024_113112 crossref_primary_10_1016_j_cma_2024_117441 crossref_primary_10_1007_s11440_023_01874_9 crossref_primary_10_1016_j_advengsoft_2021_103021 crossref_primary_10_1016_j_mtphys_2021_100429 crossref_primary_10_1007_s12551_022_01040_7 crossref_primary_10_1016_j_cma_2024_117681 crossref_primary_10_1016_j_cej_2024_158684 crossref_primary_10_3390_catal11111304 crossref_primary_10_3390_fluids7030094 crossref_primary_10_1016_j_apenergy_2024_124703 crossref_primary_10_1016_j_dt_2023_02_006 crossref_primary_10_1016_j_cma_2023_116546 crossref_primary_10_1016_j_cma_2020_113636 crossref_primary_10_3390_fluids10090226 crossref_primary_10_1016_j_cma_2023_116563 crossref_primary_10_1016_j_cma_2024_117314 crossref_primary_10_1016_j_neunet_2023_03_014 crossref_primary_10_1016_j_compind_2025_104304 crossref_primary_10_1007_s10825_024_02270_6 crossref_primary_10_1088_2632_2153_ad3a32 crossref_primary_10_1016_j_compfluid_2023_105960 crossref_primary_10_1007_s00366_022_01640_7 crossref_primary_10_1016_j_jcp_2021_110928 crossref_primary_10_1080_10618562_2023_2295286 crossref_primary_10_1016_j_engappai_2022_104953 crossref_primary_10_1109_TGRS_2023_3327781 crossref_primary_10_1016_j_compgeo_2025_107604 crossref_primary_10_1016_j_fusengdes_2025_115182 crossref_primary_10_3390_math13172746 crossref_primary_10_1016_j_cad_2022_103339 crossref_primary_10_1016_j_oceaneng_2025_122054 crossref_primary_10_1016_j_cma_2021_113927 crossref_primary_10_1016_j_cma_2023_116678 crossref_primary_10_1142_S1758825125500358 crossref_primary_10_1007_s00466_022_02251_1 crossref_primary_10_1016_j_cej_2024_149826 crossref_primary_10_1016_j_mtcomm_2021_102719 crossref_primary_10_1016_j_jcp_2023_112243 crossref_primary_10_1038_s41598_024_62117_9 crossref_primary_10_1016_j_engappai_2023_106049 crossref_primary_10_1016_j_cma_2023_116214 crossref_primary_10_3390_math13101664 crossref_primary_10_1103_PhysRevApplied_19_064049 crossref_primary_10_1016_j_engappai_2025_111084 crossref_primary_10_1007_s00158_021_02851_0 crossref_primary_10_3390_biomedicines10092157 crossref_primary_10_3390_w17071075 crossref_primary_10_1016_j_ijheatmasstransfer_2020_120417 crossref_primary_10_1371_journal_pone_0285506 crossref_primary_10_1109_ACCESS_2024_3498437 crossref_primary_10_1186_s40323_022_00221_z crossref_primary_10_5194_gmd_18_3707_2025 crossref_primary_10_1016_j_ymssp_2024_111458 crossref_primary_10_1109_TGRS_2022_3222507 crossref_primary_10_1016_j_jcp_2025_113726 crossref_primary_10_3390_fluids8020043 crossref_primary_10_1080_01431161_2023_2187723 crossref_primary_10_3390_fluids7020056 crossref_primary_10_1016_j_cep_2023_109540 crossref_primary_10_1016_j_compag_2024_109606 crossref_primary_10_1016_j_compfluid_2023_105864 crossref_primary_10_1016_j_cma_2024_117410 crossref_primary_10_1063_5_0287981 crossref_primary_10_1007_s00348_021_03180_0 crossref_primary_10_1109_TAI_2022_3192362 crossref_primary_10_2118_209420_PA crossref_primary_10_1016_j_ijheatmasstransfer_2025_127040 crossref_primary_10_1016_j_applthermaleng_2025_127955 crossref_primary_10_1016_j_mineng_2025_109424 crossref_primary_10_1016_j_energy_2022_124139 crossref_primary_10_1016_j_compstruc_2025_107839 crossref_primary_10_1080_01495739_2024_2321205 crossref_primary_10_1038_s41598_021_99037_x crossref_primary_10_1016_j_neucom_2025_130764 crossref_primary_10_1038_s42003_023_04914_y crossref_primary_10_1007_s11071_023_09173_4 crossref_primary_10_1016_j_cma_2025_117826 crossref_primary_10_1016_j_jcp_2023_112003 crossref_primary_10_1016_j_jcp_2025_113950 crossref_primary_10_3390_mca28050102 crossref_primary_10_1016_j_jcp_2023_112265 crossref_primary_10_1016_j_engappai_2024_108764 crossref_primary_10_12677_app_2025_155039 crossref_primary_10_1016_j_ijheatmasstransfer_2023_124671 crossref_primary_10_1061__ASCE_EM_1943_7889_0001947 crossref_primary_10_1002_fld_5217 crossref_primary_10_1016_j_ces_2023_119285 crossref_primary_10_3389_fsufs_2024_1363744 crossref_primary_10_1016_j_trc_2024_104928 crossref_primary_10_1007_s00366_024_01967_3 crossref_primary_10_1016_j_net_2025_103703 crossref_primary_10_1186_s40323_024_00272_4 crossref_primary_10_1016_j_est_2024_113103 crossref_primary_10_1016_j_est_2024_113104 crossref_primary_10_1080_00295639_2021_2014752 crossref_primary_10_1080_17499518_2024_2315301 crossref_primary_10_1016_j_cma_2021_113959 crossref_primary_10_1016_j_est_2023_106654 crossref_primary_10_1109_TNNLS_2025_3545967 crossref_primary_10_1109_TGRS_2021_3116618 crossref_primary_10_1016_j_strusafe_2023_102399 crossref_primary_10_1137_22M154209X crossref_primary_10_1016_j_ijheatmasstransfer_2023_124546 crossref_primary_10_1002_acs_3758 crossref_primary_10_1007_s00466_023_02328_5 crossref_primary_10_1038_s41598_022_11058_2 crossref_primary_10_1016_j_cma_2021_113722 crossref_primary_10_1016_j_physd_2025_134689 crossref_primary_10_1080_10618562_2023_2285330 crossref_primary_10_1016_j_jcp_2025_113837 crossref_primary_10_1016_j_neucom_2024_128254 crossref_primary_10_1016_j_heliyon_2024_e38799 crossref_primary_10_3390_s23146371
Cites_doi	10.1016/0021-9991(90)90007-N 10.2514/6.2018-4675 10.1016/j.jcp.2018.08.029 10.3174/ajnr.A2274 10.1007/s40304-017-0117-6 10.1109/72.712178 10.1073/pnas.1718942115 10.2514/6.2019-1884 10.1016/0020-0190(87)90114-1 10.1103/PhysRevD.97.094506 10.1109/72.870037 10.1017/jfm.2018.872 10.1137/090766498 10.1016/S0893-6080(97)00097-X 10.1109/ICCV.2015.123 10.1146/annurev.fluid.32.1.347 10.1016/j.jcp.2018.04.018 10.1016/j.jcp.2018.08.036 10.1287/ijoc.1060.0182 10.1161/STROKEAHA.113.002390 10.1016/j.engstruct.2010.04.019 10.1016/j.jcp.2018.10.045 10.1146/annurev.fluid.010908.165248 10.1615/Int.J.UncertaintyQuantification.2019027864 10.1038/nature14539 10.1609/aaai.v31i1.10934 10.1126/science.aag2302 10.1029/2018WR024638 10.1093/biomet/87.1.1 10.1016/j.neucom.2018.06.056 10.1103/PhysRevFluids.2.034603 10.1016/j.jcp.2018.12.037 10.1137/S1064827501387826 10.1016/j.jcp.2016.05.003 10.1137/130932715 10.1056/NEJMra052760 10.1007/s40304-018-0127-z
