Respecting causality for training physics-informed neural networks
While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing...
Saved in:
| Published in: | Computer methods in applied mechanics and engineering Vol. 421; no. C; p. 116813 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Netherlands
Elsevier B.V
01.03.2024
Elsevier |
| Subjects: | |
| ISSN: | 0045-7825, 1879-2138 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto–Sivashinsky equation in the chaotic regime, and the Navier–Stokes equations. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity. |
|---|---|
| AbstractList | While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto–Sivashinsky equation in the chaotic regime, and the Navier–Stokes equations. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity. |
| ArticleNumber | 116813 |
| Author | Perdikaris, Paris Sankaran, Shyam Wang, Sifan |
| Author_xml | – sequence: 1 givenname: Sifan orcidid: 0000-0002-7721-6884 surname: Wang fullname: Wang, Sifan email: sifanw@sas.upenn.edu organization: Graduate Group in Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, PA 19104, United States of America – sequence: 2 givenname: Shyam orcidid: 0000-0002-1583-2147 surname: Sankaran fullname: Sankaran, Shyam email: shyamss@seas.upenn.edu organization: Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, United States of America – sequence: 3 givenname: Paris orcidid: 0000-0002-2816-3229 surname: Perdikaris fullname: Perdikaris, Paris email: pgp@seas.upenn.edu organization: Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, United States of America |
| BackLink | https://www.osti.gov/biblio/2290430$$D View this record in Osti.gov |
| BookMark | eNp9kM1KAzEUhYNUsK0-gLvB_Yz5m06CKy1WhYIgug6ZTGJTp0lJUqVvb4Zx5aJ3c-Fwvsu5ZwYmzjsNwDWCFYJocbut1E5WGGJaIbRgiJyBKWINLzEibAKmENK6bBiuL8Asxi3MwxCegoc3HfdaJes-CyUPUfY2HQvjQ5GCtG6Q95tjtCqW1mV5p7vC6UOQfV7px4eveAnOjeyjvvrbc_CxenxfPpfr16eX5f26VJSQVNYdRQzJTknJYdfwNqfTmOTEBjKjIe0IbZWhhNUtR3WWedu0nBpWk5qblszBzXjXx2RFVDZptVHeuRxfYMwhJTCbmtGkgo8xaCOyTybr3fBPLxAUQ19iK3JfYuhLjH1lEv0j98HuZDieZO5GRue_v60OQyztlO5sGFJ13p6gfwFPMIUJ |
| CitedBy_id | crossref_primary_10_1007_s10915_024_02709_9 crossref_primary_10_1063_5_0273148 crossref_primary_10_1016_j_compbiomed_2024_109476 crossref_primary_10_1002_nme_7637 crossref_primary_10_1016_j_cma_2024_117586 crossref_primary_10_1098_rspa_2024_0270 crossref_primary_10_1088_1674_1056_adacd0 crossref_primary_10_3390_e26050396 crossref_primary_10_1016_j_matdes_2025_114284 crossref_primary_10_1016_j_engappai_2024_108156 crossref_primary_10_1016_j_cpc_2024_109422 crossref_primary_10_1016_j_neunet_2025_107983 crossref_primary_10_1016_j_apm_2025_116232 crossref_primary_10_1016_j_aei_2025_103581 crossref_primary_10_3390_math13071057 crossref_primary_10_1063_5_0252852 crossref_primary_10_1016_j_cma_2024_117691 crossref_primary_10_1063_5_0228104 crossref_primary_10_1017_jfm_2025_10448 crossref_primary_10_1016_j_buildenv_2025_113616 crossref_primary_10_1016_j_jhydrol_2024_131261 crossref_primary_10_1038_s41598_024_67597_3 crossref_primary_10_1109_ACCESS_2024_3504962 crossref_primary_10_1016_j_jcmds_2024_100107 crossref_primary_10_1016_j_addma_2025_104881 crossref_primary_10_1016_j_jcp_2025_114226 crossref_primary_10_1016_j_apenergy_2025_126040 crossref_primary_10_1088_2632_2153_ad5a60 crossref_primary_10_1016_j_commatsci_2024_113502 crossref_primary_10_1063_5_0220392 crossref_primary_10_1016_j_neunet_2024_106998 crossref_primary_10_1109_TEMC_2024_3490699 crossref_primary_10_1038_s44384_025_00021_w crossref_primary_10_1088_2632_2153_ad450f crossref_primary_10_1007_s11071_025_11158_4 crossref_primary_10_1016_j_compgeo_2025_107055 crossref_primary_10_1080_14697688_2025_2477673 crossref_primary_10_1016_j_cma_2025_117956 crossref_primary_10_1002_bit_28851 crossref_primary_10_1016_j_compind_2025_104304 crossref_primary_10_1016_j_enganabound_2025_106200 crossref_primary_10_1016_j_cma_2025_117851 crossref_primary_10_1016_j_cam_2025_116858 crossref_primary_10_1016_j_engappai_2024_109886 crossref_primary_10_1007_s00466_023_02435_3 crossref_primary_10_1007_s00521_025_11554_2 crossref_primary_10_1007_s11071_025_11577_3 crossref_primary_10_1016_j_cma_2025_118260 crossref_primary_10_1109_TGRS_2025_3581638 crossref_primary_10_1016_j_camwa_2025_09_014 crossref_primary_10_1016_j_enganabound_2025_106207 crossref_primary_10_1016_j_petsci_2025_07_008 crossref_primary_10_1137_23M1556320 crossref_primary_10_1371_journal_pone_0315762 crossref_primary_10_1016_j_cma_2025_118356 