Research
========

DeepXDE has been used in

- > 230 universities, e.g.,
  `Harvard University <https://www.harvard.edu>`_,
  `Massachusetts Institute of Technology <https://www.mit.edu>`_,
  `Stanford University <https://www.stanford.edu>`_,
  `University of California, Berkeley <https://www.berkeley.edu>`_,
  `Yale University <https://www.yale.edu>`_,
  `University of California, Los Angeles <https://www.ucla.edu>`_,
  `Imperial College London <https://www.imperial.ac.uk>`_,
  `Johns Hopkins University <https://www.jhu.edu>`_,
  `University of Pennsylvania <https://www.upenn.edu>`_,
  `Tsinghua University <https://www.tsinghua.edu.cn/en>`_,
  `University of Toronto <https://www.utoronto.ca>`_,
  `California Institute of Technology <https://www.caltech.edu>`_,
  `Princeton University <https://www.princeton.edu>`_,
  `Cornell University <https://www.cornell.edu>`_,
  `National University of Singapore <https://www.nus.edu.sg>`_,
  `Nanyang Technological University <https://www.ntu.edu.sg>`_,
  `University of California, San Diego <https://ucsd.edu>`_,
  `Peking University <https://english.pku.edu.cn>`_,
  `University of Amsterdam <https://www.uva.nl>`_,
  `New York University Abu Dhabi <https://nyuad.nyu.edu/en>`_,
  `University of British Columbia <https://www.ubc.ca>`_,
  `University of Copenhagen <https://www.ku.dk/english>`_,
  `KU Leuven <https://www.kuleuven.be/english/kuleuven>`_,
  `University of Pittsburgh <https://www.pitt.edu>`_,
  `Zhejiang University <https://www.zju.edu.cn/english>`_,
  `Shanghai Jiao Tong University <https://en.sjtu.edu.cn>`_,
  `University of Texas at Austin <utexas.edu>`_,
  `Leiden University <https://www.universiteitleiden.nl/en>`_,
  `University of Minnesota <https://system.umn.edu>`_,
  `University of Manchester <https://www.manchester.ac.uk>`_,
  `University of Chinese Academy of Sciences <https://english.ucas.ac.cn>`_,
  `Georgia Institute of Technology <https://www.gatech.edu>`_,
  `Boston University <https://www.bu.edu>`_,
  `University of Maryland <https://umd.edu>`_,
  `Universite Paris Saclay <https://www.universite-paris-saclay.fr>`_,
  `The University of Tokyo <https://www.u-tokyo.ac.jp>`_,
  `University of Science and Technology of China <https://www.ustc.edu.cn>`_,
  `University of Southern California <https://www.usc.edu>`_,
  `University of Wisconsin Madison <https://www.wisc.edu>`_,
  `Technical University of Munich <https://www.tum.de/en>`_,
  `University of California, Santa Barbara <https://www.ucsb.edu>`_,
  `University of Birmingham <https://www.birmingham.ac.uk>`_,
  `Pennsylvania State University <https://www.psu.edu>`_,
  `University of Colorado Boulder <https://www.colorado.edu>`_,
  `University of Illinois at Urbana-Champaign <https://illinois.edu>`_,
  `University of California Irvine <https://uci.edu>`_,
  `Sun Yat-sen University <https://www.sysu.edu.cn>`_,
  `King Abdullah University of Science and Technology <https://www.kaust.edu.sa/en>`_,
  `University of Oslo <https://www.uio.no/english>`_,
  `University of Florida <https://www.ufl.ed>`_,
  `Aarhus University <https://international.au.dk>`_,
  `University of Exeter <https://www.exeter.ac.uk>`_,
  `University of Southampton <https://www.southampton.ac.uk>`_,
  `University of California, Santa Cruz <https://www.ucsc.edu>`_,
  `University of Padua <https://www.unipd.it>`_,
  `Carnegie Mellon University <https://www.cmu.edu>`_,
  `Seoul National University <https://en.snu.ac.kr>`_,
  `University of Leeds <https://www.leeds.ac.uk>`_,
  `Sapienza University Rome <https://www.uniroma1.it/en/pagina-strutturale/home>`_,
  `University of Alberta <https://www.ualberta.ca/index.html>`_,
  `University of Liverpool <https://www.liverpool.ac.uk>`_,
  `Tongji University <https://en.tongji.edu.cn>`_,
  `University of Electronic Science and Technology of China <https://en.uestc.edu.cn>`_,
  `Brown University <https://www.brown.edu>`_,
  `University of Bonn <https://www.uni-bonn.de/en/university/university>`_,
  `Southern University of Science and Technology <https://www.sustech.edu.cn/en>`_,
  `Harbin Institute of Technology <https://www.hit.edu.cn>`_,
  `Purdue University <https://www.purdue.edu>`_,
  `Kyoto University <https://www.kyoto-u.ac.jp/en>`_,
  `University of Basel <https://www.unibas.ch/en.html>`_,
  `Central South University <https://en.csu.edu.cn>`_,
  `University of Massachusetts Amherst <https://www.umass.edu>`_,
  `Xi’an Jiaotong University <http://en.xjtu.edu.cn>`_,
  `Tel Aviv University <https://english.tau.ac.il>`_,
  `Texas A&M University <https://www.tamu.edu>`_,
  `Arizona State University <https://www.asu.edu>`_,
  `Beijing Institute of Technology <http://english.bit.edu.cn>`_,