ContentType	Journal Article
Copyright	2019 Elsevier B.V. Copyright Elsevier BV Apr 1, 2020
Copyright_xml	– notice: 2019 Elsevier B.V. – notice: Copyright Elsevier BV Apr 1, 2020
DBID	AAYXX CITATION 7SC 7TB 8FD FR3 JQ2 KR7 L7M L~C L~D
DOI	10.1016/j.cma.2019.112732
DatabaseName	CrossRef Computer and Information Systems Abstracts Mechanical & Transportation Engineering Abstracts Technology Research Database Engineering Research Database ProQuest Computer Science Collection Civil Engineering Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional
DatabaseTitle	CrossRef Civil Engineering Abstracts Technology Research Database Computer and Information Systems Abstracts – Academic Mechanical & Transportation Engineering Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Engineering Research Database Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional
DatabaseTitleList	Civil Engineering Abstracts
DeliveryMethod	fulltext_linktorsrc
Discipline	Applied Sciences Engineering Physics
EISSN	1879-2138
ExternalDocumentID	10_1016_j_cma_2019_112732 S004578251930622X
GroupedDBID	--K --M -~X .DC .~1 0R~ 1B1 1~. 1~5 4.4 457 4G. 5GY 5VS 7-5 71M 8P~ 9JN AABNK AACTN AAEDT AAEDW AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAXUO AAYFN ABAOU ABBOA ABFNM ABJNI ABMAC ABYKQ ACAZW ACDAQ ACGFS ACIWK ACRLP ACZNC ADBBV ADEZE ADGUI ADTZH AEBSH AECPX AEKER AENEX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AHJVU AHZHX AIALX AIEXJ AIGVJ AIKHN AITUG AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ AOUOD ARUGR AXJTR BJAXD BKOJK BLXMC CS3 DU5 EBS EFJIC EFLBG EO8 EO9 EP2 EP3 F5P FDB FIRID FNPLU FYGXN G-Q GBLVA GBOLZ IHE J1W JJJVA KOM LG9 LY7 M41 MHUIS MO0 N9A O-L O9- OAUVE OZT P-8 P-9 P2P PC. PQQKQ Q38 RNS ROL RPZ SDF SDG SDP SES SPC SPCBC SST SSV SSW SSZ T5K TN5 WH7 XPP ZMT ~02 ~G- 29F 9DU AAQXK AATTM AAXKI AAYWO AAYXX ABEFU ABWVN ABXDB ACLOT ACNNM ACRPL ACVFH ADCNI ADIYS ADJOM ADMUD ADNMO AEIPS AEUPX AFJKZ AFPUW AGQPQ AI. AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP ASPBG AVWKF AZFZN CITATION EFKBS EJD FEDTE FGOYB G-2 HLZ HVGLF HZ~ R2- SBC SET SEW VH1 VOH WUQ ZY4 ~HD 7SC 7TB 8FD AGCQF FR3 JQ2 KR7 L7M L~C L~D
ID	FETCH-LOGICAL-c434t-187db96fefd65d4c8278b627e5b9fab7c9580c0296d175c343cd8249bcbb01f73
ISICitedReferencesCount	715
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000508937500031&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	0045-7825
IngestDate	Sun Sep 07 03:48:28 EDT 2025 Sat Nov 29 07:25:42 EST 2025 Tue Nov 18 21:50:52 EST 2025 Fri Feb 23 02:49:12 EST 2024
IsDoiOpenAccess	false
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Keywords	Physics-informed machine learning Uncertainty quantification Navier-Stokes Cardiovascular flows Label-free Neural networks
Language	English
LinkModel	OpenURL
MergedId	FETCHMERGED-LOGICAL-c434t-187db96fefd65d4c8278b627e5b9fab7c9580c0296d175c343cd8249bcbb01f73
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ORCID	0000-0002-2462-362X 0000-0002-9030-1733
OpenAccessLink	https://doi.org/10.1016/j.cma.2019.112732
PQID	2371766194
PQPubID	2045269
ParticipantIDs	proquest_journals_2371766194 crossref_primary_10_1016_j_cma_2019_112732 crossref_citationtrail_10_1016_j_cma_2019_112732 elsevier_sciencedirect_doi_10_1016_j_cma_2019_112732
PublicationCentury	2000
PublicationDate	2020-04-01 2020-04-00 20200401
PublicationDateYYYYMMDD	2020-04-01
PublicationDate_xml	– month: 04 year: 2020 text: 2020-04-01 day: 01
PublicationDecade	2020
PublicationPlace	Amsterdam
PublicationPlace_xml	– name: Amsterdam
PublicationTitle	Computer methods in applied mechanics and engineering
PublicationYear	2020
Publisher	Elsevier B.V Elsevier BV
Publisher_xml	– name: Elsevier B.V – name: Elsevier BV
References	R. King, O. Hennigh, A. Mohan, M. Chertkov, From deep to physics-informed learning of turbulence: Diagnostics, arXiv preprint R. Sharma, A.B. Farimani, J. Gomes, P. Eastman, V. Pande, Weakly-supervised deep learning of heat transport via physics informed loss, arXiv preprint F. Bastien, P. Lamblin, R. Pascanu, J. Bergstra, I.J. Goodfellow, A. Bergeron, N. Bouchard, Y. Bengio, Theano: new features and speed improvements. deep learning and unsupervised feature learning, in: Neural Information Processing Systems Workshop (NIPS), 2012, pp. 1–10. Brisman, Song, Newell (b72) 2006; 355 Hamerly, Bernstein, Sludds, Soljačić, Englund (b78) 2019; 9 Raissi, Wang, Triantafyllou, Karniadakis (b39) 2019; 861 J.R. Holland, J.D. Baeder, K. Duraisamy, Towards integrated field inversion and machine learning with embedded neural networks for rans modeling, in: AIAA Scitech 2019 Forum, 2019, p. 1884. Xiu, Karniadakis (b13) 2002; 24 Zhu, Zabaras (b18) 2018; 366 C. Beck, S. Becker, P. Grohs, N. Jaafari, A. Jentzen, Solving stochastic differential equations and Kolmogorov equations by means of deep learning, arXiv preprint P. Grohs, F. Hornung, A. Jentzen, P. Von Wurstemberger, A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of black-scholes partial differential equations, arXiv preprint K. Xu, E. Darve, The neural network approach to inverse problems in differential equations, arXiv preprint Brunton, Kutz (b31) 2019 Wang, Wu, Xiao (b27) 2017; 2 D.P. Kingma, J. Ba, Adam: A method for stochastic optimization, arXiv preprint P. Márquez-Neila, M. Salzmann, P. Fua, Imposing hard constraints on deep networks: Promises and limitations, arXiv preprint Lassila, Manzoni, Quarteroni, Rozza (b4) 2014 Blumer, Ehrenfeucht, Haussler, Warmuth (b69) 1987; 24 Lee, Kang (b35) 