crossref_primary_10_1016_j_matcom_2024_10_039 crossref_primary_10_1088_1572_9494_adcc8e crossref_primary_10_1115_1_4068396 crossref_primary_10_1016_j_finel_2024_104305 crossref_primary_10_3390_app15168863 crossref_primary_10_1016_j_physd_2025_134878 crossref_primary_10_1126_sciadv_ads5236 crossref_primary_10_1016_j_jcp_2025_114156 crossref_primary_10_1016_j_neucom_2025_131589 crossref_primary_10_1016_j_engappai_2025_110547 crossref_primary_10_1016_j_jcp_2025_113860 crossref_primary_10_1063_5_0285282 crossref_primary_10_1016_j_geoderma_2024_117094 crossref_primary_10_1007_s44379_025_00015_1 crossref_primary_10_1016_j_ijmecsci_2025_110374 crossref_primary_10_1088_1402_4896_adeaf8 crossref_primary_10_1063_5_0211398 crossref_primary_10_1080_15502287_2024_2440420 crossref_primary_10_1063_5_0279565 crossref_primary_10_1103_ytyy_pvys crossref_primary_10_1063_5_0276518 crossref_primary_10_1016_j_cma_2025_117764 crossref_primary_10_1177_10812865251362165 crossref_primary_10_3390_fluids10070184 crossref_primary_10_1016_j_ocemod_2025_102601 crossref_primary_10_3390_math13091515 crossref_primary_10_1016_j_engappai_2025_111098 crossref_primary_10_1016_j_jcp_2025_114370 crossref_primary_10_1016_j_physa_2025_130759 crossref_primary_10_1016_j_neunet_2025_107841 crossref_primary_10_1016_j_watres_2025_123427 crossref_primary_10_1016_j_cpc_2025_109672 crossref_primary_10_1080_17486025_2025_2502029 crossref_primary_10_1016_j_brainresbull_2025_111318 crossref_primary_10_1007_s44379_025_00016_0 crossref_primary_10_1016_j_jocs_2025_102577 crossref_primary_10_1063_5_0287489 crossref_primary_10_1007_s42791_025_00101_8 crossref_primary_10_1134_S106456242460194X crossref_primary_10_1007_s00366_025_02174_4 crossref_primary_10_1016_j_enganabound_2025_106363 crossref_primary_10_1016_j_jcp_2025_113837 crossref_primary_10_1109_TII_2024_3507213 crossref_primary_10_3390_math13111882 crossref_primary_10_1016_j_jcp_2024_113656 crossref_primary_10_1007_s00033_025_02515_9 crossref_primary_10_1016_j_jcp_2025_113799 crossref_primary_10_1016_j_matcom_2024_10_043 |
| Cites_doi | 10.1016/j.cma.2019.112623 10.1016/j.neucom.2021.10.036 10.1137/20M1318043 10.1016/j.array.2021.100110 10.1038/s41586-020-2649-2 10.1109/JPROC.2021.3058954 10.1126/science.aaw4741 10.3389/fphy.2020.00042 10.1016/j.jcp.2021.110768 10.1016/j.jcp.2021.110242 10.1016/0094-5765(77)90096-0 10.1145/3241036 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 10.1364/OE.384875 10.4208/cicp.OA-2020-0164 10.1016/j.jcp.2019.109136 10.1111/mice.12685 10.1002/aic.690381003 10.1143/PTP.55.356 10.1103/PhysRevE.104.025205 10.1109/72.712178 10.1103/PhysRevFluids.4.124501 10.1016/j.jcp.2020.109914 10.1016/j.jcp.2018.10.045 10.1016/j.cma.2021.113938 10.1016/j.cma.2021.114474 10.1007/s10921-020-00705-1 10.1371/journal.pcbi.1007575 10.1073/pnas.2101784118 |
| ContentType | Journal Article |
| Copyright | 2024 Elsevier B.V. |
| Copyright_xml | – notice: 2024 Elsevier B.V. |
| DBID | AAYXX CITATION OTOTI |
| DOI | 10.1016/j.cma.2024.116813 |
| DatabaseName | CrossRef OSTI.GOV |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Applied Sciences Engineering |
| EISSN | 1879-2138 |
| ExternalDocumentID | 2290430 10_1016_j_cma_2024_116813 S0045782524000690 |
| 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 AAIKJ AAKOC AALRI AAOAW AAQFI AAXKI AAXUO AAYFN ABAOU ABBOA ABFNM ABJNI ABMAC ACDAQ ACGFS ACIWK ACRLP ACZNC ADBBV ADEZE ADGUI ADTZH AEBSH AECPX AEKER AENEX AFJKZ AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AHJVU AHZHX AIALX AIEXJ AIGVJ AIKHN AITUG AJOXV AKRWK ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ AOUOD ARUGR AXJTR BJAXD BKOJK BLXMC CS3 DU5 EBS EFJIC 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 RIG RNS ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SST SSV SSW SSZ T5K TN5 WH7 XPP ZMT ~02 ~G- 29F 9DU AAQXK AATTM AAYWO AAYXX ABEFU ABWVN ABXDB ACLOT ACNNM ACRPL ACVFH ADCNI ADIYS ADJOM ADMUD ADNMO AEIPS AEUPX AFPUW AGQPQ AI. AIGII AIIUN AKBMS AKYEP ANKPU APXCP ASPBG AVWKF AZFZN CITATION EFKBS EFLBG EJD FEDTE FGOYB G-2 HLZ HVGLF HZ~ R2- SBC SET VH1 VOH WUQ ZY4 ~HD AAIAV ABYKQ ACAZW OTOTI |
| ID | FETCH-LOGICAL-c433t-5d4181adcaa90d79b879e23024f08fe04d34bcf4385b91524f9b7b94f85359fb3 |
| ISICitedReferencesCount | 101 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=001179213600001&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 | Mon Feb 12 04:55:34 EST 2024 Tue Nov 18 22:25:03 EST 2025 Sat Nov 29 06:16:58 EST 2025 Sat Oct 26 15:42:17 EDT 2024 |
