PhyCRNet: Physics-informed convolutional-recurrent network for solving spatiotemporal PDEs
Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (PINNs) to solve PDEs as a basis for data-driven modeling and inve...
Saved in:
| Published in: | Computer methods in applied mechanics and engineering Vol. 389; p. 114399 |
|---|---|
| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
01.02.2022
Elsevier BV |
| Subjects: | |
| ISSN: | 0045-7825, 1879-2138 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (PINNs) to solve PDEs as a basis for data-driven modeling and inverse analysis. However, the majority of existing PINN methods, based on fully-connected NNs, pose intrinsic limitations to low-dimensional spatiotemporal parameterizations. Moreover, since the initial/boundary conditions (I/BCs) are softly imposed via penalty, the solution quality heavily relies on hyperparameter tuning. To this end, we propose the novel physics-informed convolutional-recurrent learning architectures (PhyCRNet and PhyCRNet-s) for solving PDEs without any labeled data. Specifically, an encoder–decoder convolutional long short-term memory network is proposed for low-dimensional spatial feature extraction and temporal evolution learning. The loss function is defined as the aggregated discretized PDE residuals, while the I/BCs are hard-encoded in the network to ensure forcible satisfaction (e.g., periodic boundary padding). The networks are further enhanced by autoregressive and residual connections that explicitly simulate time marching. The performance of our proposed methods has been assessed by solving three nonlinear PDEs (e.g., 2D Burgers’ equations, the λ-ω and FitzHugh Nagumo reaction–diffusion equations), and compared against the start-of-the-art baseline algorithms. The numerical results demonstrate the superiority of our proposed methodology in the context of solution accuracy, extrapolability and generalizability.
•Presented a novel physics-informed discrete learning strategy for solving PDEs without any labeled data.•Proposed an encoder–decoder convolutional-recurrent scheme for low-dimensional feature learning.•Employed hard-encoding of initial and boundary conditions.•Incorporated autoregressive and residual connections to explicitly simulate time marching.•Demonstrated excellent solution accuracy, extrapolability and generalizability of the proposed methodology. |
|---|---|
| AbstractList | Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (PINNs) to solve PDEs as a basis for data-driven modeling and inverse analysis. However, the majority of existing PINN methods, based on fully-connected NNs, pose intrinsic limitations to low-dimensional spatiotemporal parameterizations. Moreover, since the initial/boundary conditions (I/BCs) are softly imposed via penalty, the solution quality heavily relies on hyperparameter tuning. To this end, we propose the novel physics-informed convolutional-recurrent learning architectures (PhyCRNet and PhyCRNet-s) for solving PDEs without any labeled data. Specifically, an encoder–decoder convolutional long short-term memory network is proposed for low-dimensional spatial feature extraction and temporal evolution learning. The loss function is defined as the aggregated discretized PDE residuals, while the I/BCs are hard-encoded in the network to ensure forcible satisfaction (e.g., periodic boundary padding). The networks are further enhanced by autoregressive and residual connections that explicitly simulate time marching. The performance of our proposed methods has been assessed by solving three nonlinear PDEs (e.g., 2D Burgers’ equations, the λ-ω and FitzHugh Nagumo reaction–diffusion equations), and compared against the start-of-the-art baseline algorithms. The numerical results demonstrate the superiority of our proposed methodology in the context of solution accuracy, extrapolability and generalizability.
•Presented a novel physics-informed discrete learning strategy for solving PDEs without any labeled data.•Proposed an encoder–decoder convolutional-recurrent scheme for low-dimensional feature learning.•Employed hard-encoding of initial and boundary conditions.•Incorporated autoregressive and residual connections to explicitly simulate time marching.•Demonstrated excellent solution accuracy, extrapolability and generalizability of the proposed methodology. Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (PINNs) to solve PDEs as a basis for data-driven modeling and inverse analysis. However, the majority of existing PINN methods, based on fully-connected NNs, pose intrinsic limitations to low-dimensional spatiotemporal parameterizations. Moreover, since the initial/boundary conditions (I/BCs) are softly imposed via penalty, the solution quality heavily relies on hyperparameter tuning. To this end, we propose the novel physics-informed convolutional-recurrent learning architectures (PhyCRNet and PhyCRNet-s) for solving PDEs without any labeled data. Specifically, an encoder–decoder convolutional long short-term memory network is proposed for low-dimensional spatial feature extraction and temporal evolution learning. The loss function is defined as the aggregated discretized PDE residuals, while the I/BCs are hard-encoded in the network to ensure forcible satisfaction (e.g., periodic boundary padding). The networks are further enhanced by autoregressive and residual connections that explicitly simulate time marching. The performance of our proposed methods has been assessed by solving three nonlinear PDEs (e.g., 2D Burgers' equations, the |
| ArticleNumber | 114399 |
| Author | Liu, Yang Wang, Jian-Xun Sun, Hao Rao, Chengping Ren, Pu |
| Author_xml | – sequence: 1 givenname: Pu orcidid: 0000-0002-6354-385X surname: Ren fullname: Ren, Pu email: ren.pu@northeastern.edu organization: Department of Civil and Environmental Engineering, Northeastern University, Boston, MA 02115, USA – sequence: 2 givenname: Chengping surname: Rao fullname: Rao, Chengping organization: Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA – sequence: 3 givenname: Yang orcidid: 0000-0003-0127-4030 surname: Liu fullname: Liu, Yang email: yang1.liu@northeastern.edu organization: Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA – sequence: 4 givenname: Jian-Xun surname: Wang fullname: Wang, Jian-Xun organization: Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA – sequence: 5 givenname: Hao surname: Sun fullname: Sun, Hao email: haosun@ruc.edu.cn organization: Department of Civil and Environmental Engineering, Northeastern University, Boston, MA 02115, USA |
