Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data
Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by partial differential equations (PDEs). With the aim of discoveri...
Saved in:
| Published in: | Computer methods in applied mechanics and engineering Vol. 377; p. 113706 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
15.04.2021
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 | Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by partial differential equations (PDEs). With the aim of discovering hidden physics, we have previously presented a variational approach to identifying such systems of PDEs in the face of noisy data at varying fidelities (Computer Methods in Applied Mechanics and Engineering, 356:44–74, 2019). Here, we extend our variational system identification methods to address the challenges presented by image data on microstructures in materials physics. PDEs are formally posed as initial and boundary value problems over combinations of time intervals and spatial domains whose evolution is either fixed or can be tracked. However, the vast majority of microscopy techniques for evolving microstructure in a given material system deliver micrographs of pattern evolution over domains that bear no relation with each other at different time instants. The temporal resolution can rarely capture the fastest time scales that dominate the early dynamics, and noise abounds. Furthermore, data for evolution of the same phenomenon in a material system may well be obtained from different physical specimens. Against this backdrop of spatially unrelated, sparse and multi-source data, we exploit the variational framework to make judicious choices of weighting functions and identify PDE operators from the dynamics. A consistency condition arises for parsimonious inference of a minimal set of the spatial operators at steady state. It is complemented by a confirmation test that provides a sharp condition for acceptance of the inferred operators. The entire framework is demonstrated on synthetic data that reflect the characteristics of the experimental material microscopy images.
•Our study focuses on inferring the equations that govern pattern formation.•We use a variational approach to system identification.•Our approach works with spatially unrelated, sparse and multi-source data.•We introduce a Confirmation Test of Consistency with Operator Suppression.•Synthetic data are used reflecting the characteristics of experimental images. |
|---|---|
| AbstractList | Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by partial differential equations (PDEs). With the aim of discovering hidden physics, we have previously presented a variational approach to identifying such systems of PDEs in the face of noisy data at varying fidelities (Computer Methods in Applied Mechanics and Engineering, 356:44–74, 2019). Here, we extend our variational system identification methods to address the challenges presented by image data on microstructures in materials physics. PDEs are formally posed as initial and boundary value problems over combinations of time intervals and spatial domains whose evolution is either fixed or can be tracked. However, the vast majority of microscopy techniques for evolving microstructure in a given material system deliver micrographs of pattern evolution over domains that bear no relation with each other at different time instants. The temporal resolution can rarely capture the fastest time scales that dominate the early dynamics, and noise abounds. Furthermore, data for evolution of the same phenomenon in a material system may well be obtained from different physical specimens. Against this backdrop of spatially unrelated, sparse and multi-source data, we exploit the variational framework to make judicious choices of weighting functions and identify PDE operators from the dynamics. A consistency condition arises for parsimonious inference of a minimal set of the spatial operators at steady state. It is complemented by a confirmation test that provides a sharp condition for acceptance of the inferred operators. The entire framework is demonstrated on synthetic data that reflect the characteristics of the experimental material microscopy images.