| IsDoiOpenAccess | false |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | C |
| Keywords | Deep learning Computational physics Partial differential equations Chaotic systems |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c433t-5d4181adcaa90d79b879e23024f08fe04d34bcf4385b91524f9b7b94f85359fb3 |
| Notes | USDOE Advanced Research Projects Agency - Energy (ARPA-E) |
| ORCID | 0000-0002-2816-3229 0000-0002-7721-6884 0000-0002-1583-2147 0000000228163229 0000000277216884 0000000215832147 |
| OpenAccessLink | https://www.osti.gov/biblio/2339526 |
| ParticipantIDs | osti_scitechconnect_2290430 crossref_citationtrail_10_1016_j_cma_2024_116813 crossref_primary_10_1016_j_cma_2024_116813 elsevier_sciencedirect_doi_10_1016_j_cma_2024_116813 |
| PublicationCentury | 2000 |
| PublicationDate | 2024-03-01 2024-03-00 |
| PublicationDateYYYYMMDD | 2024-03-01 |
| PublicationDate_xml | – month: 03 year: 2024 text: 2024-03-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Netherlands |
| PublicationPlace_xml | – name: Netherlands |
| PublicationTitle | Computer methods in applied mechanics and engineering |
| PublicationYear | 2024 |
| Publisher | Elsevier B.V Elsevier |
| Publisher_xml | – name: Elsevier B.V – name: Elsevier |
| References | Shukla, Jagtap, Karniadakis (b61) 2021 Raissi, Perdikaris, Karniadakis (b11) 2019; 378 Kaddour, Lynch, Liu, Kusner, Silva (b44) 2022 Wang, Wang, Perdikaris (b9) 2021; 384 Bu, Karpatne (b18) 2021 Wang, Perdikaris (b39) 2021 Li, Zheng, Kovachki, Jin, Chen, Liu, Azizzadenesheli, Anandkumar (b41) 2021 Harris, Millman, van der Walt, Gommers, Virtanen, Cournapeau, Wieser, Taylor, Berg, Smith (b64) 2020; 585 Hennigh, Narasimhan, Nabian, Subramaniam, Tangsali, Fang, Rietmann, Byeon, Choudhry (b37) 2021 Liang, Lyu, Wang, Yang (b20) 2021 Jung, Tian, Bareinboim (b45) 2020; vol. 33 Byrd, Lipton (b47) 2019 McClenny, Braga-Neto (b14) 2020 Krishnapriyan, Gholami, Zhe, Kirby, Mahoney (b24) 2021 Lagaris, Likas, Fotiadis (b28) 1998; 9 Sivashinsky (b58) 1977; 4 Pearl (b42) 2019; 62 Karniadakis, Kevrekidis, Lu, Perdikaris, Wang, Yang (b10) 2021 Kuramoto, Tsuzuki (b57) 1976; 55 Jagtap, Kawaguchi, Karniadakis (b23) 2020; 404 Jagtap, Shin, Kawaguchi, Karniadakis (b19) 2022; 468 Mattey, Ghosh (b25) 2022; 390 Griewank, Walther (b31) 2008 Dong, Ni (b50) 2021; 435 Wight, Zhao (b16) 2020 Wang, Wang, Perdikaris (b40) 2021 Strauss (b26) 2007 Hunter (b63) 2007; 9 Wang, Yu, Perdikaris (b13) 2022; 449 Wang, Wang, Perdikaris (b38) 2021 Du, Zaki (b48) 2021 Evans, American Mathematical Society (b27) 1998 Shukla, Di Leoni, Blackshire, Sparkman, Karniadakis (b6) 2020; 39 Kharazmi, Zhang, Karniadakis (b30) 2019 Bettencourt, Johnson, Duvenaud (b53) 2019 Schaarschmidt, Grewe, Vytiniotis, Paszke, Schmid, Norman, Molloy, Godwin, Rink, Nair (b60) 2021 Raissi, Yazdani, Karniadakis (b1) 2020; 367 Jagtap, Karniadakis (b21) 2020; 28 Kingma, Ba (b32) 2014 Vadyala, Betgeri, Betgeri (b49) 2022; 13 Nabian, Gladstone, Meidani (b17) 2021 Li, Beirami, Sanjabi, Smith (b46) 2020 Yazdani, Lu, Raissi, Karniadakis (b4) 2020; 16 Lorenz (b56) 1963; 20 Moseley, Markham, Nissen-Meyer (b22) 2021 Xavier Glorot, Yoshua Bengio, Understanding the difficulty of training deep feedforward neural networks, in: Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, 2010, pp. 249–256. Raissi, Babaee, Givi (b29) 2019; 4 Bradbury, Frostig, Hawkins, Johnson, Leary, Maclaurin, Necula, Paszke, VanderPlas, Wanderman-Milne, Zhang (b35) 2018 Mathews, Francisquez, Hughes, Hatch, Zhu, Rogers (b2) 2021; 104 Psichogios, Ungar (b54) 1992; 38 Kissas, Yang, Hwuang, Witschey, Detre, Perdikaris (b3) 2020; 358 Jacot, Gabriel, Hongler (b34) 2018 Sukumar, Srivastava (b51) 2021 Chen, Lu, Karniadakis, Dal Negro (b7) 2020; 28 Raissi (b59) 2018; 19 Lu, Meng, Mao, Karniadakis (b36) 2019 Maddu, Sturm, Müller, Sbalzarini (b15) 2021 Iserles (b33) 2009 Schölkopf, Locatello, Bauer, Ke, Kalchbrenner, Goyal, Bengio (b43) 2021; 109 Wang, Perdikaris (b5) 2021; 428 Wang, Teng, Perdikaris (b12) 2021; 43 Sahli Costabal, Yang, Perdikaris, Hurtado, Kuhl (b8) 2020; 8 Lu, Pestourie, Yao, Wang, Verdugo, Johnson (b52) 2021 Kochkov, Smith, Alieva, Wang, Brenner, Hoyer (b62) 2021; 118 Shukla (10.1016/j.cma.2024.116813_b6) 2020; 39 Dong (10.1016/j.cma.2024.116813_b50) 2021; 435 Kochkov (10.1016/j.cma.2024.116813_b62) 2021; 118 Mathews (10.1016/j.cma.2024.116813_b2) 2021; 104 Hunter (10.1016/j.cma.2024.116813_b63) 2007; 9 Yazdani (10.1016/j.cma.2024.116813_b4) 2020; 16 Wang (10.1016/j.cma.2024.116813_b9) 2021; 384 Du (10.1016/j.cma.2024.116813_b48) 2021 Sukumar (10.1016/j.cma.2024.116813_b51) 2021 Bu (10.1016/j.cma.2024.116813_b18) 2021 Kaddour (10.1016/j.cma.2024.116813_b44) 2022 Sahli Costabal (10.1016/j.cma.2024.116813_b8) 2020; 8 McClenny (10.1016/j.cma.2024.116813_b14) 2020 Iserles (10.1016/j.cma.2024.116813_b33) 2009 Kingma (10.1016/j.cma.2024.116813_b32) 2014 Jung (10.1016/j.cma.2024.116813_b45) 2020; vol. 33 Maddu (10.1016/j.cma.2024.116813_b15) 2021 Li (10.1016/j.cma.2024.116813_b41) 2021 Nabian (10.1016/j.cma.2024.116813_b17) 2021 Psichogios (10.1016/j.cma.2024.116813_b54) 1992; 38 Griewank (10.1016/j.cma.2024.116813_b31) 2008 Lu (10.1016/j.cma.2024.116813_b52) 2021 Shukla (10.1016/j.cma.2024.116813_b61) 2021 Krishnapriyan (10.1016/j.cma.2024.116813_b24) 2021 Jagtap (10.1016/j.cma.2024.116813_b19) 2022; 468 Chen (10.1016/j.cma.2024.116813_b7) 2020; 28 Raissi (10.1016/j.cma.2024.116813_b59) 2018; 19 Jacot (10.1016/j.cma.2024.116813_b34) 2018 Schölkopf (10.1016/j.cma.2024.116813_b43) 2021; 109 Bettencourt (10.1016/j.cma.2024.116813_b53) 2019 Kissas (10.1016/j.cma.2024.116813_b3) 2020; 358 Strauss (10.1016/j.cma.2024.116813_b26) 2007 Moseley (10.1016/j.cma.2024.116813_b22) 2021 Evans (10.1016/j.cma.2024.116813_b27) 1998 Raissi (10.1016/j.cma.2024.116813_b29) 2019; 4 Hennigh (10.1016/j.cma.2024.116813_b37) 2021 Wang (10.1016/j.cma.2024.116813_b39) 2021 10.1016/j.cma.2024.116813_b55 Wang (10.1016/j.cma.2024.116813_b5) 2021; 428 Wang (10.1016/j.cma.2024.116813_b13) 2022; 449 Kharazmi (10.1016/j.cma.2024.116813_b30) 2019 Raissi (10.1016/j.cma.2024.116813_b1) 2020; 367 Harris (10.1016/j.cma.2024.116813_b64) 2020; 585 Lu (10.1016/j.cma.2024.116813_b36) 2019 Li (10.1016/j.cma.2024.116813_b46) 2020 Kuramoto (10.1016/j.cma.2024.116813_b57) 1976; 55 Raissi (10.1016/j.cma.2024.116813_b11) 2019; 378 Byrd (10.1016/j.cma.2024.116813_b47) 2019 Bradbury (10.1016/j.cma.2024.116813_b35) 2018 Sivashinsky (10.1016/j.cma.2024.116813_b58) 1977; 4 Wang (10.1016/j.cma.2024.116813_b12) 2021; 43 Karniadakis (10.1016/j.cma.2024.116813_b10) 2021 Wang (10.1016/j.cma.2024.116813_b38) 2021 Jagtap (10.1016/j.cma.2024.116813_b23) 2020; 404 Pearl (10.1016/j.cma.2024.116813_b42) 2019; 62 Vadyala (10.1016/j.cma.2024.116813_b49) 2022; 13 Lagaris (10.1016/j.cma.2024.116813_b28) 1998; 9 Wang (10.1016/j.cma.2024.116813_b40) 2021 Jagtap (10.1016/j.cma.2024.116813_b21) 2020; 28 Mattey (10.1016/j.cma.2024.116813_b25) 2022; 390 Liang (10.1016/j.cma.2024.116813_b20) 2021 Wight (10.1016/j.cma.2024.116813_b16) 2020 Schaarschmidt (10.1016/j.cma.2024.116813_b60) 2021 Lorenz (10.1016/j.cma.2024.116813_b56) 1963; 20 |
| References_xml | – year: 2021 ident: b20 article-title: Reproducing activation function for deep learning – volume: 28 start-page: 11618 year: 2020 end-page: 11633 ident: b7 article-title: Physics-informed neural networks for inverse problems in nano-optics and metamaterials publication-title: Opt. Express – volume: 404 year: 2020 ident: b23 article-title: Adaptive activation functions accelerate convergence in deep and physics-informed neural networks publication-title: J. Comput. Phys. – year: 2021 ident: b17 article-title: Efficient training of physics-informed neural networks via importance sampling publication-title: Comput.-Aided Civ. Infrastruct. Eng. – volume: 449 year: 2022 ident: b13 article-title: When and why PINNs fail to train: A neural tangent kernel perspective publication-title: J. Comput. Phys. – year: 2021 ident: b41 article-title: Physics-informed neural operator for learning partial differential equations – year: 2020 ident: b46 article-title: Tilted empirical risk minimization – year: 2009 ident: b33 article-title: A First Course in the Numerical Analysis of Differential Equations – year: 2014 ident: b32 article-title: Adam: A method for stochastic optimization – year: 2022 ident: b44 article-title: Causal machine learning: A survey and open problems – year: 2021 ident: b60 article-title: Automap: Towards ergonomic automated parallelism for ML models – volume: 118 year: 2021 ident: b62 article-title: Machine learning–accelerated computational fluid dynamics publication-title: Proc. Natl. Acad. Sci. – year: 2020 ident: b14 article-title: Self-adaptive physics-informed neural networks using a soft attention mechanism – volume: 384 year: 2021 ident: b9 article-title: On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2021 ident: b51 article-title: Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks – volume: 28 start-page: 2002 year: 2020 end-page: 2041 