| BookMark | eNp9kE1PAyEQhompiW31B3jbxPOusLssoCdT60ditDF68UKQnVXqFirQmv57aerJQ7kMk7zPZOYZoYF1FhA6JbggmDTn80IvVFHikhSE1JUQB2hIOBN5SSo-QEOMa5ozXtIjNAphjtPjpByit9nnZvL8CPEiS79gdMiN7ZxfQJtpZ9euX0XjrOpzD3rlPdiYWYg_zn9lKZYF16-N_cjCUqVchMXSedVns-tpOEaHneoDnPzVMXq9mb5M7vKHp9v7ydVDrquGx5ySpia8FC1lotGsxhreta5T21HoFMWs0tByxaGiomsp1y3GtOoawdoOBK3G6Gw3d-nd9wpClHO38mnlIMumEgyzWpQpxXYp7V0IHjqpTdzubKNXppcEy61IOZdJpNyKlDuRiST_yKU3C-U3e5nLHQPp8LUBL4M2YNMhJnmMsnVmD_0LFt2Ogw |
| CitedBy_id | crossref_primary_10_1016_j_engappai_2023_106127 crossref_primary_10_1002_nme_7637 crossref_primary_10_1016_j_cma_2024_117102 crossref_primary_10_1038_s41467_024_53165_w crossref_primary_10_1080_10618562_2023_2260763 crossref_primary_10_1063_5_0266451 crossref_primary_10_1063_5_0285013 crossref_primary_10_3389_fphy_2022_971722 crossref_primary_10_1016_j_jmps_2025_106219 crossref_primary_10_1016_j_physd_2024_134399 crossref_primary_10_1007_s11063_022_10968_6 crossref_primary_10_1016_j_neucom_2025_130578 crossref_primary_10_1016_j_jcp_2023_112438 crossref_primary_10_1016_j_jcp_2025_114235 crossref_primary_10_1038_s41524_022_00712_y crossref_primary_10_1016_j_cma_2025_118308 crossref_primary_10_1016_j_actaastro_2024_09_027 crossref_primary_10_1140_epjs_s11734_024_01263_7 crossref_primary_10_1186_s40323_025_00283_9 crossref_primary_10_3390_app132413312 crossref_primary_10_1016_j_engfracmech_2024_110675 crossref_primary_10_1016_j_expthermflusci_2025_111562 crossref_primary_10_1080_24725579_2024_2398592 crossref_primary_10_1007_s10489_024_05834_y crossref_primary_10_1016_j_compgeo_2025_107389 crossref_primary_10_1016_j_eswa_2025_127302 crossref_primary_10_1016_j_cpc_2023_109010 crossref_primary_10_12677_aam_2025_145264 crossref_primary_10_1016_j_engappai_2023_106867 crossref_primary_10_1007_s10483_023_2992_6 crossref_primary_10_2118_225442_PA crossref_primary_10_3390_math11092016 crossref_primary_10_1038_s41598_025_92900_1 crossref_primary_10_1016_j_engappai_2025_110405 crossref_primary_10_1016_j_mechrescom_2023_104087 crossref_primary_10_1007_s10462_025_11322_7 crossref_primary_10_1016_j_jer_2024_02_011 crossref_primary_10_1016_j_cma_2025_117956 crossref_primary_10_1016_j_ijmecsci_2024_109783 crossref_primary_10_1016_j_compind_2025_104304 crossref_primary_10_1016_j_cma_2024_117035 crossref_primary_10_1016_j_cma_2024_117397 crossref_primary_10_1016_j_watres_2022_118972 crossref_primary_10_1016_j_cma_2025_118025 crossref_primary_10_1016_j_cma_2024_117036 crossref_primary_10_1016_j_cpc_2025_109599 crossref_primary_10_1016_j_commatsci_2024_113518 crossref_primary_10_1016_j_jpowsour_2023_233087 crossref_primary_10_1007_s00371_025_03967_w crossref_primary_10_1016_j_cma_2024_117274 crossref_primary_10_1016_j_cpc_2025_109757 crossref_primary_10_1038_s42005_024_01521_z crossref_primary_10_1016_j_jocs_2024_102514 crossref_primary_10_1088_1402_4896_ad7dc0 crossref_primary_10_1109_TGRS_2025_3581638 crossref_primary_10_1109_TMAG_2025_3553236 crossref_primary_10_2118_219773_PA crossref_primary_10_1016_j_jcp_2022_111510 crossref_primary_10_1016_j_ress_2024_110752 crossref_primary_10_1016_j_tws_2024_112495 crossref_primary_10_1016_j_mechmat_2023_104789 crossref_primary_10_1007_s10462_022_10329_8 crossref_primary_10_1016_j_cma_2023_115902 crossref_primary_10_1371_journal_pone_0315762 crossref_primary_10_1007_s10462_024_10784_5 crossref_primary_10_1016_j_jcp_2024_112761 crossref_primary_10_1016_j_cpc_2025_109582 crossref_primary_10_7498_aps_74_20250707 crossref_primary_10_1007_s10489_024_06195_2 crossref_primary_10_1016_j_dsp_2024_104766 crossref_primary_10_1016_j_engappai_2023_106721 crossref_primary_10_1061_JENMDT_EMENG_7062 crossref_primary_10_1016_j_ast_2023_108740 crossref_primary_10_1088_1572_9494_adcc8e crossref_primary_10_1016_j_cma_2025_117782 crossref_primary_10_1088_1572_9494_ad1a0e crossref_primary_10_1016_j_cnsns_2024_107911 crossref_primary_10_3390_e27030275 crossref_primary_10_1016_j_ijmultiphaseflow_2024_104877 crossref_primary_10_1016_j_enganabound_2025_106396 crossref_primary_10_1111_mice_70045 crossref_primary_10_1007_s00466_024_02554_5 crossref_primary_10_1016_j_cpc_2024_109342 crossref_primary_10_1016_j_cpc_2024_109462 crossref_primary_10_1029_2025JH000683 crossref_primary_10_1016_j_ymssp_2024_111297 crossref_primary_10_1016_j_jcp_2024_113284 crossref_primary_10_1002_mp_17554 crossref_primary_10_1007_s00466_023_02434_4 crossref_primary_10_1007_s12206_024_0624_9 crossref_primary_10_1016_j_egyai_2025_100553 crossref_primary_10_1016_j_physa_2024_130090 crossref_primary_10_1016_j_asoc_2024_111437 crossref_primary_10_1016_j_jcp_2024_112904 crossref_primary_10_1038_s42003_023_04914_y crossref_primary_10_1016_j_jcp_2024_113557 crossref_primary_10_1088_1572_9494_ada3ce crossref_primary_10_3390_w16152138 crossref_primary_10_1016_j_cma_2023_116512 crossref_primary_10_59717_j_xinn_energy_2025_100087 crossref_primary_10_1016_j_earscirev_2025_105276 crossref_primary_10_1016_j_engstruct_2023_117235 crossref_primary_10_1016_j_cpc_2025_109601 crossref_primary_10_1016_j_advwatres_2024_104837 crossref_primary_10_1016_j_ijheatmasstransfer_2023_124392 crossref_primary_10_1038_s42256_023_00685_7 crossref_primary_10_12677_AAM_2024_132069 crossref_primary_10_1016_j_rineng_2025_105440 crossref_primary_10_1016_j_ress_2023_109822 crossref_primary_10_1016_j_ijsolstr_2024_112692 crossref_primary_10_3390_sym16101376 crossref_primary_10_1016_j_compgeo_2024_106801 crossref_primary_10_1109_TNNLS_2025_3545967 crossref_primary_10_1016_j_matcom_2023_10_011 crossref_primary_10_1016_j_cma_2024_117478 crossref_primary_10_1109_ACCESS_2024_3452160 crossref_primary_10_1080_17486025_2025_2502029 crossref_primary_10_1016_j_ces_2025_122001 crossref_primary_10_1016_j_neucom_2025_129917 crossref_primary_10_1080_10407790_2023_2264489 crossref_primary_10_1016_j_physd_2024_134304 crossref_primary_10_1016_j_cma_2023_115944 crossref_primary_10_1007_s11424_024_3349_z crossref_primary_10_1016_j_oceaneng_2024_118779 crossref_primary_10_1016_j_cja_2024_02_011 crossref_primary_10_1016_j_neucom_2024_128254 crossref_primary_10_1016_j_rse_2024_114425 crossref_primary_10_3390_en16052343 crossref_primary_10_1016_j_jcp_2025_113954 crossref_primary_10_1063_5_0258420 crossref_primary_10_1016_j_jsv_2024_118796 |