•Our study focuses on inferring the equations that govern pattern formation.•We use a variational approach to system identification.•Our approach works with spatially unrelated, sparse and multi-source data.•We introduce a Confirmation Test of Consistency with Operator Suppression.•Synthetic data are used reflecting the characteristics of experimental images. Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by partial differential equations (PDEs). With the aim of discovering hidden physics, we have previously presented a variational approach to identifying such systems of PDEs in the face of noisy data at varying fidelities (Computer Methods in Applied Mechanics and Engineering, 356:44–74, 2019). Here, we extend our variational system identification methods to address the challenges presented by image data on microstructures in materials physics. PDEs are formally posed as initial and boundary value problems over combinations of time intervals and spatial domains whose evolution is either fixed or can be tracked. However, the vast majority of microscopy techniques for evolving microstructure in a given material system deliver micrographs of pattern evolution over domains that bear no relation with each other at different time instants. The temporal resolution can rarely capture the fastest time scales that dominate the early dynamics, and noise abounds. Furthermore, data for evolution of the same phenomenon in a material system may well be obtained from different physical specimens. Against this backdrop of spatially unrelated, sparse and multi-source data, we exploit the variational framework to make judicious choices of weighting functions and identify PDE operators from the dynamics. A consistency condition arises for parsimonious inference of a minimal set of the spatial operators at steady state. It is complemented by a confirmation test that provides a sharp condition for acceptance of the inferred operators. The entire framework is demonstrated on synthetic data that reflect the characteristics of the experimental material microscopy images. |
| ArticleNumber | 113706 |
| Author | Wang, Z. Huan, X. Garikipati, K. |
| Author_xml | – sequence: 1 givenname: Z. surname: Wang fullname: Wang, Z. organization: Department of Mechanical Engineering, University of Michigan, United States of America – sequence: 2 givenname: X. surname: Huan fullname: Huan, X. organization: Department of Mechanical Engineering, Michigan Institute for Computational Discovery & Engineering, University of Michigan, United States of America – sequence: 3 givenname: K. surname: Garikipati fullname: Garikipati, K. email: krishna@umich.edu organization: Departments of Mechanical Engineering, and Mathematics, Michigan Institute for Computational Discovery & Engineering, University of Michigan, United States of America |
| BookMark | eNp9kc1u3CAUhVGVSp2kfYDukLr2lB9j7HZVRf2JFCmbqFuE4ZIysmECeKR5pr5ksKerLsLmAjrfubr3XKOrEAMg9JGSPSW0-3zYm1nvGWF0TymXpHuDdrSXQ8Mo76_QjpBWNLJn4h26zvlA6ukp26G_v3XyuvgY9ITzOReYsbcQinfebP84Olz-AD7qVHwVWe8cpFVRH_C8bKKMn-IJUvDhCc_epJhLWkxZEmA4xWnZjHzAsy5Q-035C74Lm40BvJI4V_8MWAe7Xlfz6YyXkGCqiMVWF_0evXUVhQ__6g16_PH98fZXc__w8-72231jOBOlGQRpbc-oYa2AcWRDLW4woxwHTckoieTd6KRpNbWyG0zfA3ei5ZYLI_qO36BPF9tjis8L5KIOcUl1P1kxQTvWEj7QqpIX1TpsTuCU8WXbRUnaT4oStQajDqoGo9Zg1CWYStL_yGPys07nV5mvFwbq3CcPSWXj1-VZn8AUZaN_hX4BesStDQ |
| CitedBy_id | crossref_primary_10_3390_bioengineering10020269 crossref_primary_10_1016_j_cpc_2025_109582 crossref_primary_10_1002_nme_7509 crossref_primary_10_1007_s11831_021_09643_1 crossref_primary_10_1038_s42005_024_01521_z crossref_primary_10_1016_j_jcp_2021_110525 crossref_primary_10_1080_00224065_2023_2260018 crossref_primary_10_1016_j_jmps_2021_104474 crossref_primary_10_1038_s41598_024_64730_0 crossref_primary_10_1016_j_compchemeng_2023_108320 crossref_primary_10_1016_j_ymssp_2023_110147 crossref_primary_10_3389_fsysb_2024_1333760 crossref_primary_10_1093_imanum_drae086 crossref_primary_10_1016_j_cma_2021_114399 crossref_primary_10_1016_j_cma_2022_115248 crossref_primary_10_1088_1361_6501_acabdd crossref_primary_10_1002_nme_7475 crossref_primary_10_1016_j_commatsci_2022_111493 |