1990; 91 Glorot, Bengio (b67) 2010 R. Stewart, S. Ermon, Label-free supervision of neural networks with physics and domain knowledge, in: Thirty-First AAAI Conference on Artificial Intelligence, 2017, pp. 2576–2582. Cebral, Mut, Weir, Putman (b73) 2011; 32 Sirignano, Spiliopoulos (b49) 2018; 375 Carleo, Troyer (b26) 2017; 355 M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard, et al. Tensorflow: A system for large-scale machine learning, in: 12th USENIX Symposium on Operating Systems Design and Implementation, OSDI 16, 2016, pp. 265–283. Weinan, Yu (b54) 2018; 6 . B. Peherstorfer, Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling, arXiv preprint Ling, Jones, Templeton (b28) 2016; 318 A. Hannun, C. Case, J. Casper, B. Catanzaro, G. Diamos, E. Elsen, R. Prenger, S. Satheesh, S. Sengupta, A. Coates, et al. Deep speech: Scaling up end-to-end speech recognition, arXiv preprint Berger, Jou (b71) 2000; 32 Tripathy, Bilionis (b19) 2018; 375 Y. Yang, P. Perdikaris, Adversarial uncertainty quantification in physics-informed neural networks, arXiv preprint Mo, Zabaras, Shi, Wu (b20) 2019; 55 Y. Zhu, N. Zabaras, P.-S. Koutsourelakis, P. Perdikaris, Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data, arXiv preprint B. Peherstorfer, Z. Drmač, S. Gugercin, Stabilizing discrete empirical interpolation via randomized and deterministic oversampling, arXiv preprint S. Karumuri, R. Tripathy, I. Bilionis, J. Panchal, Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks, arXiv preprint Lee, Baker (b25) 2018 Han, Jentzen, Weinan (b55) 2018; 115 M. Hutzenthaler, A. Jentzen, T. Kruse, T.A. Nguyen, A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations, arXiv preprint Lagaris, Likas, Fotiadis (b36) 1998; 9 A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, A. Lerer, Automatic differentiation in PyTorch, in: NIPS Autodiff Workshop, 2017. Kennedy, O’Hagan (b11) 2000; 87 C. Huang, K. Duraisamy, C. Merkle, Challenges in reduced order modeling of reacting flows, in: 2018 Joint Propulsion Conference, 2018, p. 4675. R. Kleinberg, Y. Li, Y. Yuan, An alternative view: When does SGD escape local minima? arXiv preprint Beck, Weinan, Jentzen (b52) 2017 LeCun, Bengio, Hinton (b32) 2015; 521 M. Hutzenthaler, A. Jentzen, T. Kruse, T.A. Nguyen, P. von Wurstemberger, Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, arXiv preprint M.A. Nabian, H. Meidani, A deep neural network surrogate for high-dimensional random partial differential equations, arXiv preprint Jasak, Jemcov, Kingdom (b70) 2007 Anderson, Wendt (b1) 1995 Yang, Wan, Lin, Huan (b15) 2017; 9 Benner, Cohen, Ohlberger, Willcox (b3) 2017 P. Ramachandran, B. Zoph, Q.V. Le, Searching for activation functions, arXiv preprint Ren, Chen (b9) 2010; 32 Baydin, Pearlmutter, Radul, Siskind (b59) 2018; 18 Wang, Roy, Xiao (b17) 2018; 4 Raissi, Perdikaris, Karniadakis (b33) 2019; 378 K. He, X. Zhang, S. Ren, J. Sun, Delving deep into rectifiers: Surpassing human-level performance on imagenet classification, in: Proceedings of the IEEE International Conference on Computer Vision, 2015, pp. 1026–1034. X. Meng, G.E. Karniadakis, A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse pde problems, arXiv preprint Weinan, Han, Jentzen (b51) 2017; 5 P. Ramachandran, B. Zoph, Q.V. Le, Swish: a self-gated activation function, arXiv preprint Atkinson, Zabaras (b12) 2019; 383 M. Raissi, A. Yazdani, G.E. Karniadakis, Hidden fluid mechanics: A navier-stokes informed deep learning framework for assimilating flow visualization data, arXiv preprint A.D. Jagtap, G.E. Karniadakis, Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, arXiv preprint Scarselli, Tsoi (b21) 1998; 11 Lagaris, Likas, Papageorgiou (b37) 2000; 11 Chalouhi, Hoh, Hasan (b74) 2013; 44 Najm (b14) 2009; 41 Le Maître, Knio (b16) 2010 A.M. Tartakovsky, C.O. Marrero, D. Tartakovsky, D. Barajas-Solano, Learning parameters and constitutive relationships with physics informed deep neural networks, arXiv preprint A.L. Maas, A.Y. Hannun, A.Y. Ng, Rectifier nonlinearities improve neural network acoustic models, in: Proceedings of the 30th Annual International Conference on Machine Learning, Vol. 30, 2013, p. 3. Benner, Gugercin, Willcox (b2) 2015; 57 Regis, Shoemaker (b10) 2007; 19 Shanahan, Trewartha, Detmold (b29) 2018; 97 Berg, Nyström (b50) 2018; 317 D. Zhang, L. Lu, L. Guo, G.E. Karniadakis, Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems, arXiv preprint M.A. Nabian, H. Meidani, Physics-informed regularization of deep neural networks, arXiv preprint Chaturantabut, Sorensen (b6) 2010; 32 10.1016/j.cma.2019.112732_b34 10.1016/j.cma.2019.112732_b77 10.1016/j.cma.2019.112732_b38 Atkinson (10.1016/j.cma.2019.112732_b12) 2019; 383 Yang (10.1016/j.cma.2019.112732_b15) 2017; 9 Baydin (10.1016/j.cma.2019.112732_b59) 2018; 18 Anderson (10.1016/j.cma.2019.112732_b1) 1995 Tripathy (10.1016/j.cma.2019.112732_b19) 2018; 375 10.1016/j.cma.2019.112732_b76 Raissi (10.1016/j.cma.2019.112732_b33) 2019; 378 10.1016/j.cma.2019.112732_b75 10.1016/j.cma.2019.112732_b30 Scarselli (10.1016/j.cma.2019.112732_b21) 1998; 11 Hamerly (10.1016/j.cma.2019.112732_b78) 2019; 9 Ling (10.1016/j.cma.2019.112732_b28) 2016; 318 Ren (10.1016/j.cma.2019.112732_b9) 2010; 32 Zhu (10.1016/j.cma.2019.112732_b18) 2018; 366 Regis (10.1016/j.cma.2019.112732_b10) 2007; 19 Mo (10.1016/j.cma.2019.112732_b20) 2019; 55 Lassila (10.1016/j.cma.2019.112732_b4) 2014 Wang (10.1016/j.cma.2019.112732_b27) 2017; 2 10.1016/j.cma.2019.112732_b24 10.1016/j.cma.2019.112732_b68 10.1016/j.cma.2019.112732_b23 Le Maître (10.1016/j.cma.2019.112732_b16) 2010 10.1016/j.cma.2019.112732_b22 10.1016/j.cma.2019.112732_b66 Lagaris (10.1016/j.cma.2019.112732_b37) 2000; 11 Glorot (10.1016/j.cma.2019.112732_b67) 2010 10.1016/j.cma.2019.112732_b61 10.1016/j.cma.2019.112732_b60 10.1016/j.cma.2019.112732_b65 LeCun (10.1016/j.cma.2019.112732_b32) 2015; 521 Lagaris (10.1016/j.cma.2019.112732_b36) 1998; 9 10.1016/j.cma.2019.112732_b64 Shanahan (10.1016/j.cma.2019.112732_b29) 2018; 97 10.1016/j.cma.2019.112732_b63 Lee (10.1016/j.cma.2019.112732_b35) 1990; 91 10.1016/j.cma.2019.112732_b62 Weinan (10.1016/j.cma.2019.112732_b51) 2017; 5 Benner (10.1016/j.cma.2019.112732_b2) 2015; 57 Jasak (10.1016/j.cma.2019.112732_b70) 2007 10.1016/j.cma.2019.112732_b58 10.1016/j.cma.2019.112732_b57 Kennedy (10.1016/j.cma.2019.112732_b11) 2000; 87 10.1016/j.cma.2019.112732_b56 10.1016/j.cma.2019.112732_b7 10.1016/j.cma.2019.112732_b53 Chalouhi (10.1016/j.cma.2019.112732_b74) 2013; 44 10.1016/j.cma.2019.112732_b5 Carleo (10.1016/j.cma.2019.112732_b26) 2017; 355 Berg (10.1016/j.cma.2019.112732_b50) 2018; 317 Weinan (10.1016/j.cma.2019.112732_b54) 2018; 6 Brisman (10.1016/j.cma.2019.112732_b72) 2006; 355 Beck (10.1016/j.cma.2019.112732_b52) 2017 Wang (10.1016/j.cma.2019.112732_b17) 2018; 4 Brunton (10.1016/j.cma.2019.112732_b31) 2019 10.1016/j.cma.2019.112732_b47 10.1016/j.cma.2019.112732_b46 10.1016/j.cma.2019.112732_b45 10.1016/j.cma.2019.112732_b44 Blumer (10.1016/j.cma.2019.112732_b69) 1987; 24 10.1016/j.cma.2019.112732_b48 Sirignano (10.1016/j.cma.2019.112732_b49) 2018; 375 Han (10.1016/j.cma.2019.112732_b55) 2018; 115 Lee (10.1016/j.cma.2019.112732_b25) 2018 10.1016/j.cma.2019.112732_b43 Xiu (10.1016/j.cma.2019.112732_b13) 2002; 24 10.1016/j.cma.2019.112732_b42 Benner (10.1016/j.cma.2019.112732_b3) 2017 10.1016/j.cma.2019.112732_b41 Raissi (10.1016/j.cma.2019.112732_b39) 2019; 861 10.1016/j.cma.2019.112732_b40 Cebral (10.1016/j.cma.2019.112732_b73) 2011; 32 10.1016/j.cma.2019.112732_b8 Najm (10.1016/j.cma.2019.112732_b14) 2009; 41 Berger (10.1016/j.cma.2019.112732_b71) 2000; 32 Chaturantabut (10.1016/j.cma.2019.112732_b6) 2010; 32
References_xml	– year: 2018 ident: b25 article-title: Basic Research Needs for Scientific Machine Learning: Core Technologies for Artificial Intelligence – reference: M. Hutzenthaler, A. Jentzen, T. Kruse, T.A. Nguyen, A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations, arXiv preprint – reference: A.M. Tartakovsky, C.O. Marrero, D. Tartakovsky, D. Barajas-Solano, Learning parameters and constitutive relationships with physics informed deep neural networks, arXiv preprint – volume: 5 start-page: 349 year: 2017 end-page: 380 ident: b51 article-title: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations publication-title: Commun. Math. Stat. – reference: P. Grohs, F. Hornung, A. Jentzen, P. Von Wurstemberger, A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of black-scholes partial differential equations, arXiv preprint – volume: 24 start-page: 377 year: 1987 end-page: 380 ident: b69 article-title: Occam’s razor publication-title: Inform. Process. Lett. – reference: P. Ramachandran, B. Zoph, Q.V. Le, Swish: a self-gated activation function, arXiv preprint – volume: 19 start-page: 497 year: 2007 end-page: 509 ident: b10 article-title: A stochastic radial basis function method for the global optimization of expensive functions publication-title: INFORMS J. Comput. – reference: X. Meng, G.E. Karniadakis, A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse pde problems, arXiv preprint – reference: A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, A. Lerer, Automatic differentiation in PyTorch, in: NIPS Autodiff Workshop, 2017. – reference: K. He, X. Zhang, S. Ren, J. Sun, Delving deep into rectifiers: Surpassing human-level performance on imagenet classification, in: Proceedings of the IEEE International Conference on Computer Vision, 2015, pp. 1026–1034. – reference: M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard, et al. Tensorflow: A system for large-scale machine learning, in: 12th USENIX Symposium on Operating Systems Design and Implementation, OSDI 16, 2016, pp. 265–283. – reference: C. Huang, K. Duraisamy, C. Merkle, Challenges in reduced order modeling of reacting flows, in: 2018 Joint Propulsion Conference, 2018, p. 4675. – reference: S. Karumuri, R. Tripathy, I. Bilionis, J. Panchal, Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks, arXiv preprint – volume: 355 start-page: 928 year: 2006 end-page: 939 ident: b72 article-title: Cerebral aneurysms publication-title: New Engl. J. Med. – volume: 32 start-page: 2455 year: 2010 end-page: 2465 ident: b9 article-title: Finite element model updating in structural dynamics by using the response surface method publication-title: Eng. Struct. – volume: 4 year: 2018 ident: b17 article-title: Propagation of input uncertainty in presence of model-form uncertainty: a multifidelity approach for computational fluid dynamics applications publication-title: ASCE-ASME J. Risk Uncertain. Eng. Syst. Part B: Mech. Eng. – volume: 318 start-page: 22 year: 2016 end-page: 35 ident: b28 article-title: Machine learning strategies for systems with invariance properties publication-title: J. Comput. Phys. – volume: 18 start-page: 1 year: 2018 end-page: 43 ident: b59 article-title: Automatic differentiation in machine learning: a survey publication-title: J. Mach. Learn. Res. – volume: 32 start-page: 347 year: 2000 end-page: 382 ident: b71 article-title: Flows in stenotic vessels publication-title: Annu. Rev. Fluid Mech. – volume: 9 start-page: 987 year: 1998 end-page: 1000 ident: b36 article-title: Artificial neural networks for solving ordinary and partial differential equations publication-title: IEEE Trans. Neural Netw. – volume: 44 start-page: 3613 year: 2013 end-page: 3622 ident: b74 article-title: Review of cerebral aneurysm formation, growth, and rupture publication-title: Stroke – volume: 32 start-page: 264 year: 2011 end-page: 270 ident: b73 article-title: Association of hemodynamic characteristics and cerebral aneurysm rupture publication-title: Am. J. Neuroradiol. – reference: B. Peherstorfer, Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling, arXiv preprint – reference: Y. Zhu, N. Zabaras, P.