  `Southeast University <https://www.seu.edu.cn/english>`_,
  `Delft University of Technology <https://www.tudelft.nl/en>`_,
  `University of Naples Federico II <http://www.international.unina.it>`_,
  `University of Waterloo <https://uwaterloo.ca>`_,
  `Tianjin University <https://www.tju.edu.cn/english/index.htm>`_,
  `Xiamen University <https://en.xmu.edu.cn>`_,
  `University of Calgary <https://www.ucalgary.ca>`_,
  `Beijing Normal University <https://english.bnu.edu.cn>`_,
  `Kapodistrian University <https://en.uoa.gr>`_,
  `University of Turin <https://www.unito.it>`_,
  `RWTH Aachen University <https://www.rwth-aachen.de/go/id/a/?lidx=1>`_,
  `National Taiwan University <https://www.ntu.edu.tw>`_,
  `China University of Geosciences <https://en.cug.edu.cn>`_,
  `Sichuan University <https://www.scu.edu.cn>`_,
  `Rice University <https://www.rice.edu>`_,
  `Beihang University <https://ev.buaa.edu.cn>`_,
  `University of Sussex <https://www.sussex.ac.uk>`_,
  `University of Bergen <https://www.uib.no/en>`_,
  `KTH Royal Institute of Technology <https://www.kth.se/en>`_,
  `Northwestern Polytechnical University <https://en.nwpu.edu.cn>`_,
  `Tufts University <https://www.tufts.edu>`_,
  `Wuhan University of Technology <http://english.whut.edu.cn>`_,
  `Universidade do Porto <https://www.up.pt/portal/en>`_,
  `Florida State University <https://www.fsu.edu>`_,
  `Karlsruhe Institute of Technology <https://www.kit.edu>`_,
  `University Duisburg-Essen <https://www.uni-due.org/category/english>`_,
  `Dartmouth College <https://home.dartmouth.edu>`_,
  `University of Western Ontario <https://www.uwo.ca/index.html>`_,
  `University of Strasbourg <https://en.unistra.fr>`_,
  `University of Surrey <https://www.surrey.ac.uk>`_,
  `Shanghai University <https://en.shu.edu.cn>`_,
  `Chalmers University of Technology <https://www.chalmers.se/en>`_,
  `Pontificia Universidad Católica de Chile <https://www.uc.cl>`_,
  `Kyushu University <https://www.kyushu-u.ac.jp/en>`_,
  `Nagoya University <https://en.nagoya-u.ac.jp>`_,
  `University of Johannesburg <https://www.uj.ac.za>`_,
  `University of Rome Tor Vergata <http://web.uniroma2.it>`_,
  `University of Kentucky <https://www.uky.edu>`_,
  `Mansoura University <https://www.mans.edu.eg/en>`_,
  `Eindhoven University of Technology <https://www.tue.nl/en>`_,
  `Friedrich Schiller University of Jena <https://www.uni-jena.de/en>`_,
  `University of Victoria <https://www.uvic.ca>`_,
  `University of Twente <https://www.utwente.nl/en>`_,
  `University of Houston <https://www.uh.edu>`_,
  `Fuzhou University <https://en.fzu.edu.cn>`_,
  `University of Delaware <https://www.udel.edu>`_,
  `University of Mississippi <https://olemiss.edu>`_,
  `Swansea University <https://www.swansea.ac.uk>`_,
  `University of Bath <https://www.bath.ac.uk>`_,
  `University of Trieste <https://www.units.it/en>`_,
  `University of the Basque Country <https://www.ehu.eus/en/en-home>`_,
  `Hong Kong Baptist University <https://www.hkbu.edu.hk>`_,
  `University of Hawaii Manoa <https://manoa.hawaii.edu>`_,
  `Federal University of Rio de Janeiro <https://ufrj.br/en>`_,
  `George Mason University <https://www.gmu.edu>`_,
  `University of Sevilla <https://www.us.es>`_,
  `International School for Advanced Studies <https://www.sissa.it>`_,
  `Beijing University of Technology <https://english.bjut.edu.cn>`_,
  `Nanjing Tech University <https://www.njtech.edu.cn>`_,
  `TU Wien <https://www.tuwien.at/en>`_,
  `Beijing Jiaotong University <http://en.njtu.edu.cn>`_,
  `University of Canterbury <https://www.canterbury.ac.nz>`_,
  `Universidade do Minho <https://www.uminho.pt>`_,
  `Nanchang University <https://english.ncu.edu.cn>`_,
  `Carleton University <https://carleton.ca>`_,
  `Ocean University of China <http://www.ouc.edu.cn>`_,
  `South China Normal University <http://english.scnu.edu.cn>`_,
  `Roma Tre University <https://www.uniroma3.it>`_,
  `AmirKabir University of Technology <https://aut.ac.ir/en>`_,
  `Sabanci University <https://www.sabanciuniv.edu/en>`_,
  `Indian Institute of Technology <https://www.iitk.ac.in>`_,
  `Concordia University <https://www.concordia.ca>`_,
  `Tarbiat Modares University <https://en.modares.ac.ir>`_,
  `National University of Colombia <https://unal.edu.co/en.html>`_,
  `Clemson University <https://www.clemson.edu>`_,
  `Dortmund University of Technology <https://www.tu-dortmund.de/en>`_,
  `University of Los Andes <https://uniandes.edu.co/en>`_,
  `University of Stuttgart <https://www.uni-stuttgart.de/en>`_,
  `Universidad de salamanca <https://www.usal.es/universidad-de-salamanca>`_,
  `Harbin Engineering University <https://english.hrbeu.edu.cn>`_,