ident: b21 article-title: Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations publication-title: Commun. Comput. Phys. – year: 2019 ident: b36 article-title: DeepXDE: A deep learning library for solving differential equations – volume: 19 start-page: 932 year: 2018 end-page: 955 ident: b59 article-title: Deep hidden physics models: Deep learning of nonlinear partial differential equations publication-title: J. Mach. Learn. Res. – year: 2021 ident: b38 article-title: Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets – start-page: 872 year: 2019 end-page: 881 ident: b47 article-title: What is the effect of importance weighting in deep learning? publication-title: International Conference on Machine Learning – year: 2008 ident: b31 article-title: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation – volume: 39 start-page: 1 year: 2020 end-page: 20 ident: b6 article-title: Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks publication-title: J. Nondestruct. Eval. – year: 1998 ident: b27 publication-title: Partial Differential Equations – volume: 468 start-page: 165 year: 2022 end-page: 180 ident: b19 article-title: Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions publication-title: Neurocomputing – volume: 358 year: 2020 ident: b3 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. – volume: 38 start-page: 1499 year: 1992 end-page: 1511 ident: b54 article-title: A hybrid neural network-first principles approach to process modeling publication-title: AIChE J. – volume: 13 year: 2022 ident: b49 article-title: Physics-informed neural network method for solving one-dimensional advection equation using PyTorch publication-title: Array – volume: 428 year: 2021 ident: b5 article-title: Deep learning of free boundary and Stefan problems publication-title: J. Comput. Phys. – reference: Xavier Glorot, Yoshua Bengio, Understanding the difficulty of training deep feedforward neural networks, in: Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, 2010, pp. 249–256. – volume: 4 start-page: 1177 year: 1977 end-page: 1206 ident: b58 article-title: Nonlinear analysis of hydrodynamic instability in laminar flames—I. derivation of basic equations publication-title: Acta Astronaut. – start-page: 675 year: 2021 end-page: 683 ident: b18 article-title: Quadratic residual networks: A new class of neural networks for solving forward and inverse problems in physics involving PDEs publication-title: Proceedings of the 2021 SIAM International Conference on Data Mining – volume: 9 start-page: 987 year: 1998 end-page: 1000 ident: b28 article-title: Artificial neural networks for solving ordinary and partial differential equations publication-title: IEEE Trans. Neural Netw. – volume: vol. 33 start-page: 12697 year: 2020 end-page: 12709 ident: b45 article-title: Learning causal effects via weighted empirical risk minimization publication-title: Advances in Neural Information Processing Systems – volume: 104 year: 2021 ident: b2 article-title: Uncovering turbulent plasma dynamics via deep learning from partial observations publication-title: Phys. Rev. E – year: 2021 ident: b48 article-title: Evolutional deep neural network – volume: 43 start-page: A3055 year: 2021 end-page: A3081 ident: b12 article-title: Understanding and mitigating gradient flow pathologies in physics-informed neural networks publication-title: SIAM J. Sci. Comput. – year: 2021 ident: b61 article-title: Parallel physics-informed neural networks via domain decomposition – volume: 20 start-page: 130 year: 1963 end-page: 141 ident: b56 article-title: Deterministic nonperiodic flow publication-title: J. Atmos. Sci. – volume: 62 start-page: 54 year: 2019 end-page: 60 ident: b42 article-title: The seven tools of causal inference, with reflections on machine learning publication-title: Commun. ACM – year: 2021 ident: b52 article-title: Physics-informed neural networks with hard constraints for inverse design – volume: 378 start-page: 686 year: 2019 end-page: 707 ident: b11 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. – year: 2018 ident: b35 article-title: JAX: composable transformations of python+numpy programs – start-page: 447 year: 2021 end-page: 461 ident: b37 article-title: NVIDIA SimNet™: An AI-accelerated multi-physics simulation framework publication-title: International Conference on Computational Science – volume: 390 year: 2022 ident: b25 article-title: A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 55 start-page: 356 year: 1976 end-page: 