-S. Koutsourelakis, P. Perdikaris, Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data, arXiv preprint – reference: M.A. Nabian, H. Meidani, A deep neural network surrogate for high-dimensional random partial differential equations, arXiv preprint – volume: 2 year: 2017 ident: b27 article-title: Physics-informed machine learning approach for reconstructing reynolds stress modeling discrepancies based on dns data publication-title: Phys. Rev. Fluids – reference: R. King, O. Hennigh, A. Mohan, M. Chertkov, From deep to physics-informed learning of turbulence: Diagnostics, arXiv preprint – reference: A. Hannun, C. Case, J. Casper, B. Catanzaro, G. Diamos, E. Elsen, R. Prenger, S. Satheesh, S. Sengupta, A. Coates, et al. Deep speech: Scaling up end-to-end speech recognition, arXiv preprint – volume: 11 start-page: 1041 year: 2000 end-page: 1049 ident: b37 article-title: Neural-network methods for boundary value problems with irregular boundaries publication-title: IEEE Trans. Neural Netw. – reference: M. Hutzenthaler, A. Jentzen, T. Kruse, T.A. Nguyen, P. von Wurstemberger, Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, arXiv preprint – volume: 6 start-page: 1 year: 2018 end-page: 12 ident: b54 article-title: The deep ritz method: A deep learning-based numerical algorithm for solving variational problems publication-title: Commun. Math. Stat. – reference: R. Kleinberg, Y. Li, Y. Yuan, An alternative view: When does SGD escape local minima? arXiv preprint – volume: 91 start-page: 110 year: 1990 end-page: 131 ident: b35 article-title: Neural algorithm for solving differential equations publication-title: J. Comput. Phys. – volume: 861 start-page: 119 year: 2019 end-page: 137 ident: b39 article-title: Deep learning of vortex-induced vibrations publication-title: J. Fluid Mech. – reference: F. Bastien, P. Lamblin, R. Pascanu, J. Bergstra, I.J. Goodfellow, A. Bergeron, N. Bouchard, Y. Bengio, Theano: new features and speed improvements. deep learning and unsupervised feature learning, in: Neural Information Processing Systems Workshop (NIPS), 2012, pp. 1–10. – volume: 375 start-page: 1339 year: 2018 end-page: 1364 ident: b49 article-title: Dgm: A deep learning algorithm for solving partial differential equations publication-title: J. Comput. Phys. – year: 2010 ident: b16 article-title: Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics – reference: J.R. Holland, J.D. Baeder, K. Duraisamy, Towards integrated field inversion and machine learning with embedded neural networks for rans modeling, in: AIAA Scitech 2019 Forum, 2019, p. 1884. – volume: 375 start-page: 565 year: 2018 end-page: 588 ident: b19 article-title: Deep uq: Learning deep neural network surrogate models for high dimensional uncertainty quantification publication-title: J. Comput. Phys. – volume: 55 start-page: 3856 year: 2019 end-page: 3881 ident: b20 article-title: Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identification publication-title: Water Resour. Res. – reference: K. Xu, E. Darve, The neural network approach to inverse problems in differential equations, arXiv preprint – volume: 383 start-page: 166 year: 2019 end-page: 195 ident: b12 article-title: Structured bayesian gaussian process latent variable model: Applications to data-driven dimensionality reduction and high-dimensional inversion publication-title: J. Comput. Phys. – volume: 378 start-page: 686 year: 2019 end-page: 707 ident: b33 article-title: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations publication-title: J. Comput. Phys. – reference: D.P. Kingma, J. Ba, Adam: A method for stochastic optimization, arXiv preprint – volume: 355 start-page: 602 year: 2017 end-page: 606 ident: b26 article-title: Solving the quantum many-body problem with artificial neural networks publication-title: Science – year: 2019 ident: b31 article-title: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control – reference: A.D. Jagtap, G.E. Karniadakis, Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, arXiv preprint – volume: 11 start-page: 15 year: 1998 end-page: 37 ident: b21 article-title: Universal approximation using feedforward neural networks: A survey of some existing methods, and some new results publication-title: Neural Netw. – volume: 41 start-page: 35 year: 2009 end-page: 52 ident: b14 article-title: Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics publication-title: Annu. Rev. Fluid Mech. – volume: 57 start-page: 483 year: 2015 end-page: 531 ident: b2 article-title: A survey of projection-based model reduction methods for parametric dynamical systems publication-title: SIAM Rev. – year: 2017 ident: b3 article-title: Model Reduction and Approximation: Theory and Algorithms, Vol. 15 – start-page: 235 year: 2014 end-page: 273 ident: b4 article-title: Model order reduction in fluid dynamics: challenges and perspectives publication-title: Reduced Order Methods for Modeling and Computational Reduction – volume: 24 start-page: 619 year: 2002 end-page: 644 ident: b13 article-title: The Wiener–Askey polynomial chaos for stochastic differential equations publication-title: SIAM J. Sci. Comput. – volume: 9 start-page: 221 year: 2017 end-page: 243 ident: b15 article-title: A general framework for enhancing sparsity of generalized polynomial chaos expansions publication-title: Int. J. Uncertain. Quantif. – reference: M.A. Nabian, H. Meidani, Physics-informed regularization of deep neural networks, arXiv preprint – volume: 87 start-page: 1 year: 2000 end-page: 13 ident: b11 article-title: Predicting the output from a complex computer code when fast approximations are available publication-title: Biometrika – volume: 97 year: 2018 ident: b29 article-title: Machine learning action parameters in lattice quantum chromodynamics publication-title: Phys. Rev. D – reference: D. Zhang, L. Lu, L. Guo, G.E. Karniadakis, Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems, arXiv preprint – reference: M. Raissi, A. Yazdani, G.E. Karniadakis, Hidden fluid mechanics: A navier-stokes informed deep learning framework for assimilating flow visualization data, arXiv preprint – volume: 317 start-page: 28 year: 2018 end-page: 41 ident: b50 article-title: A unified deep artificial neural network approach to partial differential equations in complex geometries publication-title: Neurocomputing – year: 2010 ident: b67 article-title: Understanding the difficulty of training deep feedforward neural networks publication-title: Proceedings of the International Conference on Artificial Intelligence and Statistics – reference: R. Sharma, A.B. Farimani, J. Gomes, P. Eastman, V. Pande, Weakly-supervised deep learning of heat transport via physics informed loss, arXiv preprint – year: 1995 ident: b1 article-title: Computational Fluid Dynamics, Vol. 206 – reference: P. Ramachandran, B. Zoph, Q.V. Le, Searching for activation functions, arXiv preprint – reference: Y. Yang, P. Perdikaris, Adversarial uncertainty quantification in physics-informed neural networks, arXiv preprint – volume: 521 start-page: 436 year: 2015 ident: b32 article-title: Deep learning publication-title: Nature – volume: 32 start-page: 2737 year: 2010 end-page: 2764 ident: b6 article-title: Nonlinear model reduction via discrete empirical interpolation publication-title: SIAM J. Sci. Comput. – reference: A.L. Maas, A.Y. Hannun, A.Y. Ng, Rectifier nonlinearities improve neural network acoustic models, in: Proceedings of the 30th Annual International Conference on Machine Learning, Vol. 30, 2013, p. 3. – reference: R. Stewart, S. Ermon, Label-free supervision of neural networks with physics and domain knowledge, in: Thirty-First AAAI Conference on Artificial Intelligence, 2017, pp. 2576–2582. – start-page: 1 year: 2007 end-page: 20 ident: b70 article-title: Openfoam: A c++ library for complex physics simulations publication-title: International Workshop on Coupled Methods in Numerical Dynamics – reference: . – reference: C. Beck, S. Becker, P. Grohs, N. Jaafari, A. Jentzen, Solving stochastic differential equations and Kolmogorov equations by means of deep learning, arXiv preprint – volume: 9 year: 2019 ident: b78 article-title: Large-scale optical neural networks based on photoelectric multiplication publication-title: Phys. Rev. X – volume: 366 start-page: 415 year: 2018 end-page: 447 ident: b18 article-title: Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification publication-title: J. Comput. Phys. – start-page: 1 year: 2017 end-page: 57 ident: b52 article-title: Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations publication-title: J. Nonlinear Sci. – volume: 115 start-page: 8505 year: 2018 end-page: 8510 ident: b55 article-title: Solving high-dimensional partial differential equations using deep learning publication-title: Proc. Natl. Acad. Sci. – reference: P. Márquez-Neila, M. Salzmann, P. Fua, Imposing hard constraints on deep networks: Promises and limitations, arXiv preprint – reference: B. Peherstorfer, Z. Drmač, S. Gugercin, Stabilizing discrete empirical interpolation via randomized and deterministic oversampling, arXiv preprint – volume: 91 start-page: 110 issue: 1 year: 1990 ident: 10.1016/j.cma.2019.112732_b35 article-title: Neural algorithm for solving differential equations publication-title: J. Comput. Phys. doi: 10.1016/0021-9991(90)90007-N – ident: 10.1016/j.cma.2019.112732_b60 – ident: 10.1016/j.cma.2019.112732_b5 doi: 10.2514/6.2018-4675 – year: 1995 ident: 10.1016/j.cma.2019.112732_b1 – volume: 375 start-page: 1339 year: 2018 ident: 10.1016/j.cma.2019.112732_b49 article-title: Dgm: A deep learning algorithm for solving partial differential equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.08.029 – ident: 10.1016/j.cma.2019.112732_b58 – year: 2017 ident: 10.1016/j.cma.2019.112732_b3 – volume: 32 start-page: 264 issue: 2 year: 2011 ident: 10.1016/j.cma.2019.112732_b73 article-title: Association of hemodynamic characteristics and cerebral aneurysm rupture publication-title: Am. J. Neuroradiol. doi: 10.3174/ajnr.A2274 – volume: 5 start-page: 349 issue: 4 year: 2017 ident: 10.1016/j.cma.2019.112732_b51 article-title: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations publication-title: Commun. Math. Stat. doi: 10.1007/s40304-017-0117-6 – volume: 9 start-page: 987 issue: 5 year: 1998 ident: 10.1016/j.cma.2019.112732_b36 article-title: Artificial neural networks for solving ordinary and partial differential equations publication-title: IEEE Trans. Neural Netw. doi: 10.1109/72.712178 – ident: 10.1016/j.cma.2019.112732_b41 – volume: 115 start-page: 8505 issue: 34 year: 2018 ident: 10.1016/j.cma.2019.112732_b55 article-title: Solving high-dimensional partial differential equations using deep learning publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.1718942115 – ident: 10.1016/j.cma.2019.112732_b77 – volume: 9 issue: 2 year: 2019 ident: 10.1016/j.cma.2019.112732_b78 article-title: Large-scale optical neural networks based on photoelectric multiplication publication-title: Phys. Rev. X – ident: 