  `Universiti Teknologi Petronas <https://www.utp.edu.my>`_,
  `ITMO University <https://en.itmo.ru>`_,
  `University of Nevada, Las Vegas <https://www.unlv.edu>`_,
  `University of Bayreuth <https://www.uni-bayreuth.de/en>`_,
  `Macau University of Science & Technology <https://www.must.edu.mo/en>`_,
  `Isfahan University of Technology <http://english.iut.ac.ir>`_,
  `Rensselaer Polytechnic Institute <https://www.rpi.edu>`_,
  `Missouri University of Science and Technology <https://www.mst.edu>`_,
  `AGH University of Krakow <https://www.agh.edu.pl/en>`_,
  `University of Calabria <https://www.unical.it/internazionale/intenational-students/unical-admission>`_,
  `Ulster University <https://www.ulster.ac.uk>`_,
  `University of Thessaly <https://www.uth.gr/en>`_,
  `Kuwait University <https://kuweb.ku.edu.kw>`_,
  `University of Malaga <https://www.uma.es>`_,
  `Brno University of Technology <https://www.vut.cz/en>`_,
  `Old Dominion University <https://www.odu.edu>`_,
  `Johannes Kepler University Linz <https://www.jku.at>`_,
  `University of Kragujevac <https://en.kg.ac.rs>`_,
  `California Polytechnic State University <https://www.calpoly.edu>`_,
  `Chung-Ang University <https://neweng.cau.ac.kr>`_,
  `Graz University of Technology <https://www.tugraz.at>`_,
  `Shahid Beheshti University <https://www.sbu.ac.ir>`_,
  `Shanghai Normal University <https://english.shnu.edu.cn>`_,
  `Cadi Ayyad University <https://www.uca.ma/en/page/university-cadi-ayyad-the-eminent-university-of-science-and-knowledge>`_,
  `Universidad Rey Juan Carlos <https://www.urjc.es>`_,
  `Zhejiang A&F University <http://zafu.admissions.cn>`_,
  `Universidade Federal Fluminense <https://www.uff.br>`_,
  `Pontifical Catholic University of Valparaiso <https://www.pucv.cl>`_,
  `Nazarbayev University <https://nu.edu.kz>`_,
  `University of A Coruña <https://www.udc.es>`_, 
  `Worcester Polytechnic Institute <https://www.wpi.edu>`_,
  `Xinjiang University <https://english.xju.edu.cn>`_,
  `LUT University <https://www.lut.fi>`_,
  `University of Las Palmas de Gran Canaria <https://www.ulpgc.es>`_,
  `Nanjing University of Aeronautics and Astronautics <https://www.nuaa.edu.cn>`_,
  `Lahore University of Management Sciences <https://www.lums.edu.pk>`_,
  `Hangzhou Dianzi University <http://hdu.admissions.cn>`_,
  `London South Bank University <https://www.lsbu.ac.uk>`_,
  `Taras Shevchenko National University Kiev <https://knu.ua/en>`_,
  `University of Calcutta <https://www.caluniv.ac.in>`_,
  `University of Kaiserslautern <https://www.uni-kl.de/en/startseite>`_,
  `Wuhan Textile University <https://www.wtu.edu.cn>`_,
  `San Francisco State University <https://www.sfsu.edu/index.html>`_,
  `Anhui University of Science and Technology <https://www.aust.edu.cn>`_,
  `Boise State University <https://www.boisestate.edu>`_,
  `Necmettin Erbakan University <https://www.erbakan.edu.tr>`_,
  `Shahrekord University <https://www.sku.ac.ir/en>`_,
  `Shahrood University of Technology <https://www.shahroodut.ac.ir>`_,
  `Yangtze University <https://www.yangtzeu.edu.cn>`_,
  `Technical University of Cartagena <https://www.upct.es/english/content/departments>`_,
  `Adolfo Ibáñez University <https://www.uai.cl/en>`_,
  `Bundeswehr University Munich <https://www.unibw.de/home-en>`_,
  `Universidad de Burgos <https://www.ubu.es>`_,
  `Dong A University <https://english.donga.ac.kr>`_,
  `East China University of Science and Technology Shanghai <https://www.ecust.edu.cn/>`_,
  `Bauhaus-Universität Weimar <https://www.uni-weimar.de/en/university/start>`_,
  `Henan Institute of Economics and Trade <https://www.hnjmxy.cn/info/1107/1116.htm>`_
  `National University of Defence Technology <https://english.nudt.edu.cn/About/index.htm>`_,
  `University of Applied Sciences and Arts Northwestern Switzerland <https://www.fhnw.ch/en/startseiteu>`_,
  `University of Engineering and Management <https://uem.edu.in>`_,
  `Pontifical Catholic University of Rio de Janeiro <https://www.puc-rio.br>`_,
  `Fujian Agriculture and Forestry University <https://www.fafu.edu.cn>`_,
  `Central South University of Forestry and Technology <http://www.csuft.edu.cn>`_,
  `Cho Chun Shik Graduate School of Mobility <https://www.kaist.ac.kr/en/>`_,
  `Adama Science and Technology University <https://www.astu.edu.et>`_,
  `The University of Waikato <https://www.waikato.ac.nz>`_,
  `Lishui University <http://www.lsu.edu.cn>`_,
  `Westphalian University <https://www.w-hs.de>`_,
  `Shanghai Jian Qiao University <https://en.gench.edu.cn>`_,
  `Tel-Aviv University <https://english.tau.ac.il>`_,
- > 40 national labs and research institutes, e.g.,
  `Pacific Northwest National Laboratory <https://www.pnnl.gov>`_,