| Cites_doi | 10.1016/j.cma.2020.113127 10.1109/72.712178 10.1016/j.jcp.2020.109951 10.1016/j.cma.2020.113603 10.1103/PhysRevFluids.4.034602 10.23915/distill.00003 10.1016/j.jcp.2018.10.045 10.1016/j.cma.2021.113706 10.1109/MSP.2017.2693418 10.1016/0893-6080(89)90020-8 10.1016/j.jcp.2019.07.048 10.1126/science.aaw4741 10.1137/19M1274067 10.1615/Int.J.UncertaintyQuantification.2020032978 10.1017/jfm.2018.872 10.1073/pnas.1922210117 10.1145/1409060.1409106 10.1007/s00466-019-01740-0 10.1073/pnas.2100697118 10.1016/j.jcp.2019.05.026 10.1016/j.taml.2020.01.039 10.1016/j.cma.2019.112623 10.1007/s10921-020-00705-1 10.1126/sciadv.1602614 10.1016/j.jcp.2018.04.018 10.1016/j.cma.2021.113722 10.1016/j.cma.2019.112732 10.1137/19M1260141 10.1038/s42256-021-00302-5 10.1109/72.870037 10.1016/j.cma.2020.113226 10.1073/pnas.1302752110 10.1016/j.cma.2018.01.045 10.1016/j.jcp.2012.08.013 10.1016/j.probengmech.2018.10.002 10.1016/j.engstruct.2020.110704 10.1016/j.cma.2019.112791 10.1016/j.jcp.2019.06.042 10.1109/CVPR.2016.207 10.1146/annurev-fluid-010719-060214 10.1007/BF01589116 10.1016/j.cma.2004.10.008 10.1016/S0166-4115(97)80111-2 10.1162/neco.1997.9.8.1735 10.1073/pnas.1718942115 10.1016/j.cma.2019.112790 10.1016/j.cma.2021.113741 10.1007/s40304-018-0127-z 10.1016/j.cma.2020.113500 10.1016/j.cma.2019.112789 10.1109/CVPR.2017.243 10.1073/pnas.1814058116 10.1016/j.jcp.2019.05.024 10.1016/j.jcp.2019.109056 10.1016/j.taml.2020.01.031 |
| ContentType | Journal Article |
| Copyright | 2021 Elsevier B.V. Copyright Elsevier BV Feb 1, 2022 |
| Copyright_xml | – notice: 2021 Elsevier B.V. – notice: Copyright Elsevier BV Feb 1, 2022 |
| DBID | AAYXX CITATION 7SC 7TB 8FD FR3 JQ2 KR7 L7M L~C L~D |
| DOI | 10.1016/j.cma.2021.114399 |
| 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_2021_114399 S0045782521006514 |
| 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 FR3 JQ2 KR7 L7M L~C L~D |
| ID | FETCH-LOGICAL-c368t-51641829d5796c740cebcc4d57f5efa5073ced8a8e359fd58cd0053f697dfe953 |
| ISICitedReferencesCount | 169 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000740320100007&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 Nov 09 08:52:13 EST 2025 Sat Nov 29 07:30:40 EST 2025 Tue Nov 18 21:52:56 EST 2025 Fri Feb 23 02:41:00 EST 2024 |
| IsDoiOpenAccess | false |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Encoder–decoder Residual connection Convolutional-recurrent learning Partial differential equations Physics-informed deep learning Hard-encoding of I/BCs |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c368t-51641829d5796c740cebcc4d57f5efa5073ced8a8e359fd58cd0053f697dfe953 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0002-6354-385X 0000-0003-0127-4030 |
| OpenAccessLink | https://doi.org/10.1016/j.cma.2021.114399 |
| PQID | 2639707492 |
| PQPubID | 2045269 |
| ParticipantIDs | proquest_journals_2639707492 crossref_citationtrail_10_1016_j_cma_2021_114399 crossref_primary_10_1016_j_cma_2021_114399 elsevier_sciencedirect_doi_10_1016_j_cma_2021_114399 |
| PublicationCentury | 2000 |
| PublicationDate | 2022-02-01 2022-02-00 20220201 |
| PublicationDateYYYYMMDD | 2022-02-01 |
| PublicationDate_xml | – month: 02 year: 2022 text: 2022-02-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Amsterdam |
| PublicationPlace_xml | – name: Amsterdam |
| PublicationTitle | Computer methods in applied mechanics and engineering |
| PublicationYear | 2022 |
| Publisher | Elsevier B.V Elsevier BV |
| Publisher_xml | – name: Elsevier B.V – name: Elsevier BV |
| References | Bar-Sinai, Hoyer, Hickey, Brenner (b10) 2019; 116 Jin, Cai, Li, Karniadakis (b26) 2021; 426 Cai, Li, Zheng, Kong, Dao, Karniadakis, Suresh (b33) 2021; 118 Haghighat, Raissi, Moure, Gomez, Juanes (b29) 2021; 379 Raissi, Yazdani, Karniadakis (b18) 2020; 367 Paszke, Gross, Chintala, Chanan, Yang, DeVito, Lin, Desmaison, Antiga, Lerer (b75) 2017 Lagaris, Likas, Papageorgiou (b5) 2000; 11 Defferrard, Bresson, Vandergheynst (b84) 2016 Long, Lu, Ma, Dong (b52) 2018 Rao, Sun, Liu (b74) 2021 Qin, Wu, Xiu (b19) 2019; 395 Bresson, Laurent (b81) 2017 Gorodetsky, Jakeman, Geraci, Eldred (b14) 2020; 10 Jordan (b65) 1997 Zhang, Liu, Sun (b38) 2020; 369 Zhang, Lu, Guo, Karniadakis (b30) 2019; 397 T.Q. Chen, Y. Rubanova, J. Bettencourt, D. Duvenaud, Neural ordinary differential equations, in: NeurIPS, 2018, pp. 6572–6583. Kissas, Yang, Hwuang, Witschey, Detre, Perdikaris (b32) 2020; 358 Yin, Zheng, Humphrey, Karniadakis (b35) 2021; 375 Li, Kovachki, Azizzadenesheli, Liu, Bhattacharya, Stuart, Anandkumar (b54) 2020 Raissi, Wang, Triantafyllou, Karniadakis (b17) 2019; 861 Pfaff, Fortunato, Sanchez-Gonzalez, Battaglia (b59) 2020 Raissi (b16) 2018; 19 Hochreiter, Schmidhuber (b62) 1997; 9 G. Huang, Z. Liu, L. Van Der Maaten, K.Q. Weinberger, Densely connected convolutional networks, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp. 4700–4708. Lu, Meng, Mao, Karniadakis (b43) 2021; 63 Huan, Marzouk (b6) 2013; 232 Trask, Patel, Gross, Atzberger (b56) 2019 Samaniego, Anitescu, Goswami, Nguyen-Thanh, Guo, Hamdia, Zhuang, Rabczuk (b20) 2020; 362 Sun, Wang (b21) 2020; 10 Zhu, Zabaras (b12) 2018; 366 Brunton, Noack, Koumoutsakos (b13) 2020; 52 Bronstein, Bruna, LeCun, Szlam, Vandergheynst (b82) 2017; 34 Raissi, Perdikaris, Karniadakis (b9) 2019; 378 Li, Kovachki, Azizzadenesheli, Liu, Bhattacharya, Stuart, Anandkumar (b57) 2020 Rao, Sun, Liu (b73) 2021 Zhu, Zabaras, Koutsourelakis, Perdikaris (b50) 2019; 394 Sun, Gao, Pan, Wang (b23) 2020; 361 Lagaris, Likas, Fotiadis (b4) 1998; 9 Zhang, Guo, Karniadakis (b31) 2020; 42 Odena, Dumoulin, Olah (b69) 2016; 1 Rao, Sun, Liu (b22) 2020; 10 Hornik, Stinchcombe, White (b3) 1989; 2 Gao, Sun, Wang (b41) 2020 Graves (b63) 2013 Schaeffer, Caflisch, Hauck, Osher (b61) 2013; 110 Rao, Sun, Liu (b24) 2021; 147 Bhattacharya, Hosseini, Kovachki, Stuart (b42) 2020 Geneva, Zabaras (b47) 2020; 403 Rudy, Brunton, Proctor, Kutz (b79) 2017; 3 Hughes (b1) 2012 Sorteberg, Garasto, Pouplin, Cantwell, Bharath (b51) 2018 Sutskever, Vinyals, Le (b64) 2014 Liu, Nocedal (b77) 1989; 45 Ranade, Hill, Pathak (b48) 2021; 378 Winovich, Ramani, Lin (b46) 2019; 394 Bhatnagar, Afshar, Pan, Duraisamy, Kaushik (b49) 2019; 64 Han, Jentzen, Weinan (b7) 2018; 115 Salimans, Kingma (b72) 2016 Shan, Li, Jia, Tang (b68) 2008; 27 Kipf, Welling (b80) 2016 Zhang, Shields (b11) 2019; 55 Wang, Huan, Garikipati (b15) 2021; 377 Yang, Zafar, Wang, Xiao (b25) 2019; 4 Yu, Fan, Yang, Xu, Wang, Wang, Huang (b71) 2018 Patel, Trask, Wood, Cyr (b45) 2021; 373 Lu, Dao, Kumar, Ramamurty, Karniadakis, Suresh (b36) 2020; 117 S. Seo, C. Meng, Y. Liu, Physics-aware difference graph networks for sparsely-observed dynamics, in: International Conference on Learning Representations, 2019. Ioffe, Szegedy (b70) 2015 Hughes, Cottrell, Bazilevs (b2) 2005; 194 Battaglia, Hamrick, Bapst, Sanchez-Gonzalez, Zambaldi, Malinowski, Tacchetti, Raposo, Santoro, Faulkner (b83) 2018 Goswami, Yin, Yu, Karniadakis (b86) 2021 Geneva, Zabaras (b53) 2020 Kingma, Ba (b76) 2014 Mao, Jagtap, Karniadakis (b27) 2020; 360 Lu, Jin, Pang, Zhang, Karniadakis (b44) 2021; 3 Zhang, Liu, Sun (b37) 2020; 215 He, Chen (b39) 2020; 363 Shi, Chen, Wang, Yeung, Wong, Woo (b60) 2015; 28 Weinan, Yu (b85) 2018; 6 Wessels, Weißenfels, Wriggers (b28) 2020; 368 Belbute-Peres, Economon, Kolter (b58) 2020 Shukla, Di Leoni, Blackshire, Sparkman, Karniadakis (b34) 2020; 39 Chen, Liu, Sun (b40) 2020 W. Shi, J. Caballero, F. Huszár, J. Totz, A.P. Aitken, R. Bishop, D. Rueckert, Z. Wang, Real-time single image and video super-resolution using an efficient sub-pixel convolutional neural network, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 1874–1883. Zhang, Shields (b8) 2018; 334 Gao (10.1016/j.cma.2021.114399_b41) 2020 Hochreiter (10.1016/j.cma.2021.114399_b62) 1997; 9 He (10.1016/j.cma.2021.114399_b39) 2020; 363 10.1016/j.cma.2021.114399_b55 Li (10.1016/j.cma.2021.114399_b57) 2020 Graves (10.1016/j.cma.2021.114399_b63) 2013 Paszke (10.1016/j.cma.2021.114399_b75) 2017 Bar-Sinai (10.1016/j.cma.2021.114399_b10) 2019; 116 Rao (10.1016/j.cma.2021.114399_b73) 2021 Bhatnagar (10.1016/j.cma.2021.114399_b49) 2019; 64 Battaglia (10.1016/j.cma.2021.114399_b83) 2018 Chen (10.1016/j.cma.2021.114399_b40) 2020 Qin (10.1016/j.cma.2021.114399_b19) 2019; 395 Bhattacharya (10.1016/j.cma.2021.114399_b42) 2020 Li (10.1016/j.cma.2021.114399_b54) 2020 Patel (10.1016/j.cma.2021.114399_b45) 2021; 373 Long (10.1016/j.cma.2021.114399_b52) 2018 Odena (10.1016/j.cma.2021.114399_b69) 2016; 1 Yang (10.1016/j.cma.2021.114399_b25) 2019; 4 Gorodetsky (10.1016/j.cma.2021.114399_b14) 2020; 10 Shukla (10.1016/j.cma.2021.114399_b34) 2020; 39 Defferrard (10.1016/j.cma.2021.114399_b84) 2016 Ioffe (10.1016/j.cma.2021.114399_b70) 2015 Goswami (10.1016/j.cma.2021.114399_b86) 2021 Yu (10.1016/j.cma.2021.114399_b71) 2018 Lagaris (10.1016/j.cma.2021.114399_b4) 1998; 9 Raissi (10.1016/j.cma.2021.114399_b18) 2020; 367 Weinan (10.1016/j.cma.2021.114399_b85) 2018; 6 Lu (10.1016/j.cma.2021.114399_b44) 2021; 3 Rao (10.1016/j.cma.2021.114399_b74) 2021 Raissi (10.1016/j.cma.2021.114399_b9) 2019; 378 Raissi (10.1016/j.cma.2021.114399_b17) 2019; 861 Wessels (10.1016/j.cma.2021.114399_b28) 2020; 368 Zhang (10.1016/j.cma.2021.114399_b11) 2019; 55 Yin (10.1016/j.cma.2021.114399_b35) 2021; 375 Liu (10.1016/j.cma.2021.114399_b77) 1989; 45 Kingma (10.1016/j.cma.2021.114399_b76) 2014 Bresson (10.1016/j.cma.2021.114399_b81) 2017 Rao (10.1016/j.cma.2021.114399_b22) 2020; 10 Sorteberg (10.1016/j.cma.2021.114399_b51) 2018 Shan (10.1016/j.cma.2021.114399_b68) 2008; 27 Trask (10.1016/j.cma.2021.114399_b56) 2019 Mao (10.1016/j.cma.2021.114399_b27) 2020; 360 Sun (10.1016/j.cma.2021.114399_b23) 2020; 361 Winovich (10.1016/j.cma.2021.114399_b46) 2019; 394 Zhang (10.1016/j.cma.2021.114399_b38) 2020; 369 Zhu (10.1016/j.cma.2021.114399_b50) 2019; 394 10.1016/j.cma.2021.114399_b78 Hughes (10.1016/j.cma.2021.114399_b1) 2012 Wang (10.1016/j.cma.2021.114399_b15) 2021; 377 Kissas (10.1016/j.cma.2021.114399_b32) 2020; 358 Geneva (10.1016/j.cma.2021.114399_b47) 2020; 403 Jin (10.1016/j.cma.2021.114399_b26) 2021; 426 Hornik (10.1016/j.cma.2021.114399_b3) 1989; 2 Geneva (10.1016/j.cma.2021.114399_b53) 2020 Haghighat (10.1016/j.cma.2021.114399_b29) 2021; 379 Brunton (10.1016/j.cma.2021.114399_b13) 2020; 52 Hughes (10.1016/j.cma.2021.114399_b2) 2005; 194 Zhang (10.1016/j.cma.2021.114399_b8) 2018; 334 Salimans (10.1016/j.cma.2021.114399_b72) 2016 Pfaff (10.1016/j.cma.2021.114399_b59) 2020 Ranade (10.1016/j.cma.2021.114399_b48) 2021; 378 Sutskever (10.1016/j.cma.2021.114399_b64) 2014 Zhu (10.1016/j.cma.2021.114399_b12) 2018; 366 Rudy (10.1016/j.cma.2021.114399_b79) 2017; 3 Lu (10.1016/j.cma.2021.114399_b36) 2020; 117 Zhang (10.1016/j.cma.2021.114399_b31) 2020; 42 10.1016/j.cma.2021.114399_b66 10.1016/j.cma.2021.114399_b67 Shi (10.1016/j.cma.2021.114399_b60) 2015; 28 Kipf (10.1016/j.cma.2021.114399_b80) 2016 Samaniego (10.1016/j.cma.2021.114399_b20) 2020; 362 Raissi (10.1016/j.cma.2021.114399_b16) 2018; 19 Huan (10.1016/j.cma.2021.114399_b6) 2013; 232 Sun (10.1016/j.cma.2021.114399_b21) 2020; 10 Cai (10.1016/j.cma.2021.114399_b33) 2021; 118 Lu (10.1016/j.cma.2021.114399_b43) 2021; 63 Lagaris (10.1016/j.cma.2021.114399_b5) 2000; 11 Zhang (10.1016/j.cma.2021.114399_b37) 2020; 215 Bronstein (10.1016/j.cma.2021.114399_b82) 2017; 34 Han (10.1016/j.cma.2021.114399_b7) 2018; 115 Belbute-Peres (10.1016/j.cma.2021.114399_b58) 2020 Zhang (10.1016/j.cma.2021.114399_b30) 2019; 397 Jordan (10.1016/j.cma.2021.114399_b65) 1997 Schaeffer (10.1016/j.cma.2021.114399_b61) 2013; 110 Rao (10.1016/j.cma.2021.114399_b24) 2021; 147 |
| References_xml | – volume: 377 year: 2021 ident: b15 article-title: Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2019 ident: b56 article-title: GMLS-Nets: A framework for learning from unstructured data – year: 2016 ident: b72 article-title: Weight normalization: A simple reparameterization to accelerate training of deep neural networks – year: 2014 ident: b64 article-title: Sequence to sequence learning with neural networks – volume: 55 start-page: 54 year: 2019 end-page: 66 ident: b11 article-title: Efficient Monte Carlo resampling for probability measure changes from Bayesian updating publication-title: Probab. Eng. Mech. – year: 2018 ident: b71 article-title: Wide activation for efficient and accurate image super-resolution – year: 2021 ident: b74 article-title: Hard encoding of physics for learning spatiotemporal dynamics – volume: 426 year: 2021 ident: b26 article-title: NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations publication-title: J. Comput. Phys. – volume: 116 start-page: 15344 year: 2019 end-page: 15349 ident: b10 article-title: Learning data-driven discretizations for partial differential equations publication-title: Proc. Natl. Acad. Sci. – volume: 362 year: 2020 ident: b20 article-title: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 403 year: 2020 ident: b47 article-title: Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks publication-title: J. Comput. Phys. – year: 2014 ident: b76 article-title: Adam: A method for stochastic optimization – volume: 4 year: 2019 ident: b25 article-title: Predictive large-eddy-simulation wall modeling via physics-informed neural networks publication-title: Phys. Rev. Fluids – year: 2020 ident: b42 article-title: Model reduction and neural networks for parametric pdes – volume: 19 start-page: 932 year: 2018 end-page: 955 ident: b16 article-title: Deep hidden physics models: Deep learning of nonlinear partial differential equations publication-title: J. Mach. Learn. Res. – start-page: 471 year: 1997 end-page: 495 ident: b65 article-title: Serial order: A parallel distributed processing approach publication-title: Advances in Psychology, Vol. 121 – volume: 861 start-page: 119 year: 2019 end-page: 137 ident: b17 article-title: Deep learning of vortex-induced vibrations publication-title: J. Fluid Mech. – volume: 118 year: 2021 ident: b33 article-title: Artificial intelligence velocimetry and microaneurysm-on-a-chip for three-dimensional analysis of blood flow in physiology and disease publication-title: Proc. Natl. Acad. Sci. – volume: 363 year: 2020 ident: b39 article-title: A physics-constrained data-driven approach based on locally convex reconstruction for noisy database publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 2 start-page: 359 year: 1989 end-page: 366 ident: b3 article-title: Multilayer feedforward networks are universal approximators publication-title: Neural Netw. – reference: S. Seo, C. Meng, Y. Liu, Physics-aware difference graph networks for sparsely-observed dynamics, in: International Conference on Learning Representations, 2019. – volume: 3 year: 2017 ident: b79 article-title: Data-driven discovery of partial differential equations publication-title: Sci. Adv. – volume: 194 start-page: 4135 year: 2005 end-page: 4195 ident: b2 article-title: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2020 ident: b54 article-title: Fourier neural operator for parametric partial differential equations – reference: T.Q. Chen, Y. Rubanova, J. Bettencourt, D. Duvenaud, Neural ordinary differential equations, in: NeurIPS, 2018, pp. 6572–6583. – volume: 42 start-page: A639 year: 2020 end-page: A665 ident: b31 article-title: Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks publication-title: SIAM J. Sci. Comput. – year: 2020 ident: b57 article-title: Neural operator: Graph kernel network for partial differential equations – volume: 369 year: 2020 ident: b38 article-title: Physics-informed multi-LSTM networks for metamodeling of nonlinear structures publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2020 ident: b41 article-title: PhyGeoNet: PHysics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain publication-title: J. Comput. Phys. – start-page: 3208 year: 2018 end-page: 3216 ident: b52 article-title: PDE-net: Learning PDEs from data publication-title: International Conference on Machine Learning – year: 2018 ident: b83 article-title: Relational inductive biases, deep learning, and graph networks – volume: 39 start-page: 1 year: 2020 end-page: 20 ident: b34 article-title: Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks publication-title: J. Nondestruct. Eval. – year: 2020 ident: b40 article-title: Physics-informed learning of governing equations from scarce data – year: 2016 ident: b84 article-title: Convolutional neural networks on graphs with fast localized spectral filtering – volume: 373 year: 2021 ident: b45 article-title: A physics-informed operator regression framework for extracting data-driven continuum models publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 52 start-page: 477 year: 2020 end-page: 508 ident: b13 article-title: Machine learning for fluid mechanics publication-title: Annu. Rev. Fluid Mech. – volume: 367 start-page: 1026 year: 2020 end-page: 1030 ident: b18 article-title: Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations publication-title: Science – volume: 63 start-page: 208 year: 2021 end-page: 228 ident: b43 article-title: DeepXDE: A Deep learning library for solving differential equations publication-title: SIAM Rev. – reference: G. Huang, Z. Liu, L. Van Der Maaten, K.Q. Weinberger, Densely connected convolutional networks, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp. 4700–4708. – volume: 334 start-page: 483 year: 2018 end-page: 506 ident: b8 article-title: The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 360 year: 2020 ident: b27 article-title: Physics-informed neural networks for high-speed flows publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 64 start-page: 525 year: 2019 end-page: 545 ident: b49 article-title: Prediction of aerodynamic flow fields using convolutional neural networks publication-title: Comput. Mech. – year: 2017 ident: b75 article-title: Automatic differentiation in pytorch – volume: 11 start-page: 1041 year: 2000 end-page: 1049 ident: b5 article-title: Neural-network methods for boundary value problems with irregular boundaries publication-title: IEEE Trans. Neural Netw. – volume: 397 year: 2019 ident: b30 article-title: Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems publication-title: J. Comput. Phys. – volume: 358 year: 2020 ident: b32 article-title: Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2020 ident: b59 article-title: Learning mesh-based simulation with graph networks – volume: 115 start-page: 8505 year: 2018 end-page: 8510 ident: b7 article-title: Solving high-dimensional partial differential equations using deep learning publication-title: Proc. Natl. Acad. Sci. – volume: 378 start-page: 686 year: 2019 end-page: 707 ident: b9 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. – volume: 368 year: 2020 ident: b28 article-title: The neural particle method–an updated Lagrangian physics informed neural network for computational fluid dynamics publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2021 ident: b73 article-title: Embedding physics to learn spatiotemporal dynamics from sparse data – volume: 10 start-page: 161 year: 2020 end-page: 169 ident: b21 article-title: Physics-constrained bayesian neural network for fluid flow reconstruction with sparse and noisy data publication-title: Theor. Appl. Mech. Lett. – volume: 232 start-page: 288 year: 2013 end-page: 317 ident: b6 article-title: Simulation-based optimal Bayesian experimental design for nonlinear systems publication-title: J. Comput. Phys. – volume: 10 start-page: 207 year: 2020 end-page: 212 ident: b22 article-title: Physics-informed deep learning for incompressible laminar flows publication-title: Theor. Appl. Mech. Lett. – volume: 215 year: 2020 ident: b37 article-title: Physics-guided convolutional neural network (PhyCNN) for data-driven seismic response modeling publication-title: Eng. Struct. – start-page: 448 year: 2015 end-page: 456 ident: b70 article-title: Batch normalization: Accelerating deep network training by reducing internal covariate shift publication-title: International Conference on Machine Learning – volume: 9 start-page: 987 year: 1998 end-page: 1000 ident: b4 article-title: Artificial neural networks for solving ordinary and partial differential equations publication-title: IEEE Trans. Neural Netw. – start-page: 2402 year: 2020 end-page: 2411 ident: b58 article-title: Combining differentiable PDE solvers and graph neural networks for fluid flow prediction publication-title: International Conference on Machine Learning – volume: 361 year: 2020 ident: b23 article-title: Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 394 start-page: 56 year: 2019 end-page: 81 ident: b50 