| Cites_doi | 10.1103/PhysRevE.79.031908 10.1016/j.tree.2007.10.013 10.1007/s00285-008-0215-x 10.1016/0001-6160(79)90196-2 10.1126/sciadv.1602614 10.1109/TMBMC.2016.2633265 10.1098/rsfs.2011.0113 10.1109/TIT.2005.862083 10.1021/acs.jpcc.6b09775 10.1007/s00466-013-0958-0 10.1016/j.jcp.2018.10.045 10.1142/S0218202510004313 10.1016/j.jcp.2015.07.002 10.1016/j.jtbi.2014.11.024 10.1006/bulm.1998.0093 10.1016/j.cma.2019.07.007 10.1073/pnas.1517384113 10.1103/PhysRevE.79.031926 10.1103/RevModPhys.48.571 10.1109/TIT.2006.871582 10.1098/rspa.2017.0009 10.1063/1.1744102 10.1073/pnas.1620045114 10.1088/0951-7715/23/1/R01 10.1088/1478-3975/8/5/055011 10.1007/BF00289234 10.1103/PhysRevE.101.010203 10.1002/cnm.2552 10.1016/j.taml.2020.01.028 10.1016/j.jmps.2016.11.008 10.1890/0012-9658(2001)082[0050:VPFISA]2.0.CO;2 10.1016/j.jtbi.2008.03.027 10.1016/j.jmps.2016.11.013 |
| ContentType | Journal Article |
| Copyright | 2021 Elsevier B.V. Copyright Elsevier BV Apr 15, 2021 |
| Copyright_xml | – notice: 2021 Elsevier B.V. – notice: Copyright Elsevier BV Apr 15, 2021 |
| DBID | AAYXX CITATION 7SC 7TB 8FD FR3 JQ2 KR7 L7M L~C L~D |
| DOI | 10.1016/j.cma.2021.113706 |
| 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_113706 S0045782521000426 |
| GroupedDBID | --K --M -~X .DC .~1 0R~ 1B1 1~. 1~5 4.4 457 4G. 5GY 5VS 7-5 71M 8P~ 9JN AABNK AACTN AAEDT AAEDW AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAXUO AAYFN ABAOU ABBOA ABFNM ABJNI ABMAC ABYKQ ACAZW ACDAQ ACGFS ACIWK ACRLP ACZNC ADBBV ADEZE ADGUI ADTZH AEBSH AECPX AEKER AENEX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AHJVU AHZHX AIALX AIEXJ AIGVJ AIKHN AITUG AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ AOUOD ARUGR AXJTR BJAXD BKOJK BLXMC CS3 DU5 EBS EFJIC EFLBG EO8 EO9 EP2 EP3 F5P FDB FIRID FNPLU FYGXN G-Q GBLVA GBOLZ IHE J1W JJJVA KOM LG9 LY7 M41 MHUIS MO0 N9A O-L O9- OAUVE OZT P-8 P-9 P2P PC. PQQKQ Q38 RNS ROL RPZ SDF SDG SDP SES SPC SPCBC SST SSV SSW SSZ T5K TN5 WH7 XPP ZMT ~02 ~G- 29F 9DU AAQXK AATTM AAXKI AAYWO AAYXX ABEFU ABWVN ABXDB ACLOT ACNNM ACRPL ACVFH ADCNI ADIYS ADJOM ADMUD ADNMO AEIPS AEUPX AFJKZ AFPUW AGQPQ AI. AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP ASPBG AVWKF AZFZN CITATION EFKBS EJD FEDTE FGOYB G-2 HLZ HVGLF HZ~ R2- SBC SET SEW VH1 VOH WUQ ZY4 ~HD 7SC 7TB 8FD AGCQF FR3 JQ2 KR7 L7M L~C L~D |
| ID | FETCH-LOGICAL-c325t-9504d821c245ebb2945ef9cb7b9a10b70736bf7c4a1d769c88e3f543d35c5863 |
| ISICitedReferencesCount | 21 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000657581100020&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0045-7825 |
| IngestDate | Sun Sep 07 03:53:37 EDT 2025 Sat Nov 29 07:26:25 EST 2025 Tue Nov 18 20:51:24 EST 2025 Fri Feb 23 02:47:07 EST 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Pattern formation Incomplete data Inverse problems System identification |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c325t-9504d821c245ebb2945ef9cb7b9a10b70736bf7c4a1d769c88e3f543d35c5863 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| PQID | 2516240391 |
| PQPubID | 2045269 |
| ParticipantIDs | proquest_journals_2516240391 crossref_citationtrail_10_1016_j_cma_2021_113706 crossref_primary_10_1016_j_cma_2021_113706 elsevier_sciencedirect_doi_10_1016_j_cma_2021_113706 |
| PublicationCentury | 2000 |
| PublicationDate | 2021-04-15 |
| PublicationDateYYYYMMDD | 2021-04-15 |
| PublicationDate_xml | – month: 04 year: 2021 text: 2021-04-15 day: 15 |
| PublicationDecade | 2020 |
| PublicationPlace | Amsterdam |
| PublicationPlace_xml | – name: Amsterdam |
| PublicationTitle | Computer methods in applied mechanics and engineering |
| PublicationYear | 2021 |
| Publisher | Elsevier B.V Elsevier BV |
| Publisher_xml | – name: Elsevier B.V – name: Elsevier BV |
| References | Wise, Lowengrub, Frieboes, Cristini (b15) 2008; 253 Brooks, Gelman, Jones, Meng (b25) 2011 Messenger, Bortz (b36) 2020 Lowengrub, Rätz, Voigt (b18) 2009; 79 Garikipati (b14) 2017; 99 Turing (b5) 1952; 237 Wang, Wu, Garikipati, Huan (b31) 2020; 10 Rietkerk, van de Koppel (b24) 2008; 23 Rudy, Brunton, Proctor, Kutz (b29) 2017; 3 Vilanova, Colominas, Gomez (b20) 2014; 53 Brunton, Proctor, Kutz (b26) 2016; 113 Korvasová, Gaffney, Maini, Ferreira, Klika (b13) 2015; 367 Teichert, Rudraraju, Garikipati (b3) 2017; 99 Rudraraju, Van der Ven, Garikipati (b2) 2016; 2 Oden, Hawkins, Prudhomme (b21) 2010; 20 Schmidt, Vallabhajosyula, Jenkins, Hood, Soni, Wikswo, Lipson (b44) 2011; 8 Cottrell, Hughes, Bazilevs (b37) 2009 Allen, Cahn (b45) 1979; 27 Barrio, Varea, Aragon (b9) 1999; 61 Mangan, Kutz, Brunton, Proctor (b41) 2017; 473 