  `Sandia National Laboratories <https://www.sandia.gov>`_,
  `Argonne National Laboratory <https://www.anl.gov>`_,
  `Los Alamos National Laboratory <https://www.lanl.gov>`_,
  `Oak Ridge National Laboratory <https://www.ornl.gov>`_,
  `Idaho National Laboratory <https://inl.gov>`_,
  `Institute of Applied Physics and Computational Mathematics <http://www.iapcm.ac.cn>`_,
  `Institute of Computational Mathematics and Scientific/Engineering Computing <http://icmsec.cc.ac.cn>`_,
  `China Academy of Engineering Physics <https://unitracker.aspi.org.au/universities/chinese-academy-of-engineering-physics>`_,
  `National Key Laboratory for Remanufacturing <https://unitracker.aspi.org.au/universities/army-academy-of-armored-forces>`_,
  `Laboratory of Web Science <https://www.ffhs.ch/en/research/lws>`_,
  `Associate Laboratory LSRE-LCM <https://lsre-lcm.fe.up.pt>`_,
  `Center of Applied Ecology and Sustainability <https://investigacion.uc.cl/en/centros-de-excelencia/center-of-applied-ecology-and-sustainability-capes>`_,
  `NEC Lab Europe <https://www.neclab.eu>`_,
  `CSIRO's Data61 <https://data61.csiro.au>`_,
  `Zienkiewicz Institute for Modelling, Data and AI <https://www.swansea.ac.uk/science-and-engineering/research/zienkiewicz-institute-for-modelling-data-ai>`_,
  `Erich Schmid Institute of Materials Science <https://www.oeaw.ac.at/esi/erich-schmid-institute>`_,
  `Athinoula A. Martinos Center for Biomedical Imaging <https://www.martinos.org>`_,
  `Friedrich-Alexander-Universität Erlangen-Nürnberg Research Center for Mathematics of Data <https://mod.fau.eu>`_,
  `Data Observatory Foundation <https://dataobservatory.net>`_,
  `Fraunhofer Institute for Integrated Systems and Device Technology IISB <https://www.iisb.fraunhofer.de>`_,
  `ONERA, The French Aerospace Lab <https://www.onera.fr/en>`_,
  `Ministry of Natural Resources (China) <https://www.mnr.gov.cn>`_,
  `Zhejiang Lab <https://en.zhejianglab.com>`_,
  `United Kingdom Atomic Energy Authority <https://www.gov.uk/government/organisations/uk-atomic-energy-authority>`_,
  `Russian Academy of Sciences <https://new.ras.ru/en>`_,
  `Chinese Academy of Sciences <https://english.cas.cn>`_,
  `AppliedAI Institute for Europe <https://www.appliedai.de>`_,
  `Rutherford Appleton Laboratory <https://www.ukri.org/who-we-are/stfc/locations/rutherford-appleton-laboratory>`_,
  `Mathematical Sciences Research Laboratory <http://math.uth.gr/laboratory/?lang=en>`_,
  `German Aerospace Center <https://www.dlr.de/en>`_,
  `Materials Center Leoben Forschung GmbH <https://www.mcl.at>`_,
  `Ecole Polytechnique <https://www.polytechnique.edu/en>`_,
  `Scuola Superiore Meridionale <https://www.ssm.unina.it>`_,
  `Mitsubishi Electric Research Laboratories <https://www.merl.com>`_,
  `Forschungszentrum Jülich <https://fz-juelich.de/de>`_,
  `China Ship Scientific Research Center <http://www.cssrc.com>`_,
  `Yanqi Lake Beijing Institute of Mathematical Sciences and Applications <https://www.bimsa.cn>`_,
  `Korea Institute of Fusion Energy <https://www.kfe.re.kr/eng>`_, 
  `Fraunhofer Heinrich Hertz Institute <https://www.hhi.fraunhofer.de>`_,   
  `Northwest Institute of Nuclear Technology <https://unitracker.aspi.org.au/universities/northwest-institute-of-nuclear-technology/>`_, 
  `Bay Area Environmental Research Institute <https://baeri.org>`_,   
  `Lockheed Martin Solar and Astrophysics Laboratory <https://www.atlasobscura.com/places/lockheed-martin-solar-astrophysics-laboratory>`_,   
  `CSIRO, Space & Astronomy <https://www.csiro.au/en/research/technology-space/astronomy-space>`_,   
  `Fujian Special Equipment Inspection and Research Institute <http://www.cbpvi.org>`_,   
  `Centrale Lille Institute <https://centralelille.fr/en/>`_, 
  `Science and Technology Facilities Council Scientific Computing Department <https://www.scd.stfc.ac.uk>`_,   
  `Children’s Hospital of Philadelphia <https://www.chop.edu>`_, 
- > 10 industry, e.g.,
  `Anailytica <https://anailytica.com>`_,
  `Ansys <https://www.ansys.com>`_,
  `BioME <https://biome-science.com>`_,
  `BirenTech Research <https://www.birentech.com>`_,
  `Bosch <https://www.bosch.us>`_,
  `ExxonMobil <https://corporate.exxonmobil.com>`_,
  `General Motors <https://www.gm.com>`_,
  `Intel Corporation <https://www.intel.com>`_,
  `RocketML <https://www.rocketml.net>`_,
  `Saudi Aramco <https://www.aramco.com>`_,
  `Shell <https://www.shell.com>`_,
  `SoftServe <https://www.softserveinc.com/en-us>`_,
  `Quantiph <https://quantiphi.com>`_,
  `Moldex3D <https://www.moldex3d.com>`_,

Here is a list of research papers that used DeepXDE. If you would like your paper to appear here, open an issue in the GitHub "Issues" section.