article-title: Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data publication-title: J. Comput. Phys. – year: 2020 ident: b53 article-title: Transformers for modeling physical systems – volume: 27 start-page: 1 year: 2008 end-page: 7 ident: b68 article-title: Fast image/video upsampling publication-title: ACM Trans. Graph. – volume: 6 start-page: 1 year: 2018 end-page: 12 ident: b85 article-title: The deep ritz method: a deep learning-based numerical algorithm for solving variational problems publication-title: Commun. Math. Stat. – volume: 45 start-page: 503 year: 1989 end-page: 528 ident: b77 article-title: On the limited memory BFGS method for large scale optimization publication-title: Math. Program. – volume: 147 year: 2021 ident: b24 article-title: Physics-informed deep learning for computational elastodynamics without labeled data publication-title: J. Eng. Mech. – volume: 117 start-page: 7052 year: 2020 end-page: 7062 ident: b36 article-title: Extraction of mechanical properties of materials through deep learning from instrumented indentation publication-title: Proc. Natl. Acad. Sci. – year: 2016 ident: b80 article-title: Semi-supervised classification with graph convolutional networks – year: 2018 ident: b51 article-title: Approximating the solution to wave propagation using deep neural networks – year: 2021 ident: b86 article-title: A physics-informed variational DeepONet for predicting the crack path in brittle materials – volume: 34 start-page: 18 year: 2017 end-page: 42 ident: b82 article-title: Geometric deep learning: going beyond euclidean data publication-title: IEEE Signal Process. Mag. – volume: 1 year: 2016 ident: b69 article-title: Deconvolution and checkerboard artifacts publication-title: Distill – volume: 379 year: 2021 ident: b29 article-title: A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2012 ident: b1 article-title: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis – volume: 395 start-page: 620 year: 2019 end-page: 635 ident: b19 article-title: Data driven governing equations approximation using deep neural networks publication-title: J. Comput. Phys. – volume: 375 year: 2021 ident: b35 article-title: Non-invasive inference of thrombus material properties with physics-informed neural networks publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2017 ident: b81 article-title: Residual gated graph convnets – volume: 394 start-page: 263 year: 2019 end-page: 279 ident: b46 article-title: ConvPDE-UQ: COnvolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains publication-title: J. Comput. Phys. – volume: 3 start-page: 218 year: 2021 end-page: 229 ident: b44 article-title: Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators publication-title: Nat. Mach. Intell. – year: 2013 ident: b63 article-title: Generating sequences with recurrent neural networks – volume: 366 start-page: 415 year: 2018 end-page: 447 ident: b12 article-title: BayesIan deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification publication-title: J. Comput. Phys. – volume: 110 start-page: 6634 year: 2013 end-page: 6639 ident: b61 article-title: Sparse dynamics for partial differential equations publication-title: Proc. Natl. Acad. Sci. – reference: W. Shi, J. Caballero, F. Huszár, J. Totz, A.P. Aitken, R. Bishop, D. Rueckert, Z. Wang, Real-time single image and video super-resolution using an efficient sub-pixel convolutional neural network, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 1874–1883. – volume: 10 year: 2020 ident: b14 article-title: MFNets: MUlti-fidelity data-driven networks for bayesian learning and prediction publication-title: Int. J. Uncertain. Quantif. – volume: 378 year: 2021 ident: b48 article-title: DiscretizationNet: A Machine-learning based solver for Navier–Stokes equations using finite volume discretization publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 9 start-page: 1735 year: 1997 end-page: 1780 ident: b62 article-title: Long short-term memory publication-title: Neural Comput. – volume: 28 start-page: 802 year: 2015 end-page: 810 ident: b60 article-title: Convolutional LSTM network: A machine learning approach for precipitation nowcasting publication-title: Adv. Neural Inf. Process. Syst. – year: 2018 ident: 10.1016/j.cma.2021.114399_b83 – volume: 368 year: 2020 ident: 10.1016/j.cma.2021.114399_b28 article-title: The neural particle method–an updated Lagrangian physics informed neural network for computational fluid dynamics publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2020.113127 – year: 2020 ident: 10.1016/j.cma.2021.114399_b59 – volume: 9 start-page: 987 issue: 5 year: 1998 ident: 10.1016/j.cma.2021.114399_b4 article-title: Artificial neural networks for solving ordinary and partial differential equations publication-title: IEEE Trans. Neural Netw. doi: 10.1109/72.712178 – volume: 426 year: 2021 ident: 10.1016/j.cma.2021.114399_b26 article-title: NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2020.109951 – volume: 375 year: 2021 ident: 10.1016/j.cma.2021.114399_b35 article-title: Non-invasive inference of thrombus material properties with physics-informed neural networks publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2020.113603 – year: 2021 ident: 10.1016/j.cma.2021.114399_b86 – volume: 4 issue: 3 year: 2019 ident: 10.1016/j.cma.2021.114399_b25 article-title: Predictive large-eddy-simulation wall modeling via physics-informed neural networks publication-title: Phys. Rev. Fluids doi: 10.1103/PhysRevFluids.4.034602 – year: 2020 ident: 10.1016/j.cma.2021.114399_b42 – volume: 1 issue: 10 year: 2016 ident: 10.1016/j.cma.2021.114399_b69 article-title: Deconvolution and checkerboard artifacts publication-title: Distill doi: 10.23915/distill.00003 – start-page: 3208 year: 2018 ident: 10.1016/j.cma.2021.114399_b52 article-title: PDE-net: Learning PDEs from data – year: 2014 ident: 10.1016/j.cma.2021.114399_b64 – volume: 378 start-page: 686 year: 2019 ident: 10.1016/j.cma.2021.114399_b9 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: 377 year: 2021 ident: 10.1016/j.cma.2021.114399_b15 article-title: Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2021.113706 – ident: 10.1016/j.cma.2021.114399_b66 – volume: 34 start-page: 18 issue: 4 year: 2017 ident: 10.1016/j.cma.2021.114399_b82 article-title: Geometric deep learning: going beyond euclidean data publication-title: IEEE Signal Process. Mag. doi: 10.1109/MSP.2017.2693418 – volume: 2 start-page: 359 issue: 5 year: 1989 ident: 10.1016/j.cma.2021.114399_b3 article-title: Multilayer feedforward networks are universal approximators publication-title: Neural Netw. doi: 10.1016/0893-6080(89)90020-8 – volume: 397 year: 2019 ident: 