369 ident: b57 article-title: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium publication-title: Prog. Theor. Phys. – volume: 16 year: 2020 ident: b4 article-title: Systems biology informed deep learning for inferring parameters and hidden dynamics publication-title: PLoS Comput. Biol. – volume: 9 start-page: 90 year: 2007 end-page: 95 ident: b63 article-title: Matplotlib: A 2D graphics environment publication-title: IEEE Ann. Hist. Comput. – volume: 367 start-page: 1026 year: 2020 end-page: 1030 ident: b1 article-title: Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations publication-title: Science – year: 2021 ident: b39 article-title: Long-time integration of parametric evolution equations with physics-informed DeepONets – volume: 4 year: 2019 ident: b29 article-title: Deep learning of turbulent scalar mixing publication-title: Phys. Rev. Fluids – start-page: 8571 year: 2018 end-page: 8580 ident: b34 article-title: Neural tangent kernel: Convergence and generalization in neural networks publication-title: Advances in Neural Information Processing Systems – year: 2007 ident: b26 article-title: Partial Differential Equations: An Introduction – year: 2019 ident: b30 article-title: Variational physics-informed neural networks for solving partial differential equations – year: 2019 ident: b53 article-title: Taylor-mode automatic differentiation for higher-order derivatives in JAX – year: 2020 ident: b16 article-title: Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks – year: 2021 ident: b40 article-title: Improved architectures and training algorithms for deep operator networks – volume: 8 start-page: 42 year: 2020 ident: b8 article-title: Physics-informed neural networks for cardiac activation mapping publication-title: Front. Phys. – year: 2021 ident: b24 article-title: Characterizing possible failure modes in physics-informed neural networks – volume: 435 year: 2021 ident: b50 article-title: A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks publication-title: J. Comput. Phys. – year: 2021 ident: b15 article-title: Inverse Dirichlet weighting enables reliable training of physics informed neural networks publication-title: Mach. Learn.: Sci. Technol. – volume: 585 start-page: 357 year: 2020 end-page: 362 ident: b64 article-title: Array programming with NumPy publication-title: Nature – start-page: 1 year: 2021 end-page: 19 ident: b10 article-title: Physics-informed machine learning publication-title: Nat. Rev. Phys. – year: 2021 ident: b22 article-title: Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations – volume: 109 start-page: 612 year: 2021 end-page: 634 ident: b43 article-title: Toward causal representation learning publication-title: Proc. IEEE – year: 2021 ident: 10.1016/j.cma.2024.116813_b39 – volume: 358 year: 2020 ident: 10.1016/j.cma.2024.116813_b3 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 – ident: 10.1016/j.cma.2024.116813_b55 – start-page: 447 year: 2021 ident: 10.1016/j.cma.2024.116813_b37 article-title: NVIDIA SimNet™: An AI-accelerated multi-physics simulation framework – volume: 468 start-page: 165 year: 2022 ident: 10.1016/j.cma.2024.116813_b19 article-title: Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions publication-title: Neurocomputing doi: 10.1016/j.neucom.2021.10.036 – volume: 43 start-page: A3055 issue: 5 year: 2021 ident: 10.1016/j.cma.2024.116813_b12 article-title: Understanding and mitigating gradient flow pathologies in physics-informed neural networks publication-title: SIAM J. Sci. Comput. doi: 10.1137/20M1318043 – year: 2021 ident: 10.1016/j.cma.2024.116813_b61 – year: 1998 ident: 10.1016/j.cma.2024.116813_b27 – year: 2021 ident: 10.1016/j.cma.2024.116813_b24 – volume: 13 year: 2022 ident: 10.1016/j.cma.2024.116813_b49 article-title: Physics-informed neural network method for solving one-dimensional advection equation using PyTorch publication-title: Array doi: 10.1016/j.array.2021.100110 – volume: 585 start-page: 357 issue: 7825 year: 2020 ident: 10.1016/j.cma.2024.116813_b64 article-title: Array programming with NumPy publication-title: Nature doi: 10.1038/s41586-020-2649-2 – start-page: 8571 year: 2018 ident: 10.1016/j.cma.2024.116813_b34 article-title: Neural tangent kernel: Convergence and generalization in neural networks – year: 2021 ident: 10.1016/j.cma.2024.116813_b15 article-title: Inverse Dirichlet weighting enables reliable training of physics informed neural networks publication-title: Mach. Learn.: Sci. Technol. – volume: 109 