PINN
----

#. L\. Yin & X. Lv. `Adapting physics-informed neural networks for bifurcation detection in ecological migration models <https://arxiv.org/abs/2409.00651>`_. *arXiv preprint arXiv:2409.00651*, 2024.
#. K\.-L\. Lu, Y.-M. Su, Z. Bi, C. Qiu, & W.-J. Zhang. `Characteristic performance study on solving oscillator ODEs via soft-constrained physics-informed neural network with small data <https://arxiv.org/abs/2408.11077>`_. *arXiv preprint arXiv:2408.11077*, 2024.
#. H\. Gangloff & N. Jouvin. `jinns: a JAX library for physics-informed neural networks <https://arxiv.org/abs/2412.14132>`_. *arXiv preprint arXiv:2412.14132*, 2024.
#. M\. J. Choi. `Leveraging turbulence data with physics-informed neural networks <https://arxiv.org/abs/2412.20130>`_. *arXiv preprint arXiv:2412.20130*, 2024.
#. P\. Kumar & R. Ranjan. `Evaluation of physics-informed machine learning approach for computation of fluid flows <https://www.researchgate.net/publication/384293048>`_. *Proceedings of the 10th International and 50th National Conference on Fluid Mechanics and Fluid Power (FMFP), FMFP2023-FCS-395, December 20–22, IIT Jodhpur, Rajasthan, India*, 2024.
#. K\. Leng, M. Shankar, & J. Thiyagalingam. `Zero coordinate shift: Whetted automatic differentiation for physics-informed neural operators <https://www.sciencedirect.com/science/article/pii/S0021999124001530>`_. *Journal of Computational Physics*, Volume 505, 112904, 2024.
#. R\. Fang, K. Zhang, K. Song, Y. Kai, Y. Li, & B. Zheng. `A deep learning method for solving thermoelastic coupling problem <https://doi.org/10.1515/zna-2024-0009>`_. *Zeitschrift für Naturforschung A*, 79(8), 851–871, 2024.
#. S\. Schoder. `Physics-informed neural networks for modal wave field predictions in 3D room acoustics <https://doi.org/10.20944/preprints202411.1848.v1>`_. *Institute of Fundamentals and Theory in Electrical Engineering, Graz University of Technology, Inffeldgasse 18/I, 8010 Graz, Austria*, 2024.
#. L\. Vu-Quoc & A. Humer. `Partial-differential-algebraic equations of nonlinear dynamics by physics-informed neural-network: (I) Operator splitting and framework assessment <https://doi.org/10.1002/nme.7586>`_. *Neural Methods in Engineering*, First published: 17 October, 2024.
#. A\. Noorizadegan, R. Cavoretto, D.L. Young, & C.S. Chen. `Stable weight updating: A key to reliable PDE solutions using deep learning <https://www.sciencedirect.com/science/article/pii/S0955799724004077>`_. *Engineering Analysis with Boundary Elements*, Volume 168, 105933, 2024.
#. C\. Soyarslan & M. Pradas. `Physics-informed machine learning in asymptotic homogenization of elliptic equations <https://www.sciencedirect.com/science/article/pii/S0045782524002998>`_. *Computer Methods in Applied Mechanics and Engineering*, Volume 427, Part 2, 117043, 2024.
#. Y\. Wu, J. Guo, G. Gopalakrishna, & Z. Poulos. `Deep-MacroFin: Informed equilibrium neural network for continuous time economic models <https://arxiv.org/abs/2408.10368>`_. *arXiv preprint arXiv:2408.10368*, 2024.
#. J\. Kurz, B. Bowman, M. Seman, et al. `A physics-informed kernel approach to learning the operator for parametric PDEs <https://doi.org/10.1007/s00521-024-10460-3>`_. *Neural Computing and Applications*, 36, 22773–22787, 2024.
#. A\. Newa, A. S. Gearhart, R. A. Darragh, & M. Villafañe-Delgado. `Physics-informed neural networks for scientific modeling: uses, implementations, and directions <https://doi.org/10.1117/12.3013520>`_. *Artificial Intelligence and Machine Learning for Multi-Domain Operations Applications VI*, Vol. 13051, 130511J, 2024.
#. J\. Seo. `Past rewinding of fluid dynamics from noisy observation via physics-informed neural computing <https://link.aps.org/doi/10.1103/PhysRevE.110.025302>`_. *Phys. Rev. E*, 110(2), 025302, 2024.
#. S\. Mtshali, B. A. Jacobs. `Machine learning-based prediction of pharmacokinetic parameters for individualized drug dosage optimization <https://doi.org/10.1007/s41870-024-02314-x>`_. *Int. J. Inf. Tecnol.*, 2024.
#. W\. O. Pedruzzi, C. E. R. Dalla, W. B. D. Silva, D. Guimarães, V. A. Leão, J. C. S. Dutra. `Physics-Informed Neural Network for monitoring the sulfate ion adsorption process using particle filter <https://doi.org/10.1590/0001-3765202420240262>`_. *An. Acad. Bras. Ciênc.*, 96(4), e20240262, 2024.
#. X\. Wang, M. Sun, Y. Guo, C. Yuan, X. Sun, Z. Wei, X. Jin. `Octree-based hierarchical sampling optimization for the volumetric super-resolution of scientific data <https://doi.org/10.1016/j.jcp.2024.112804>`_. *Journal of Computational Physics*, Volume 502, 112804, 2024.