10.1016/j.cma.2021.114399_b30 article-title: Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.07.048 – year: 2017 ident: 10.1016/j.cma.2021.114399_b81 – volume: 367 start-page: 1026 issue: 6481 year: 2020 ident: 10.1016/j.cma.2021.114399_b18 article-title: Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations publication-title: Science doi: 10.1126/science.aaw4741 – volume: 63 start-page: 208 issue: 1 year: 2021 ident: 10.1016/j.cma.2021.114399_b43 article-title: DeepXDE: A Deep learning library for solving differential equations publication-title: SIAM Rev. doi: 10.1137/19M1274067 – volume: 10 issue: 6 year: 2020 ident: 10.1016/j.cma.2021.114399_b14 article-title: MFNets: MUlti-fidelity data-driven networks for bayesian learning and prediction publication-title: Int. J. Uncertain. Quantif. doi: 10.1615/Int.J.UncertaintyQuantification.2020032978 – year: 2013 ident: 10.1016/j.cma.2021.114399_b63 – year: 2019 ident: 10.1016/j.cma.2021.114399_b56 – volume: 861 start-page: 119 year: 2019 ident: 10.1016/j.cma.2021.114399_b17 article-title: Deep learning of vortex-induced vibrations publication-title: J. Fluid Mech. doi: 10.1017/jfm.2018.872 – volume: 117 start-page: 7052 issue: 13 year: 2020 ident: 10.1016/j.cma.2021.114399_b36 article-title: Extraction of mechanical properties of materials through deep learning from instrumented indentation publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.1922210117 – volume: 27 start-page: 1 issue: 5 year: 2008 ident: 10.1016/j.cma.2021.114399_b68 article-title: Fast image/video upsampling publication-title: ACM Trans. Graph. doi: 10.1145/1409060.1409106 – volume: 64 start-page: 525 issue: 2 year: 2019 ident: 10.1016/j.cma.2021.114399_b49 article-title: Prediction of aerodynamic flow fields using convolutional neural networks publication-title: Comput. Mech. doi: 10.1007/s00466-019-01740-0 – volume: 118 issue: 13 year: 2021 ident: 10.1016/j.cma.2021.114399_b33 article-title: Artificial intelligence velocimetry and microaneurysm-on-a-chip for three-dimensional analysis of blood flow in physiology and disease publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.2100697118 – volume: 394 start-page: 263 year: 2019 ident: 10.1016/j.cma.2021.114399_b46 article-title: ConvPDE-UQ: COnvolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.05.026 – year: 2020 ident: 10.1016/j.cma.2021.114399_b53 – volume: 10 start-page: 207 issue: 3 year: 2020 ident: 10.1016/j.cma.2021.114399_b22 article-title: Physics-informed deep learning for incompressible laminar flows publication-title: Theor. Appl. Mech. Lett. doi: 10.1016/j.taml.2020.01.039 – volume: 358 year: 2020 ident: 10.1016/j.cma.2021.114399_b32 article-title: Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2019.112623 – year: 2016 ident: 10.1016/j.cma.2021.114399_b84 – volume: 39 start-page: 1 issue: 3 year: 2020 ident: 10.1016/j.cma.2021.114399_b34 article-title: Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks publication-title: J. Nondestruct. Eval. doi: 10.1007/s10921-020-00705-1 – volume: 3 issue: 4 year: 2017 ident: 10.1016/j.cma.2021.114399_b79 article-title: Data-driven discovery of partial differential equations publication-title: Sci. Adv. doi: 10.1126/sciadv.1602614 – volume: 366 start-page: 415 year: 2018 ident: 10.1016/j.cma.2021.114399_b12 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 – volume: 378 year: 2021 ident: 10.1016/j.cma.2021.114399_b48 article-title: DiscretizationNet: A Machine-learning based solver for Navier–Stokes equations using finite volume discretization publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2021.113722 – volume: 361 year: 2020 ident: 10.1016/j.cma.2021.114399_b23 article-title: Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2019.112732 – volume: 42 start-page: A639 issue: 2 year: 2020 ident: 10.1016/j.cma.2021.114399_b31 article-title: Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks publication-title: SIAM J. Sci. Comput. doi: 10.1137/19M1260141 – year: 2020 ident: 10.1016/j.cma.2021.114399_b40 – volume: 3 start-page: 218 issue: 3 year: 2021 ident: 10.1016/j.cma.2021.114399_b44 article-title: Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators publication-title: Nat. Mach. Intell. doi: 10.1038/s42256-021-00302-5 – volume: 11 start-page: 1041 issue: 5 year: 2000 ident: 10.1016/j.cma.2021.114399_b5 article-title: Neural-network methods for boundary value problems with irregular boundaries publication-title: IEEE Trans. Neural Netw. doi: 10.1109/72.870037 – volume: 369 year: 2020 ident: 10.1016/j.cma.2021.114399_b38 article-title: Physics-informed multi-LSTM networks for metamodeling of nonlinear structures publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2020.113226 – volume: 110 start-page: 6634 issue: 17 year: 2013 ident: 10.1016/j.cma.2021.114399_b61 article-title: Sparse dynamics for partial differential equations publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.1302752110 – volume: 334 start-page: 483 year: 2018 ident: 10.1016/j.cma.2021.114399_b8 article-title: The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2018.01.045 – volume: 232 start-page: 288 issue: 1 year: 2013 ident: 10.1016/j.cma.2021.114399_b6 article-title: Simulation-based optimal Bayesian experimental design for nonlinear systems publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2012.08.013 – start-page: 2402 year: 2020 ident: 10.1016/j.cma.2021.114399_b58 article-title: Combining differentiable PDE solvers and graph neural networks for fluid flow prediction – volume: 55 start-page: 54 year: 2019 ident: 10.1016/j.cma.2021.114399_b11 article-title: Efficient Monte Carlo resampling for probability measure changes from Bayesian updating publication-title: Probab. Eng. Mech. doi: 10.1016/j.probengmech.2018.10.002 – volume: 19 start-page: 932 issue: 1 year: 2018 ident: 10.1016/j.cma.2021.114399_b16 article-title: Deep hidden physics models: Deep learning of nonlinear partial differential equations publication-title: J. Mach. Learn. Res. – volume: 215 year: 2020 ident: 10.1016/j.cma.2021.114399_b37 article-title: Physics-guided convolutional neural network (PhyCNN) for data-driven seismic response modeling publication-title: Eng. Struct. doi: 10.1016/j.engstruct.2020.110704 – volume: 363 year: 2020 ident: 10.1016/j.cma.2021.114399_b39 article-title: A physics-constrained data-driven approach based on locally convex reconstruction for noisy database publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2019.112791 – volume: 147 issue: 8 year: 2021 ident: 10.1016/j.cma.2021.114399_b24 article-title: Physics-informed deep learning for computational elastodynamics without labeled data publication-title: J. Eng. Mech. – ident: 