start-page: 612 issue: 5 year: 2021 ident: 10.1016/j.cma.2024.116813_b43 article-title: Toward causal representation learning publication-title: Proc. IEEE doi: 10.1109/JPROC.2021.3058954 – year: 2007 ident: 10.1016/j.cma.2024.116813_b26 – year: 2021 ident: 10.1016/j.cma.2024.116813_b41 – volume: 367 start-page: 1026 issue: 6481 year: 2020 ident: 10.1016/j.cma.2024.116813_b1 article-title: Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations publication-title: Science doi: 10.1126/science.aaw4741 – year: 2021 ident: 10.1016/j.cma.2024.116813_b52 – volume: 8 start-page: 42 year: 2020 ident: 10.1016/j.cma.2024.116813_b8 article-title: Physics-informed neural networks for cardiac activation mapping publication-title: Front. Phys. doi: 10.3389/fphy.2020.00042 – volume: 449 year: 2022 ident: 10.1016/j.cma.2024.116813_b13 article-title: When and why PINNs fail to train: A neural tangent kernel perspective publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2021.110768 – year: 2019 ident: 10.1016/j.cma.2024.116813_b53 – year: 2020 ident: 10.1016/j.cma.2024.116813_b16 – volume: 9 start-page: 90 issue: 03 year: 2007 ident: 10.1016/j.cma.2024.116813_b63 article-title: Matplotlib: A 2D graphics environment publication-title: IEEE Ann. Hist. Comput. – year: 2020 ident: 10.1016/j.cma.2024.116813_b14 – volume: 435 year: 2021 ident: 10.1016/j.cma.2024.116813_b50 article-title: A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2021.110242 – volume: 4 start-page: 1177 issue: 11 year: 1977 ident: 10.1016/j.cma.2024.116813_b58 article-title: Nonlinear analysis of hydrodynamic instability in laminar flames—I. derivation of basic equations publication-title: Acta Astronaut. doi: 10.1016/0094-5765(77)90096-0 – start-page: 1 year: 2021 ident: 10.1016/j.cma.2024.116813_b10 article-title: Physics-informed machine learning publication-title: Nat. Rev. Phys. – volume: 62 start-page: 54 issue: 3 year: 2019 ident: 10.1016/j.cma.2024.116813_b42 article-title: The seven tools of causal inference, with reflections on machine learning publication-title: Commun. ACM doi: 10.1145/3241036 – volume: 20 start-page: 130 issue: 2 year: 1963 ident: 10.1016/j.cma.2024.116813_b56 article-title: Deterministic nonperiodic flow publication-title: J. Atmos. Sci. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 – year: 2021 ident: 10.1016/j.cma.2024.116813_b20 – year: 2021 ident: 10.1016/j.cma.2024.116813_b22 – volume: 28 start-page: 11618 issue: 8 year: 2020 ident: 10.1016/j.cma.2024.116813_b7 article-title: Physics-informed neural networks for inverse problems in nano-optics and metamaterials publication-title: Opt. Express doi: 10.1364/OE.384875 – year: 2008 ident: 10.1016/j.cma.2024.116813_b31 – year: 2022 ident: 10.1016/j.cma.2024.116813_b44 – year: 2021 ident: 10.1016/j.cma.2024.116813_b40 – volume: 28 start-page: 2002 issue: 5 year: 2020 ident: 10.1016/j.cma.2024.116813_b21 article-title: Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations publication-title: Commun. Comput. Phys. doi: 10.4208/cicp.OA-2020-0164 – volume: 404 year: 2020 ident: 10.1016/j.cma.2024.116813_b23 article-title: Adaptive activation functions accelerate convergence in deep and physics-informed neural networks publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.109136 – year: 2021 ident: 10.1016/j.cma.2024.116813_b17 article-title: Efficient training of physics-informed neural networks via importance sampling publication-title: Comput.-Aided Civ. Infrastruct. Eng. doi: 10.1111/mice.12685 – volume: 38 start-page: 1499 issue: 10 year: 1992 ident: 10.1016/j.cma.2024.116813_b54 article-title: A hybrid neural network-first principles approach to process modeling publication-title: AIChE J. doi: 10.1002/aic.690381003 – volume: 55 start-page: 356 issue: 2 year: 1976 ident: 10.1016/j.cma.2024.116813_b57 article-title: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium publication-title: Prog. Theor. Phys. doi: 10.1143/PTP.55.356 – start-page: 872 year: 2019 ident: 10.1016/j.cma.2024.116813_b47 article-title: What is the effect of importance weighting in deep learning? – volume: 19 start-page: 932 issue: 1 year: 2018 ident: 10.1016/j.cma.2024.116813_b59 article-title: Deep hidden physics models: Deep learning of nonlinear partial differential equations publication-title: J. Mach. Learn. Res. – year: 2021 ident: 10.1016/j.cma.2024.116813_b38 – volume: 104 issue: 2 year: 2021 ident: 10.1016/j.cma.2024.116813_b2 article-title: Uncovering turbulent plasma dynamics via deep