#. L\. Santos. `Deep and Physics-Informed Neural Networks as a Substitute for Finite Element Analysis <https://doi.org/10.1145/3674029.3674043>`_. *ICMLT '24: Proceedings of the 2024 9th International Conference on Machine Learning Technologies*, Pages 84–90, 2024.
#. Y\. Tong, S. Xiong, X. He, et al. `RoeNet: Predicting discontinuity of hyperbolic systems from continuous data <https://doi.org/10.1002/nme.7406>`_. *Int J Numer Methods Eng*, 125(6), e7406, 2024.
#. H\. Kikumoto, Y. Wang, B. Zhang, H. Jia. `Enhanced Wind Velocity and Pressure Measurement Around Buildings Using Physics-Informed Neural Networks: A Case Study with a Two-Dimensional Urban Street Canyon <https://doi.org/10.1007/978-981-97-8309-0_53>`_. *Lecture Notes in Civil Engineering*, Volume 553. Springer, Singapore, 2025.
#. C\. B. Ribeiro. `Advanced Numerical Solution of Navier-Stokes Equations with Energy Conservation: A Physics-Informed Neural Networks Approach to Revolutionize Computational Fluid Dynamics <https://www.researchgate.net/publication/386323233>`_. December 2024.
#. L\. Shang, Y. Zhao, S. Zheng, J. Wang, T. Zhang, J. Wang. `Quantification of gradient energy coefficients using physics-informed neural networks <https://doi.org/10.1016/j.ijmecsci.2024.109210>`_. *International Journal of Mechanical Sciences*, Volume 273, 109210, 2024.
#. Z\. Hu, A. Yang, S. Xu, N. Li, Q. Wu, Y. Sun. `Prediction of soliton evolution and parameters evaluation for a high-order nonlinear Schrödinger–Maxwell–Bloch equation in the optical fiber <https://doi.org/10.1016/j.physleta.2024.130182>`_. *Physics Letters A*, Volume 531, 130182, 2025.
#. D\. Bonnet-Eymard, A. Persoons, M. Faes, D. Moens. `Separable Physics-Informed Neural Networks for Robust Inverse Quantification in Solid Mechanics <https://doi.org/10.5281/zenodo.14039660>`_. *International Symposium on Reliability Engineering and Risk Management (ISRERM)*, October 2024.
#. Z\.-Q. Zhang, et al. `Physics-Informed Neural Network Approaches in Quantum Simulations <https://doi.org/10.1088/1742-6596/2891/6/062023>`_. *J. Phys.: Conf. Ser.*, 2891, 062023, 2024.
#. J\. R. Naujoks, A. Krasowski, M. Weckbecker, T. Wiegand, S. Lapuschkin, W. Samek, R. P. Klausen. `PINNfluence: Influence Functions for Physics-Informed Neural Networks <https://arxiv.org/abs/2409.08958>`_. *arXiv preprint arXiv:2409.08958*, 2024.
#. C\. J. McDevitt, J. Arnaud, X. Z. Tang. `An Efficient Surrogate Model of Secondary Electron Formation and Evolution <https://arxiv.org/abs/2412.13044>`_. *arXiv preprint arXiv:2412.13044*, 2024.
#. Z\. Wu, L. J. Jiang, S. Sun, P. Li. `A Hard Constraint and Domain Decomposition Based Physics-Informed Neural Network Framework for Nonhomogeneous Transient Thermal Analysis <https://ieeexplore.ieee.org/document/10562356>`_. *IEEE Transactions on Components, Packaging and Manufacturing Technology*, 2024.
#. S\. Song, H. Jin. `Identifying constitutive parameters for complex hyperelastic materials using physics-informed neural networks <https://pubs.rsc.org/en/content/articlelanding/2024/sm/d4sm00001c>`_. *Soft Matter*, 20(30), 5915–5926, 2024.
#. A\. Ahmad, A. Khan. `Pricing Rainbow Options Using Deep Learning <https://doi.org/10.20944/preprints202408.2226.v2>`_. *Preprints*, 2024.
#. T\. Sahin, D. Wolff, M. von Danwitz, A. Popp. `Towards a Hybrid Digital Twin: Fusing Sensor Information and Physics in Surrogate Modeling of a Reinforced Concrete Beam <https://doi.org/10.1109/SDF63218.2024.10773885>`_. *2024 Sensor Data Fusion: Trends, Solutions, Applications (SDF)*, Bonn, Germany, pp. 1–8, 2024.
#. A\. W. Corrêa do Lago, D. H. Braz de Sousa, P. H. Domingues, M. Daneker, L. Lu, H. V. H. Ayala. `Physics-informed and black-box identification of robotic actuator with a flexible joint <https://doi.org/10.1016/j.ifacol.2024.08.538>`_. *IFAC-PapersOnLine*, 58(15), Pages 259–264, 2024.
#. W\. Hu, S. Zheng, C. Dong, M. Chen, J.-X. Fei, R. Gao. `High-Order Partial Differential Equations Solved by the Improved Self-Adaptive PINNs <https://ssrn.com/abstract=5056812>`_. *SSRN*, 2024.
#. H\. Mertens, F. Zhu. `Comparative Analysis of Uncertainty Quantification Models in Active Learning for Efficient System Identification of Dynamical Systems <https://doi.org/10.1109/CASE59546.2024.10711530>`_. *2024 IEEE 20th International Conference on Automation Science and Engineering (CASE)*, Bari, Italy, pp. 1869–1876, 2024.
#. H\. Zhang, L. Liu, L. Lu. `Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity <https://arxiv.org/abs/2410.13141>`_. *arXiv preprint arXiv:2410.13141*, 2024.