10.1016/j.cma.2019.112732_b47 doi: 10.2514/6.2019-1884 – ident: 10.1016/j.cma.2019.112732_b44 – volume: 24 start-page: 377 issue: 6 year: 1987 ident: 10.1016/j.cma.2019.112732_b69 article-title: Occam’s razor publication-title: Inform. Process. Lett. doi: 10.1016/0020-0190(87)90114-1 – volume: 97 issue: 9 year: 2018 ident: 10.1016/j.cma.2019.112732_b29 article-title: Machine learning action parameters in lattice quantum chromodynamics publication-title: Phys. Rev. D doi: 10.1103/PhysRevD.97.094506 – volume: 11 start-page: 1041 issue: 5 year: 2000 ident: 10.1016/j.cma.2019.112732_b37 article-title: Neural-network methods for boundary value problems with irregular boundaries publication-title: IEEE Trans. Neural Netw. doi: 10.1109/72.870037 – volume: 861 start-page: 119 year: 2019 ident: 10.1016/j.cma.2019.112732_b39 article-title: Deep learning of vortex-induced vibrations publication-title: J. Fluid Mech. doi: 10.1017/jfm.2018.872 – start-page: 1 year: 2007 ident: 10.1016/j.cma.2019.112732_b70 article-title: Openfoam: A c++ library for complex physics simulations – ident: 10.1016/j.cma.2019.112732_b57 – volume: 32 start-page: 2737 issue: 5 year: 2010 ident: 10.1016/j.cma.2019.112732_b6 article-title: Nonlinear model reduction via discrete empirical interpolation publication-title: SIAM J. Sci. Comput. doi: 10.1137/090766498 – ident: 10.1016/j.cma.2019.112732_b40 – ident: 10.1016/j.cma.2019.112732_b61 – ident: 10.1016/j.cma.2019.112732_b53 – volume: 11 start-page: 15 issue: 1 year: 1998 ident: 10.1016/j.cma.2019.112732_b21 article-title: Universal approximation using feedforward neural networks: A survey of some existing methods, and some new results publication-title: Neural Netw. doi: 10.1016/S0893-6080(97)00097-X – ident: 10.1016/j.cma.2019.112732_b68 doi: 10.1109/ICCV.2015.123 – volume: 32 start-page: 347 issue: 1 year: 2000 ident: 10.1016/j.cma.2019.112732_b71 article-title: Flows in stenotic vessels publication-title: Annu. Rev. Fluid Mech. doi: 10.1146/annurev.fluid.32.1.347 – volume: 366 start-page: 415 year: 2018 ident: 10.1016/j.cma.2019.112732_b18 article-title: Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.04.018 – ident: 10.1016/j.cma.2019.112732_b22 – start-page: 235 year: 2014 ident: 10.1016/j.cma.2019.112732_b4 article-title: Model order reduction in fluid dynamics: challenges and perspectives – ident: 10.1016/j.cma.2019.112732_b43 – ident: 10.1016/j.cma.2019.112732_b64 – volume: 375 start-page: 565 year: 2018 ident: 10.1016/j.cma.2019.112732_b19 article-title: Deep uq: Learning deep neural network surrogate models for high dimensional uncertainty quantification publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.08.036 – volume: 19 start-page: 497 issue: 4 year: 2007 ident: 10.1016/j.cma.2019.112732_b10 article-title: A stochastic radial basis function method for the global optimization of expensive functions publication-title: INFORMS J. Comput. doi: 10.1287/ijoc.1060.0182 – ident: 10.1016/j.cma.2019.112732_b56 – ident: 10.1016/j.cma.2019.112732_b62 – year: 2010 ident: 10.1016/j.cma.2019.112732_b16 – volume: 44 start-page: 3613 issue: 12 year: 2013 ident: 10.1016/j.cma.2019.112732_b74 article-title: Review of cerebral aneurysm formation, growth, and rupture publication-title: Stroke doi: 10.1161/STROKEAHA.113.002390 – volume: 32 start-page: 2455 issue: 8 year: 2010 ident: 10.1016/j.cma.2019.112732_b9 article-title: Finite element model updating in structural dynamics by using the response surface method publication-title: Eng. Struct. doi: 10.1016/j.engstruct.2010.04.019 – volume: 378 start-page: 686 year: 2019 ident: 10.1016/j.cma.2019.112732_b33 article-title: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.10.045 – volume: 41 start-page: 35 year: 2009 ident: 10.1016/j.cma.2019.112732_b14 article-title: Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics publication-title: Annu. Rev. Fluid Mech. doi: 10.1146/annurev.fluid.010908.165248 – volume: 9 start-page: 221 issue: 3 year: 2017 ident: 10.1016/j.cma.2019.112732_b15 article-title: A general framework for enhancing sparsity of generalized polynomial chaos expansions publication-title: Int. J. Uncertain. Quantif. doi: 10.1615/Int.J.UncertaintyQuantification.2019027864 – volume: 521 start-page: 436 issue: 7553 year: 2015 ident: 10.1016/j.cma.2019.112732_b32 article-title: Deep learning publication-title: Nature doi: 10.1038/nature14539 – ident: 10.1016/j.cma.2019.112732_b75 – ident: 10.1016/j.cma.2019.112732_b48 doi: 10.1609/aaai.v31i1.10934 – ident: 10.1016/j.cma.2019.112732_b23 – ident: 10.1016/j.cma.2019.112732_b46 – volume: 355 start-page: 602 issue: 6325 year: 2017 ident: 10.1016/j.cma.2019.112732_b26 article-title: Solving the quantum many-body problem with artificial neural networks publication-title: Science doi: 10.1126/science.aag2302 – volume: 55 start-page: 3856 issue: 5 year: 2019 ident: 10.1016/j.cma.2019.112732_b20 article-title: Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identification publication-title: Water Resour. Res. doi: 10.1029/2018WR024638 – year: 2019 ident: 10.1016/j.cma.2019.112732_b31 – ident: 10.1016/j.cma.2019.112732_b42 – ident: 10.1016/j.cma.2019.112732_b65 – ident: 10.1016/j.cma.2019.112732_b30 – volume: 18 start-page: 1 year: 2018 ident: 10.1016/j.cma.2019.112732_b59 article-title: Automatic differentiation in machine learning: a survey publication-title: J. Mach. Learn. Res. – volume: 87 start-page: 1 issue: 1 year: 2000 ident: 10.1016/j.cma.2019.112732_b11 article-title: Predicting the output from a complex computer code when fast approximations are available publication-title: Biometrika doi: 10.1093/biomet/87.1.1 – start-page: 1 year: 2017 ident: 10.1016/j.cma.2019.112732_b52 