Maini, Woolley, Baker, Gaffney, Seirin Lee (b11) 2012; 2 Messenger, Bortz (b35) 2020 Cahn, Hilliard (b4) 1958; 28 Xu, Vilanova, Gomez (b22) 2016; 11 Candès, Romberg, Tao (b38) 2006; 52 Gierer, Meinhardt (b6) 1972; 12 Donoho (b39) 2006; 52 James, Witten, Hastie, Tibshirani (b40) 2013 Murray (b7) 1981; 295 Cristini, Li, Lowengrub, Wise (b16) 2009; 58 HilleRisLambers, Rietkerk, van den Bosch, Prins, de Kroon (b23) 2001; 82 Reinbold, Gurevich, Grigoriev (b34) 2020; 101 Barrio, Baker, Vaughan, Tribuzy, de Carvalho, Bassanezi, Maini (b10) 2009; 79 Yair, Talmon, Coifman, Kevrekidis (b33) 2017; 114 Lowengrub, Frieboes, Jin, Chuang, Li, Macklin, Cristini (b17) 2010; 23 Mangan, Brunton, Proctor, Kutz (b28) 2016; 2 Schnakenberg (b42) 1976; 48 Jiang, Rudraraju, Roy, Van der Ven, Garikipati, Falk (b1) 2016; 120 Vilanova, Colominas, Gomez (b19) 2013; 29 Wang, Huan, Garikipati (b27) 2019; 356 Raissi, Perdikaris, Karniadakis (b30) 2019; 378 Atkinson, Wang, Subber, Khan, Hawi, Ghanem (b32) 2019 Spill, Guerrero, Alarcon, Maini, Byrne (b12) 2015; 299 Schmidt, Lipson (b43) 2009; 03 Dillon, Maini, Othmer (b8) 1994; 32 Schmidt (10.1016/j.cma.2021.113706_b43) 2009; 03 Schmidt (10.1016/j.cma.2021.113706_b44) 2011; 8 Cahn (10.1016/j.cma.2021.113706_b4) 1958; 28 Lowengrub (10.1016/j.cma.2021.113706_b18) 2009; 79 Schnakenberg (10.1016/j.cma.2021.113706_b42) 1976; 48 Dillon (10.1016/j.cma.2021.113706_b8) 1994; 32 Xu (10.1016/j.cma.2021.113706_b22) 2016; 11 HilleRisLambers (10.1016/j.cma.2021.113706_b23) 2001; 82 Korvasová (10.1016/j.cma.2021.113706_b13) 2015; 367 Wise (10.1016/j.cma.2021.113706_b15) 2008; 253 Garikipati (10.1016/j.cma.2021.113706_b14) 2017; 99 Mangan (10.1016/j.cma.2021.113706_b28) 2016; 2 Cottrell (10.1016/j.cma.2021.113706_b37) 2009 Allen (10.1016/j.cma.2021.113706_b45) 1979; 27 Rudraraju (10.1016/j.cma.2021.113706_b2) 2016; 2 Atkinson (10.1016/j.cma.2021.113706_b32) 2019 Yair (10.1016/j.cma.2021.113706_b33) 2017; 114 Brooks (10.1016/j.cma.2021.113706_b25) 2011 Mangan (10.1016/j.cma.2021.113706_b41) 2017; 473 Teichert (10.1016/j.cma.2021.113706_b3) 2017; 99 Murray (10.1016/j.cma.2021.113706_b7) 1981; 295 Cristini (10.1016/j.cma.2021.113706_b16) 2009; 58 Oden (10.1016/j.cma.2021.113706_b21) 2010; 20 Messenger (10.1016/j.cma.2021.113706_b36) 2020 Vilanova (10.1016/j.cma.2021.113706_b19) 2013; 29 Barrio (10.1016/j.cma.2021.113706_b9) 1999; 61 Brunton (10.1016/j.cma.2021.113706_b26) 2016; 113 Rudy (10.1016/j.cma.2021.113706_b29) 2017; 3 Lowengrub (10.1016/j.cma.2021.113706_b17) 2010; 23 Vilanova (10.1016/j.cma.2021.113706_b20) 2014; 53 Turing (10.1016/j.cma.2021.113706_b5) 1952; 237 Spill (10.1016/j.cma.2021.113706_b12) 2015; 299 Wang (10.1016/j.cma.2021.113706_b27) 2019; 356 Donoho (10.1016/j.cma.2021.113706_b39) 2006; 52 James (10.1016/j.cma.2021.113706_b40) 2013 Reinbold (10.1016/j.cma.2021.113706_b34) 2020; 101 Gierer (10.1016/j.cma.2021.113706_b6) 1972; 12 Raissi (10.1016/j.cma.2021.113706_b30) 2019; 378 Barrio (10.1016/j.cma.2021.113706_b10) 2009; 79 Wang (10.1016/j.cma.2021.113706_b31) 2020; 10 Jiang (10.1016/j.cma.2021.113706_b1) 2016; 120 Maini (10.1016/j.cma.2021.113706_b11) 2012; 2 Candès (10.1016/j.cma.2021.113706_b38) 2006; 52 Rietkerk (10.1016/j.cma.2021.113706_b24) 2008; 23 Messenger (10.1016/j.cma.2021.113706_b35) 2020 |
| References_xml | – volume: 53 year: 2014 ident: b20 article-title: Coupling of discrete random walks and continuous modeling for three-dimensional tumor-induced angiogenesis publication-title: Comput. Mech. – volume: 61 year: 1999 ident: b9 article-title: A two-dimensional numerical study of spatial pattern formation in interacting turing systems publication-title: Bull. Math. Biol. – volume: 79 year: 2009 ident: b18 article-title: Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission publication-title: Phys. Rev. E – volume: 99 year: 2017 ident: b14 article-title: Perspectives on the mathematics of biological patterning and morphogenesis publication-title: J. Mech. Phys. Solids – volume: 3 year: 2017 ident: b29 article-title: Data-driven discovery of partial differential equations publication-title: Sci. Adv. – volume: 48 year: 1976 ident: b42 article-title: Network theory of microscopic and macroscopic behavior of master equation systems publication-title: Rev. Modern Phys. – volume: 20 year: 2010 ident: b21 article-title: General