#. C\. J. McDevitt, J. Arnaud, X.-Z. Tang. `A Physics-Constrained Deep Learning Treatment of Runaway Electron Dynamics <https://arxiv.org/abs/2412.12980>`_. *arXiv preprint arXiv:2412.12980*, 2024.
#. W\. Quan, X. Ma, Z. Shang, K. Zhao, M. Su, Z. Dong. `Hybrid Physics-Data-Driven Model for Temperature Field Prediction of Asphalt Pavement Based on Physics-Informed Neural Network <http://dx.doi.org/10.2139/ssrn.5080097>`_. *SSRN*, 2024.
#. S\. Savović, M. Ivanović, B. Drljača, A. Simović. `Numerical Solution of the Sine–Gordon Equation by Novel Physics-Informed Neural Networks and Two Different Finite Difference Methods <https://doi.org/10.3390/axioms13120872>`_. *Axioms*, 13(12), 872, 2024.
#. C\.-E. Chiu, A. Roy, S. Cechnicka, A. Gupta, A. Levy Pinto, C. Galazis, K. Christensen, D. Mandic, M. Varela. `Physics-Informed Neural Networks can accurately model cardiac electrophysiology in 3D geometries and fibrillatory conditions <https://arxiv.org/abs/2409.12712>`_. *arXiv preprint arXiv:2409.12712*, 2024.
#. B\. Bhaumik, S. Changdar, S. Chakraverty, S. De. `Effects of viscosity and induced magnetic fields on weakly nonlinear wave transmission in a viscoelastic tube using physics-informed neural networks <https://doi.org/10.1063/5.0235391>`_. *Physics of Fluids*, 36(12), 121902, 2024.
#. T\. Sahin, M. von Danwitz, A. Popp. `Solving forward and inverse problems of contact mechanics using physics-informed neural networks <https://doi.org/10.1186/s40323-024-00265-3>`_. *Advances in Modeling and Simulation in Engineering Sciences*, 11, 11, 2024.
#. V\. Kungurtsev, Y. Peng, J. Gu, S. Vahidian, A. Quinn, F. Idlahcen, Y. Chen. `Dataset Distillation from First Principles: Integrating Core Information Extraction and Purposeful Learning <https://arxiv.org/abs/2409.01410>`_. *arXiv preprint arXiv:2409.01410*, 2024.
#. J\. Duan, H. Zhao, J. Song. `Spatial domain decomposition-based physics-informed neural networks for practical acoustic propagation estimation under ocean dynamics <https://doi.org/10.1121/10.0026025>`_. *Journal of the Acoustical Society of America*, 155(5), 3306–3321, 2024.
#. S\. Changdar, B. Bhaumik, N. Sadhukhan, S. Pandey, S. Mukhopadhyay, S. De, S. Bakalis. `A Hybridized Approach on Physics-Informed Neural Networks and Symbolic Regression for Simulating Nonlinear Wave Dynamics in Arterial Blood Flow <http://dx.doi.org/10.2139/ssrn.4952356>`_. *SSRN*, 2024.
#. W\. Wu, M. Daneker, C. Herz, H. Dewey, J. A. Weiss, A. M. Pouch, L. Lu, M. A. Jolley. `ADEPT: A Noninvasive Method for Determining Elastic Properties of Valve Tissue <https://arxiv.org/abs/2409.19081>`_. *arXiv preprint arXiv:2409.19081*, 2024.
#. S\. Changdar, B. Bhaumik, N. Sadhukhan, S. Pandey, S. Mukhopadhyay, S. De, S. Bakalis. `Integrating symbolic regression with physics-informed neural networks for simulating nonlinear wave dynamics in arterial blood flow <https://doi.org/10.1063/5.0247888>`_. *Physics of Fluids*, 36(12), 121924, 2024.
#. H\.-Q. Yang, C. Shi, L. Zhang. `Ensemble learning of soil–water characteristic curve for unsaturated seepage using physics-informed neural networks <https://doi.org/10.1016/j.sandf.2024.101556>`_. *Soils and Foundations*, 65(1), 101556, 2025.
#. M\. Peng, H. Tang, Y. Kou. `Adversarial and self-adaptive domain decomposition physics-informed neural networks for high-order differential equations with discontinuities <http://dx.doi.org/10.2139/ssrn.5037580>`_. *SSRN*, 2024.
#. H\. Wang, G. Fang, B. Gao, B. Wang, S. Meng. `Inversion of spatially distributed elastic moduli of 2.5D woven composites based on DIC strain field using PINN method <http://dx.doi.org/10.2139/ssrn.4851306>`_. *SSRN Electronic Journal*, 2024.
#. L\. Novák, H. Sharma, M. D. Shields. `Physics-informed polynomial chaos expansions <https://doi.org/10.1016/j.jcp.2024.112926>`_. *Journal of Computational Physics*, Volume 506, 112926, 2024.
#. J\.-J. Zhang, N. Cheng, F\.-P. Li, X\.-C. Wang, J\.-N. Chen, L\.-G. Pang, D. Meng. `Symmetry Breaking in Neural Network Optimization: Insights from Input Dimension Expansion <https://arxiv.org/abs/2409.06402>`_. *arXiv preprint arXiv:2409.06402*, 2024.
#. D\. Sitalo, A. Ogueda-Oliva, P. Seshaiyer. `Data-Driven Mathematical Modeling and Simulation of Migration Dynamics During the Russian-Ukrainian War <https://doi.org/10.61403/2473-5493.1093>`_. *Spora: A Journal of Biomathematics*, Vol. 10, 83–90, 2024.