article-title: Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations publication-title: J. Nonlinear Sci. – ident: 10.1016/j.cma.2019.112732_b8 – year: 2018 ident: 10.1016/j.cma.2019.112732_b25 – year: 2010 ident: 10.1016/j.cma.2019.112732_b67 article-title: Understanding the difficulty of training deep feedforward neural networks – volume: 317 start-page: 28 year: 2018 ident: 10.1016/j.cma.2019.112732_b50 article-title: A unified deep artificial neural network approach to partial differential equations in complex geometries publication-title: Neurocomputing doi: 10.1016/j.neucom.2018.06.056 – ident: 10.1016/j.cma.2019.112732_b38 – ident: 10.1016/j.cma.2019.112732_b63 – volume: 2 issue: 3 year: 2017 ident: 10.1016/j.cma.2019.112732_b27 article-title: Physics-informed machine learning approach for reconstructing reynolds stress modeling discrepancies based on dns data publication-title: Phys. Rev. Fluids doi: 10.1103/PhysRevFluids.2.034603 – volume: 383 start-page: 166 year: 2019 ident: 10.1016/j.cma.2019.112732_b12 article-title: Structured bayesian gaussian process latent variable model: Applications to data-driven dimensionality reduction and high-dimensional inversion publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.12.037 – ident: 10.1016/j.cma.2019.112732_b34 – ident: 10.1016/j.cma.2019.112732_b76 – volume: 24 start-page: 619 issue: 2 year: 2002 ident: 10.1016/j.cma.2019.112732_b13 article-title: The Wiener–Askey polynomial chaos for stochastic differential equations publication-title: SIAM J. Sci. Comput. doi: 10.1137/S1064827501387826 – volume: 318 start-page: 22 year: 2016 ident: 10.1016/j.cma.2019.112732_b28 article-title: Machine learning strategies for systems with invariance properties publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2016.05.003 – volume: 57 start-page: 483 issue: 4 year: 2015 ident: 10.1016/j.cma.2019.112732_b2 article-title: A survey of projection-based model reduction methods for parametric dynamical systems publication-title: SIAM Rev. doi: 10.1137/130932715 – volume: 355 start-page: 928 issue: 9 year: 2006 ident: 10.1016/j.cma.2019.112732_b72 article-title: Cerebral aneurysms publication-title: New Engl. J. Med. doi: 10.1056/NEJMra052760 – volume: 4 issue: 1 year: 2018 ident: 10.1016/j.cma.2019.112732_b17 article-title: Propagation of input uncertainty in presence of model-form uncertainty: a multifidelity approach for computational fluid dynamics applications publication-title: ASCE-ASME J. Risk Uncertain. Eng. Syst. Part B: Mech. Eng. – ident: 10.1016/j.cma.2019.112732_b45 – ident: 10.1016/j.cma.2019.112732_b24 – ident: 10.1016/j.cma.2019.112732_b7 – ident: 10.1016/j.cma.2019.112732_b66 – volume: 6 start-page: 1 issue: 1 year: 2018 ident: 10.1016/j.cma.2019.112732_b54 article-title: The deep ritz method: A deep learning-based numerical algorithm for solving variational problems publication-title: Commun. Math. Stat. doi: 10.1007/s40304-018-0127-z
SSID	ssj0000812
Score	2.7281373
Snippet	Numerical simulations on fluid dynamics problems primarily rely on spatially or/and temporally discretization of the governing equation using polynomials into...
SourceID	proquest crossref elsevier
SourceType	Aggregation Database Enrichment Source Index Database Publisher
StartPage	112732
SubjectTerms	Artificial neural networks Boundary conditions Cardiovascular flows Computational fluid dynamics Computer simulation Data points Deep learning Design optimization First principles Fluid dynamics Fluid flow Hemodynamics Internal flow Label-free Machine learning Mathematical models Meshing Navier-Stokes Navier-Stokes equations Neural networks Partial differential equations Physics Physics-informed machine learning Polynomials Simulation Training Uncertainty Uncertainty quantification
Title	Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data
URI	https://dx.doi.org/10.1016/j.cma.2019.112732 https://www.proquest.com/docview/2371766194
Volume	361
WOSCitedRecordID	wos000508937500031&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1879-2138 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000812 issn: 0045-7825 databaseCode: AIEXJ dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Lb9QwELaWlgMceBQqCgX5wInKVR5OHB8rVCgcqkoUaW9R_IjYapusNpu2P4QfzDi212GhFT1wsSLLsbw7X8bj8cw3CL2PakaF0JToPK-IoXQjokoUyWrFdF6pRIpoKDbBTk-L6ZSfTSY_fS7M1Zw1TXFzwxf_VdTQB8I2qbP3EPd6UuiAZxA6tCB2aP9J8N_65bI1zjFb5cZHStbzfqagba-7A7N1KXNNYP0aHZHGSjTFIqBbab3wxSSsn9aELnezS1fo68ClswWCA1cYwlWjHgJsK2fcXmqTWeyZoHUgPwyXUTb3oW9GnZ-rwYF7EpB75vy0P6r2ug-XAC6cGCBOpn0z9mAk0SjwxWllmhGwVLKxVk4tR7vTq2AVMusH_UPlW-_DxaEcaKRifhjG_k6vvbHtrYMRfZzbRQlTlGaK0k7xAG0nLOOgK7ePvhxPv4YdvogtC71bt78tH-IGN9Zxm72zsfMP5sz5M_TEnUPwkcXPczTRzQ566s4k2Gn8bgc9HhFWvkB6DS7swYUBXHgAFx7AhQdw4bbBfwEXNuDCHlzYgQsHcGEDrpfo-6fj848nxBXqIJKmdEXiginB81rXKs8UlUXCCpEnTGeC15VgkmdFJKOE5wqsVZnSVKoCzv1CChHFNUt30VbTNvoVwipLYWSsTcY4lVHKVV3FwtRkoLpOVbKHIv93ltKx2JufMC9vFeMe-rB-ZWEpXO4aTL2MSmeDWtuyBLzd9dq-l2fpdEFXJikz9Ksxp6_vs4Q36FH4SvbR1mrZ67foobxazbrlO4fFXxvYtQQ
linkProvider	Elsevier
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Surrogate+modeling+for+fluid+flows+based+on+physics-constrained+deep+learning+without+simulation+data&rft.jtitle=Computer+methods+in+applied+mechanics+and+engineering&rft.au=Sun%2C+Luning&rft.au=Gao%2C+Han&rft.au=Pan%2C+Shaowu&rft.au=Wang%2C+Jian-Xun&rft.date=2020-04-01&rft.issn=0045-7825&rft.volume=361&rft.spage=112732&rft_id=info:doi/10.1016%2Fj.cma.2019.112732&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_cma_2019_112732
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0045-7825&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0045-7825&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0045-7825&client=summon