diffuse-interface theories and an approach to predictive tumor growth modeling publication-title: Math. Models Methods Appl. Sci. – year: 2019 ident: b32 article-title: Data-driven discovery of free-form governing differential equations publication-title: Second Workshop on Machine Learning and the Physical Sciences – year: 2020 ident: b35 article-title: Weak sindy: Galerkin-based data-driven model selection – volume: 11 year: 2016 ident: b22 article-title: A mathematical model coupling tumor growth and angiogenesis publication-title: PLoS ONE – volume: 82 start-page: 50 year: 2001 end-page: 61 ident: b23 article-title: Vegetation pattern formation in semi-arid grazing systems publication-title: Ecol. – volume: 113 year: 2016 ident: b26 article-title: Discovering governing equations from data by sparse identification of nonlinear dynamical systems publication-title: Proc. Natl. Acad. Sci. – volume: 299 year: 2015 ident: b12 article-title: Hybrid approaches for multiple-species stochastic reaction–diffusion models publication-title: J. Comput. Phys. – volume: 101 year: 2020 ident: b34 article-title: Using noisy or incomplete data to discover models of spatiotemporal dynamics publication-title: Phys. Rev. E – volume: 03 year: 2009 ident: b43 article-title: Distilling free-form natural laws from experimental data publication-title: Science – volume: 2 year: 2016 ident: b2 article-title: Mechano-chemical spinodal decomposition: A phenomenological theory of phase transformations in multi-component crystalline solids publication-title: Nat. Comput. Mater. – volume: 28 year: 1958 ident: b4 article-title: Free energy of a nonuniform system. i interfacial energy publication-title: J. Chem. Phys. – year: 2009 ident: b37 article-title: Isogeometric Analysis: Toward Integration of Cad and Fea – volume: 23 start-page: 169 year: 2008 end-page: 175 ident: b24 article-title: Regular pattern formation in real ecosystems publication-title: Trends Ecol. Evol. – volume: 58 year: 2009 ident: b16 article-title: Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching publication-title: J. Math. Biol. – volume: 52 start-page: 489 year: 2006 end-page: 509 ident: b38 article-title: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information publication-title: IEEE Trans. Inform. Theory – volume: 120 year: 2016 ident: b1 article-title: Multi-physics simulations of lithiation-induced stress in litio electrode particles publication-title: J. Phys. Chem. C – volume: 32 year: 1994 ident: b8 article-title: Pattern formation in generalized turing systems i: Steady-state patterns in systems with mixed boundary conditions publication-title: J. Math. Biol. – volume: 10 start-page: 188 year: 2020 ident: b31 article-title: A perspective on regression and bayesian approaches for system identification of pattern formation dynamics publication-title: Theoret. Appl. Mech. Lett. – volume: 12 year: 1972 ident: b6 article-title: A theory of biological pattern formation publication-title: Kybernetik – volume: 356 start-page: 44 year: 2019 end-page: 74 ident: b27 article-title: Variational system identification of the partial differential equations governing the physics of pattern-formation: Inference under varying fidelity and noise publication-title: Comput. Methods Appl. Mech. Engrg. – year: 2011 ident: b25 article-title: Handbook of Markov Chain Monte Carlo – volume: 29 year: 2013 ident: b19 article-title: Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis publication-title: Numer. Methods Biomed. Eng. – volume: 114 start-page: E7865 year: 2017 end-page: E7874 ident: b33 article-title: Reconstruction of normal forms by learning informed observation geometries from data publication-title: Proc. Natl. Acad. Sci. – volume: 27 year: 1979 ident: b45 article-title: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening publication-title: Acta Metall. – volume: 367 year: 2015 ident: b13 article-title: Investigating the turing conditions for diffusion-driven instability in the presence of a binding immobile substrate publication-title: J. Theoret. Biol. – volume: 8 year: 2011 ident: b44 article-title: Automated refinement and inference of analytical models for metabolic networks publication-title: Phys. Biol. – year: 2020 ident: b36 article-title: Weak sindy for partial differential equations – volume: 473 year: 2017 ident: b41 article-title: Model selection for dynamical systems via sparse regression and information criteria publication-title: Proc. R. Soc. A – volume: 253 year: 2008 ident: b15 article-title: Three-dimensional multispecies nonlinear tumor growth–model and numerical method publication-title: J. Theoret. Biol. – volume: 378 year: 2019 ident: b30 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: 79 year: 2009 ident: b10 article-title: Modeling the skin pattern of fishes publication-title: Phys. Rev. E – volume: 99 year: 2017 ident: b3 article-title: A variational treatment of material configurations with application to interface motion and microstructural evolution publication-title: J. Mech. Phys. Solids – volume: 237 year: 1952 ident: b5 article-title: The chemical basis of morphogenesis publication-title: Phil. Trans. R. Soc. Lond. Ser. B – volume: 295 year: 1981 ident: b7 article-title: On pattern formation mechanisms for lepidopteran wing patterns and mammalian coat markings publication-title: Phil. Trans. R. Soc. Lond. Ser. B – volume: 2 start-page: 487 year: 2012 end-page: 496 ident: b11 article-title: Turing’s model for biological pattern formation and the robustness problem publication-title: Interface Focus – volume: 2 year: 2016 ident: b28 article-title: Inferring biological networks by sparse identification of nonlinear dynamics publication-title: IEEE Trans. Mol. Biol. Multi-Scale Commun. – volume: 23 year: 2010 ident: b17 article-title: Nonlinear modelling of cancer: bridging the gap between cells and tumours publication-title: Nonlinearity – volume: 52 start-page: 1289 year: 2006 end-page: 1306 ident: b39 article-title: Compressed sensing publication-title: IEEE Trans. Inform. Theory – year: 2013 ident: b40 article-title: An Introduction to Statistical Learning – volume: 79 year: 2009 ident: 10.1016/j.cma.2021.113706_b10 article-title: Modeling the skin pattern of fishes publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.79.031908 – volume: 23 start-page: 169 year: 2008 ident: 10.1016/j.cma.2021.113706_b24 article-title: Regular pattern formation in real ecosystems publication-title: Trends Ecol. Evol. doi: 10.1016/j.tree.2007.10.013 – year: 2011 ident: 10.1016/j.cma.2021.113706_b25 – volume: 58 year: 2009 ident: 10.1016/j.cma.2021.113706_b16 article-title: Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching publication-title: J. Math. Biol. doi: 10.1007/s00285-008-0215-x – volume: 27 year: 1979 ident: 10.1016/j.cma.2021.113706_b45 article-title: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening publication-title: Acta Metall. doi: 10.1016/0001-6160(79)90196-2 – volume: 3 year: 2017 ident: 10.1016/j.cma.2021.113706_b29 article-title: Data-driven discovery of partial differential equations publication-title: Sci. Adv. doi: 10.1126/sciadv.1602614 – volume: 2 year: 2016 ident: 10.1016/j.cma.2021.113706_b28 article-title: Inferring biological networks by sparse identification of nonlinear dynamics publication-title: IEEE Trans. Mol. Biol. Multi-Scale Commun. doi: 10.1109/TMBMC.2016.2633265 – volume: 2 start-page: 487 issue: 4 year: 2012 ident: 10.1016/j.cma.2021.113706_b11 article-title: Turing’s model for biological pattern formation and the robustness problem publication-title: Interface Focus doi: 10.1098/rsfs.2011.0113 – volume: 52 start-page: 489 issn: 0018-9448 issue: 2 year: 2006 ident: 10.1016/j.cma.2021.113706_b38 article-title: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.2005.862083 – volume: 32 year: 1994 ident: 10.1016/j.cma.2021.113706_b8 article-title: Pattern formation in generalized turing systems i: Steady-state patterns in systems with mixed boundary conditions publication-title: J. Math. Biol. – year: 2020 ident: 10.1016/j.cma.2021.113706_b35 – volume: 03 year: 2009 ident: 10.1016/j.cma.2021.113706_b43 article-title: Distilling free-form natural laws from experimental data publication-title: Science – volume: 120 year: 2016 ident: 10.1016/j.cma.2021.113706_b1 article-title: Multi-physics simulations of lithiation-induced stress in litio electrode particles publication-title: J. Phys. Chem. C doi: 10.1021/acs.jpcc.6b09775 – volume: 53 year: 2014 ident: 10.1016/j.cma.2021.113706_b20 article-title: Coupling of discrete random walks and continuous modeling for three-dimensional