10.1016/j.cma.2021.114399_b55 – year: 2020 ident: 10.1016/j.cma.2021.114399_b54 – year: 2014 ident: 10.1016/j.cma.2021.114399_b76 – year: 2021 ident: 10.1016/j.cma.2021.114399_b74 – volume: 395 start-page: 620 year: 2019 ident: 10.1016/j.cma.2021.114399_b19 article-title: Data driven governing equations approximation using deep neural networks publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.06.042 – year: 2020 ident: 10.1016/j.cma.2021.114399_b57 – ident: 10.1016/j.cma.2021.114399_b67 doi: 10.1109/CVPR.2016.207 – volume: 52 start-page: 477 year: 2020 ident: 10.1016/j.cma.2021.114399_b13 article-title: Machine learning for fluid mechanics publication-title: Annu. Rev. Fluid Mech. doi: 10.1146/annurev-fluid-010719-060214 – volume: 28 start-page: 802 year: 2015 ident: 10.1016/j.cma.2021.114399_b60 article-title: Convolutional LSTM network: A machine learning approach for precipitation nowcasting publication-title: Adv. Neural Inf. Process. Syst. – year: 2021 ident: 10.1016/j.cma.2021.114399_b73 – volume: 45 start-page: 503 issue: 1 year: 1989 ident: 10.1016/j.cma.2021.114399_b77 article-title: On the limited memory BFGS method for large scale optimization publication-title: Math. Program. doi: 10.1007/BF01589116 – volume: 194 start-page: 4135 issue: 39–41 year: 2005 ident: 10.1016/j.cma.2021.114399_b2 article-title: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2004.10.008 – start-page: 471 year: 1997 ident: 10.1016/j.cma.2021.114399_b65 article-title: Serial order: A parallel distributed processing approach doi: 10.1016/S0166-4115(97)80111-2 – year: 2017 ident: 10.1016/j.cma.2021.114399_b75 – volume: 9 start-page: 1735 issue: 8 year: 1997 ident: 10.1016/j.cma.2021.114399_b62 article-title: Long short-term memory publication-title: Neural Comput. doi: 10.1162/neco.1997.9.8.1735 – volume: 115 start-page: 8505 issue: 34 year: 2018 ident: 10.1016/j.cma.2021.114399_b7 article-title: Solving high-dimensional partial differential equations using deep learning publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.1718942115 – volume: 362 year: 2020 ident: 10.1016/j.cma.2021.114399_b20 article-title: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2019.112790 – volume: 379 year: 2021 ident: 10.1016/j.cma.2021.114399_b29 article-title: A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2021.113741 – volume: 6 start-page: 1 issue: 1 year: 2018 ident: 10.1016/j.cma.2021.114399_b85 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 – year: 2012 ident: 10.1016/j.cma.2021.114399_b1 – start-page: 448 year: 2015 ident: 10.1016/j.cma.2021.114399_b70 article-title: Batch normalization: Accelerating deep network training by reducing internal covariate shift – volume: 373 year: 2021 ident: 10.1016/j.cma.2021.114399_b45 article-title: A physics-informed operator regression framework for extracting data-driven continuum models publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2020.113500 – volume: 360 year: 2020 ident: 10.1016/j.cma.2021.114399_b27 article-title: Physics-informed neural networks for high-speed flows publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2019.112789 – year: 2020 ident: 10.1016/j.cma.2021.114399_b41 article-title: PhyGeoNet: PHysics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain publication-title: J. Comput. Phys. – year: 2016 ident: 10.1016/j.cma.2021.114399_b80 – ident: 10.1016/j.cma.2021.114399_b78 doi: 10.1109/CVPR.2017.243 – volume: 116 start-page: 15344 issue: 31 year: 2019 ident: 10.1016/j.cma.2021.114399_b10 article-title: Learning data-driven discretizations for partial differential equations publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.1814058116 – year: 2016 ident: 10.1016/j.cma.2021.114399_b72 – year: 2018 ident: 10.1016/j.cma.2021.114399_b71 – year: 2018 ident: 10.1016/j.cma.2021.114399_b51 – volume: 394 start-page: 56 year: 2019 ident: 10.1016/j.cma.2021.114399_b50 article-title: Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.05.024 – volume: 403 year: 2020 ident: 10.1016/j.cma.2021.114399_b47 article-title: Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.109056 – volume: 10 start-page: 161 issue: 3 year: 2020 ident: 10.1016/j.cma.2021.114399_b21 article-title: Physics-constrained bayesian neural network for fluid flow reconstruction with sparse and noisy data publication-title: Theor. Appl. Mech. Lett. doi: 10.1016/j.taml.2020.01.031 |
| SSID | ssj0000812 |
| Score | 2.7094104 |
| Snippet | Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep... |
| SourceID | proquest crossref elsevier |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 114399 |
| SubjectTerms | Boundary conditions Burgers equation Coders Convolutional-recurrent learning Deep learning Encoder–decoder Feature extraction Hard-encoding of I/BCs Machine learning Mathematical models Modelling Neural networks Partial differential equations Physics Physics-informed deep learning Residual connection Time marching |
| Title | PhyCRNet: Physics-informed convolutional-recurrent network for solving spatiotemporal PDEs |
| URI | https://dx.doi.org/10.1016/j.cma.2021.114399 https://www.proquest.com/docview/2639707492 |
| Volume | 389 |
| WOSCitedRecordID | wos000740320100007&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: ScienceDirect database 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/eLvHCXMwtV1bb9MwFLbKBhI8cClDDAbyA-IBZJQmcePwNo1OMFWlQh1EvFiJL9qmEkrTTuMf8LM5vqRJK1bBAy9R6zoX9ftyfHx8_B2EXggDcpJqEkgVk1jGISmkjAgtdBHTJBBaMVtsIhmNWJal407nV70X5nKalCW7ukpn_xVqaAOwzdbZf4B7dVFogM8AOhwBdjj-FfDjs59Hn0bKhvxsfqeoiJNHVdImmfu751MyN8F2K89Uumxwm3QIj-eiDDbZ2mtXTV-P3w2qtitb14PwRahtXm3ufdpvymworgWgVaN5uFrfcdZuvGzWmVzM9gw6z1odh-dLO0rkTdMXH-I-AWaTbFm2Axcw5w3WkkCaHTWf2wY6pgScFto20JErMuRNbO-Pht_FIC7eCCsmFfaMBnLkSi-ti2yPPvLj0-GQTwbZ5OXsBzH1x8w6vS_GcgPthglNwT7uHn4YZCfNqM56TnneP2C9Qm5zBTfuep2PszHaWxdmch_d9XMPfOg48wB1VNlF9_w8BHsrX3XRnZZIZRfd8iR6iL7W1HqLN4mFryEW9sTC0A17YuF1YmFDrD10ejyYHL0nvjYHEVGfLQiFaTZMTVNp9jKLJA6EKoSI4aumSucwy4iEkixnKqKplpQJaey97qeJ1Cql0SO0U34v1WOE-1r0KDMStWEE19G5LDSck4iCBQXL430U1P8mF1643tRPmfI6Q_GCAwDcAMAdAPvo1eqUmVNt2dY5riHi3u107iQHcm077aCGk_vXv-KhWScHrzwNn2z_-Sm63bwTB2hnMV-qZ-imuFycV_Pnnny_ATk2rqk |
| 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=PhyCRNet%3A+Physics-informed+convolutional-recurrent+network+for+solving+spatiotemporal+PDEs&rft.jtitle=Computer+methods+in+applied+mechanics+and+engineering&rft.au=Ren%2C+Pu&rft.au=Rao%2C+Chengping&rft.au=Liu%2C+Yang&rft.au=Wang%2C+Jian-Xun&rft.date=2022-02-01&rft.pub=Elsevier+BV&rft.issn=0045-7825&rft.volume=389&rft.spage=1&rft_id=info:doi/10.1016%2Fj.cma.2021.114399&rft.externalDBID=NO_FULL_TEXT |
| 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 |