learning from partial observations publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.104.025205 – volume: 9 start-page: 987 issue: 5 year: 1998 ident: 10.1016/j.cma.2024.116813_b28 article-title: Artificial neural networks for solving ordinary and partial differential equations publication-title: IEEE Trans. Neural Netw. doi: 10.1109/72.712178 – volume: 4 issue: 12 year: 2019 ident: 10.1016/j.cma.2024.116813_b29 article-title: Deep learning of turbulent scalar mixing publication-title: Phys. Rev. Fluids doi: 10.1103/PhysRevFluids.4.124501 – year: 2019 ident: 10.1016/j.cma.2024.116813_b30 – year: 2021 ident: 10.1016/j.cma.2024.116813_b60 – year: 2019 ident: 10.1016/j.cma.2024.116813_b36 – year: 2021 ident: 10.1016/j.cma.2024.116813_b51 – year: 2018 ident: 10.1016/j.cma.2024.116813_b35 – volume: vol. 33 start-page: 12697 year: 2020 ident: 10.1016/j.cma.2024.116813_b45 article-title: Learning causal effects via weighted empirical risk minimization – volume: 428 year: 2021 ident: 10.1016/j.cma.2024.116813_b5 article-title: Deep learning of free boundary and Stefan problems publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2020.109914 – year: 2020 ident: 10.1016/j.cma.2024.116813_b46 – volume: 378 start-page: 686 year: 2019 ident: 10.1016/j.cma.2024.116813_b11 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 – year: 2014 ident: 10.1016/j.cma.2024.116813_b32 – volume: 384 year: 2021 ident: 10.1016/j.cma.2024.116813_b9 article-title: On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2021.113938 – year: 2021 ident: 10.1016/j.cma.2024.116813_b48 – volume: 390 year: 2022 ident: 10.1016/j.cma.2024.116813_b25 article-title: A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2021.114474 – volume: 39 start-page: 1 issue: 3 year: 2020 ident: 10.1016/j.cma.2024.116813_b6 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 – start-page: 675 year: 2021 ident: 10.1016/j.cma.2024.116813_b18 article-title: Quadratic residual networks: A new class of neural networks for solving forward and inverse problems in physics involving PDEs – year: 2009 ident: 10.1016/j.cma.2024.116813_b33 – volume: 16 issue: 11 year: 2020 ident: 10.1016/j.cma.2024.116813_b4 article-title: Systems biology informed deep learning for inferring parameters and hidden dynamics publication-title: PLoS Comput. Biol. doi: 10.1371/journal.pcbi.1007575 – volume: 118 issn: 0027-8424 issue: 21 year: 2021 ident: 10.1016/j.cma.2024.116813_b62 article-title: Machine learning–accelerated computational fluid dynamics publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.2101784118 |
| SSID | ssj0000812 |
| Score | 2.718097 |
| Snippet | While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical... |
| SourceID | osti crossref elsevier |
| SourceType | Open Access Repository Enrichment Source Index Database Publisher |
| StartPage | 116813 |
| SubjectTerms | Chaotic systems Computational physics Deep learning Partial differential equations |
| Title | Respecting causality for training physics-informed neural networks |
| URI | https://dx.doi.org/10.1016/j.cma.2024.116813 https://www.osti.gov/biblio/2290430 |
| Volume | 421 |
| WOSCitedRecordID | wos001179213600001&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/eLvHCXMwtV1NbxMxELUg5QAHPgqIUkA-cEIy2g87to-lKgIOFYIi5bayvbZIS7dVNkXl3zNej7MhgggOXFbRJl6tZl5mn2dn3hDy0payBSLhmW6FY9xMBVNeBibbKO82rX0p7DBsQh4fq9lMf8RS3n4YJyC7Tl1f68v_6mo4B86OrbP_4O7VReEEfAanwxHcDse_cvwnn7onYzOtueoTzx6KCXEaBGYzepZEU-P7fz-Ib3SpJLxfJ6x56gOOmh6qZw0y13Mf24azzLMflQ3HLD0mo-dhBOFn052ZBeZdv_4w52N0BrCexaGIidou5r_kJCo-FmWlRBk-1dcDLxcMyEh6ge1TrFVSs6pM4i45GPPUL42oO_xtkE_5htPXbhCOqjiE_alKHa0b2tlRzJ7XxU2yU0mhIejtHLw_mn0YH9WqTHLyeHf5tfdQALhx-T8Rl8kFxOI1TnJyn9zFzQQ9SCB4QG74bpfcw40FxbDd75I7a6qTD8mbESF0hRAKUKAZIXQTITQhhGaEPCJf3h6dHL5jOEqDOV7XSyZaDlTOtM4YXbRSW7C9h91nxUOhgi94W3PrAq-VsBooHQ_aSqt5ADYndLD1YzLpLjr_hFAwFvyPXTGtvOeKFybISoQKgjm3OpT1HimynRqHOvPx9r81uaDwtAHTNtG0TTLtHnm1WnKZRFa2_Zhn4zfIEhP7awAf25btR0fFJVEd2cUyMliDCHm69dt9cntE-TMyWS6u_HNyy31fzvvFCwTVTw33jwc |
| 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=Respecting+causality+for+training+physics-informed+neural+networks&rft.jtitle=Computer+methods+in+applied+mechanics+and+engineering&rft.au=Wang%2C+Sifan&rft.au=Sankaran%2C+Shyam&rft.au=Perdikaris%2C+Paris&rft.date=2024-03-01&rft.pub=Elsevier&rft.issn=0045-7825&rft.eissn=1879-2138&rft.volume=421&rft.issue=C&rft_id=info:doi/10.1016%2Fj.cma.2024.116813&rft.externalDocID=2290430 |
| 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 |