tumor-induced angiogenesis publication-title: Comput. Mech. doi: 10.1007/s00466-013-0958-0 – year: 2020 ident: 10.1016/j.cma.2021.113706_b36 – volume: 378 year: 2019 ident: 10.1016/j.cma.2021.113706_b30 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: 20 year: 2010 ident: 10.1016/j.cma.2021.113706_b21 article-title: General diffuse-interface theories and an approach to predictive tumor growth modeling publication-title: Math. Models Methods Appl. Sci. doi: 10.1142/S0218202510004313 – volume: 299 year: 2015 ident: 10.1016/j.cma.2021.113706_b12 article-title: Hybrid approaches for multiple-species stochastic reaction–diffusion models publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2015.07.002 – year: 2019 ident: 10.1016/j.cma.2021.113706_b32 article-title: Data-driven discovery of free-form governing differential equations – volume: 367 year: 2015 ident: 10.1016/j.cma.2021.113706_b13 article-title: Investigating the turing conditions for diffusion-driven instability in the presence of a binding immobile substrate publication-title: J. Theoret. Biol. doi: 10.1016/j.jtbi.2014.11.024 – volume: 61 year: 1999 ident: 10.1016/j.cma.2021.113706_b9 article-title: A two-dimensional numerical study of spatial pattern formation in interacting turing systems publication-title: Bull. Math. Biol. doi: 10.1006/bulm.1998.0093 – volume: 356 start-page: 44 issn: 0045-7825 year: 2019 ident: 10.1016/j.cma.2021.113706_b27 article-title: Variational system identification of the partial differential equations governing the physics of pattern-formation: Inference under varying fidelity and noise publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/j.cma.2019.07.007 – volume: 113 year: 2016 ident: 10.1016/j.cma.2021.113706_b26 article-title: Discovering governing equations from data by sparse identification of nonlinear dynamical systems publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.1517384113 – volume: 79 year: 2009 ident: 10.1016/j.cma.2021.113706_b18 article-title: Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.79.031926 – volume: 48 year: 1976 ident: 10.1016/j.cma.2021.113706_b42 article-title: Network theory of microscopic and macroscopic behavior of master equation systems publication-title: Rev. Modern Phys. doi: 10.1103/RevModPhys.48.571 – volume: 52 start-page: 1289 issn: 00189448 issue: 4 year: 2006 ident: 10.1016/j.cma.2021.113706_b39 article-title: Compressed sensing publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.2006.871582 – volume: 473 issue: 2204 year: 2017 ident: 10.1016/j.cma.2021.113706_b41 article-title: Model selection for dynamical systems via sparse regression and information criteria publication-title: Proc. R. Soc. A doi: 10.1098/rspa.2017.0009 – volume: 28 year: 1958 ident: 10.1016/j.cma.2021.113706_b4 article-title: Free energy of a nonuniform system. i interfacial energy publication-title: J. Chem. Phys. doi: 10.1063/1.1744102 – volume: 295 year: 1981 ident: 10.1016/j.cma.2021.113706_b7 article-title: On pattern formation mechanisms for lepidopteran wing patterns and mammalian coat markings publication-title: Phil. Trans. R. Soc. Lond. Ser. B – volume: 114 start-page: E7865 issue: 38 year: 2017 ident: 10.1016/j.cma.2021.113706_b33 article-title: Reconstruction of normal forms by learning informed observation geometries from data publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.1620045114 – volume: 23 year: 2010 ident: 10.1016/j.cma.2021.113706_b17 article-title: Nonlinear modelling of cancer: bridging the gap between cells and tumours publication-title: Nonlinearity doi: 10.1088/0951-7715/23/1/R01 – year: 2009 ident: 10.1016/j.cma.2021.113706_b37 – volume: 8 year: 2011 ident: 10.1016/j.cma.2021.113706_b44 article-title: Automated refinement and inference of analytical models for metabolic networks publication-title: Phys. Biol. doi: 10.1088/1478-3975/8/5/055011 – volume: 12 year: 1972 ident: 10.1016/j.cma.2021.113706_b6 article-title: A theory of biological pattern formation publication-title: Kybernetik doi: 10.1007/BF00289234 – volume: 101 year: 2020 ident: 10.1016/j.cma.2021.113706_b34 article-title: Using noisy or incomplete data to discover models of spatiotemporal dynamics publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.101.010203 – volume: 237 year: 1952 ident: 10.1016/j.cma.2021.113706_b5 article-title: The chemical basis of morphogenesis publication-title: Phil. Trans. R. Soc. Lond. Ser. B – volume: 29 year: 2013 ident: 10.1016/j.cma.2021.113706_b19 article-title: Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis publication-title: Numer. Methods Biomed. Eng. doi: 10.1002/cnm.2552 – volume: 10 start-page: 188 issn: 2095-0349 issue: 3 year: 2020 ident: 10.1016/j.cma.2021.113706_b31 article-title: A perspective on regression and bayesian approaches for system identification of pattern formation dynamics publication-title: Theoret. Appl. Mech. Lett. doi: 10.1016/j.taml.2020.01.028 – volume: 11 year: 2016 ident: 10.1016/j.cma.2021.113706_b22 article-title: A mathematical model coupling tumor growth and angiogenesis publication-title: PLoS ONE – volume: 2 year: 2016 ident: 10.1016/j.cma.2021.113706_b2 article-title: Mechano-chemical spinodal decomposition: A phenomenological theory of phase transformations in multi-component crystalline solids publication-title: Nat. Comput. Mater. – volume: 99 year: 2017 ident: 10.1016/j.cma.2021.113706_b3 article-title: A variational treatment of material configurations with application to interface motion and microstructural evolution publication-title: J. Mech. Phys. Solids doi: 10.1016/j.jmps.2016.11.008 – volume: 82 start-page: 50 year: 2001 ident: 10.1016/j.cma.2021.113706_b23 article-title: Vegetation pattern formation in semi-arid grazing systems publication-title: Ecol. doi: 10.1890/0012-9658(2001)082[0050:VPFISA]2.0.CO;2 – year: 2013 ident: 10.1016/j.cma.2021.113706_b40 – volume: 253 year: 2008 ident: 10.1016/j.cma.2021.113706_b15 article-title: Three-dimensional multispecies nonlinear tumor growth–model and numerical method publication-title: J. Theoret. Biol. doi: 10.1016/j.jtbi.2008.03.027 – volume: 99 year: 2017 ident: 10.1016/j.cma.2021.113706_b14 article-title: Perspectives on the mathematics of biological patterning and morphogenesis publication-title: J. Mech. Phys. Solids doi: 10.1016/j.jmps.2016.11.013 |
| SSID | ssj0000812 |
| Score | 2.4916255 |
| Snippet | Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The... |
| SourceID | proquest crossref elsevier |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 113706 |
| SubjectTerms | Acceptance Boundary value problems Developmental biology Domains Evolution Identification methods Incomplete data Inference Inverse problems Mathematical analysis Microscopy Microstructure Operators (mathematics) Partial differential equations Pattern formation Photomicrographs Physics System identification Temporal resolution Weighting functions |
| Title | Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data |
| URI | https://dx.doi.org/10.1016/j.cma.2021.113706 https://www.proquest.com/docview/2516240391 |
| Volume | 377 |
| WOSCitedRecordID | wos000657581100020&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/eLvHCXMwtV3LbtNAFB2FFCRY8AggCgXdBWJBZOTXxB52FUpFURRYhCpiMxqPJ1VK6qZ1G5Vv4h_4Nu48PE6rEsGCTR5OJnFyjmfO3Cchr_FaFioUZZAzSoO0ZCrIpcoCMYuEpBIXfWGQHmXjcT6dsi-dzq8mF2a1yKoqv7xky_8KNR5DsHXq7D_A7T8UD-BjBB1vEXa8_SvgD3D321j4bJ3m_rx0MUFeH5oEKT3Y-GhsjxTzRJ1euOC4Q9OFV1sSjnXQni00q90NauXOX9tKUPDaH6VNC_tN9mBfj-3jZHVWW_dErQO3xWLxo39RmfQZ1LkuLa4tlOAaTLiu1iZQVziRfKx0hnJTUVq1RRRbh4CdtL61PLWm3amPMML_5bsJIG-tu87cEUfac2MTPq0NzufhHKxP6ykNUOrQ9Wk9ce1h7MQc3bhcWMvF0TtpSlDFke5wk4U3lOYef-Z7X0cjPhlOJ2-Wp4HuWqa9-66Fyy2yFWeU5V2ytbs_nH5qtUAe2Xr17gQbv7qJMLz2rX9SRtc0ghE-k4fkvtuxwK5l2iPSUVWPPHC7F3BrQ90j99ZKW_bIHRNaLOvH5OcaIcESEq4SEk5mgIQER0hYJyR4QoInJFwlJHhCwrwCT8j34OkIeiRYOgISCDwdwdMRNB2fkMnecPLhY-BahAQyiel5wGiYlnkcyTilqihihnczJousYCIKiwwXsEExy2QqojIbMJnnKpnRNCkTKmk-SJ6SbnVSqWcEULkmIhODEkVrmqq4YLi5QHXMmIrKVNBtEjbocOnK5-suLgvexEkecQSUa0C5BXSbvPVDlrZ2zKY3pw3k3IlfK2o5knXTsJ2GHtxNQjXHPctA19lk0fPNL78gd9trbId0ETj1ktyWq_N5ffbKkfk32V_jlA |
| 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=Variational+system+identification+of+the+partial+differential+equations+governing+microstructure+evolution+in+materials%3A+Inference+over+sparse+and+spatially+unrelated+data&rft.jtitle=Computer+methods+in+applied+mechanics+and+engineering&rft.au=Wang%2C+Z&rft.au=Huan%2C+X&rft.au=Garikipati%2C+K&rft.date=2021-04-15&rft.pub=Elsevier+BV&rft.issn=0045-7825&rft.volume=377&rft.spage=1&rft_id=info:doi/10.1016%2Fj.cma.2021.113706&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 |