#. J\. Zhao, Z. Tian, X. Zhang, Z. Duan, J. Lu. `Kinetics Parameter Identification of Chain Shuttling Polymerization Based on Physics-Informed Neural Networks <https://doi.org/10.1016/j.ifacol.2024.08.334>`_. *IFAC-PapersOnLine*, 58(14), 184–191, 2024.
#. K\. Yuan, C. Bauinger, X. Zhang, P. Baehr, M. Kirchhart, D. Dabert, A. Tousnakhoff, P. Boudier, M. Paulitsch. `Fully-fused Multi-Layer Perceptrons on Intel Data Center GPUs <https://arxiv.org/abs/2403.17607>`_. *arXiv preprint arXiv:2403.17607*, 2024.
#. Z\. Huang, L. An, Y. Ye, X. Wang, H. Cao, Y. Du, M. Zhang. `A broadband modeling method for range-independent underwater acoustic channels using physics-informed neural networks <https://doi.org/10.1121/10.0034458>`_. *J. Acoust. Soc. Am.*, 156(5), 3523–3533, 2024.
#. P\. Xiao, M. Zheng, A. Jiao, X. Yang, L. Lu. `Quantum DeepONet: Neural operators accelerated by quantum computing <https://arxiv.org/abs/2409.15683>`_. *arXiv preprint arXiv:2409.15683*, 2024.
#. Y\. Yang, P. He, X. Peng, Q. He. `A number-theoretic method sampling neural network for solving partial differential equations <https://arxiv.org/abs/2411.17039>`_. *arXiv preprint arXiv:2411.17039*, 2025.
#. J\. Cho, S. Nam, H. Yang, S\.-B. Yun, Y. Hong, E. Park. `Separable Physics-Informed Neural Networks <https://proceedings.neurips.cc/paper_files/paper/2023/file/4af827e7d0b7bdae6097d44977e87534-Paper-Conference.pdf>`_. *Advances in Neural Information Processing Systems*, 36, 23761–23788, 2023.
#. C\. Galazis, C\.-E. Chiu, T. Arichi, A\. A. Bharath, M. Varela. `PINNing Cerebral Blood Flow: Analysis of Perfusion MRI in Infants using Physics-Informed Neural Networks <https://arxiv.org/abs/2410.19759>`_. *arXiv preprint arXiv:2410.19759*, 2024.
#. W\. Hu. `A new method to solve the forward and inverse problems for the spatial Solow model by using Physics Informed Neural Networks (PINNs) <https://doi.org/10.1016/j.enganabound.2024.106013>`_. *Engineering Analysis with Boundary Elements*, 169(Part B), 106013, 2024.
#. X\. Wang, C. Luo, D. Jiang, H. Wang, Z. Wang. `Improved design method for gas carburizing process through data-driven and physical information <https://doi.org/10.1016/j.commatsci.2024.113507>`_. *Computational Materials Science*, Volume 247, 113507, 2025.
#. M\. Xie, X. Zhao, D. Zhao, J. Fu, C. Shelton, B. Semlitsch. `Predicting bifurcation and amplitude death characteristics of thermoacoustic instabilities from PINNs-derived van der Pol oscillators <https://doi.org/10.1017/jfm.2024.800>`_. *Journal of Fluid Mechanics*, 998, A46, 2024.
#. A\. Serebrennikova, R. Teubler, L. Hoffellner, E. Leitner, U. Hirn, K. Zojer. `Physics informed neural networks reveal valid models for reactive diffusion of volatiles through paper <https://doi.org/10.1016/j.ces.2023.119636>`_. *Chemical Engineering Science*, Volume 285, 119636, 2024.
#. C\.A. Molina Catricheo, F. Lambert, J. Salomon, et al. `Modeling global surface dust deposition using physics-informed neural networks <https://doi.org/10.1038/s43247-024-01942-2>`_. *Communications Earth & Environment*, 5, 778, 2024.
#. A\. Deresse, T. Dufera. `A deep learning approach: Physics-informed neural networks for solving the 2D nonlinear Sine–Gordon equation <https://doi.org/10.1016/j.rinam.2024.100532>`_. *Results in Applied Mathematics*, 25, 2024.
#. N\. Patel, A. Aykutalp, P. Laguna. `Novel approach to solving Schwarzschild black hole perturbation equations via physics informed neural networks <https://doi.org/10.1007/s10714-024-03322-9>`_. *Gen Relativ Gravit*, 56, 137, 2024.
#. A\. Jesser, K. Krycki, R. G. McClarren, & M. Frank. `Numerical Robustness of PINNs for Multiscale Transport Equations <https://arxiv.org/abs/2412.14683>`_. *arXiv preprint arXiv:2412.14683*, 2024.
#. H\. Wu, H. Luo, Y. Ma, J. Wang, & M. Long. `RoPINN: Region Optimized Physics-Informed Neural Networks <https://arxiv.org/abs/2405.14369>`_. *arXiv preprint arXiv:2405.14369*, 2024.
#. Y\. Zhao, Y. Fei, R. P. Singh, & D. Fu. `Experimental and Numerical Simulation of the High Hydrological Performance of Root-Zone Mixture in Sports Turf <http://dx.doi.org/10.2139/ssrn.5007512>`_. *SSRN*, 2024.
#. H\. Wang, Y. Pu, S. Song, & G. Huang. `Physics-informed Dynamics Representation Learning for Parametric PDEs <https://openreview.net/forum?id=nlHEfTRo0b>`_. *OpenReview*, 2024.
#. J\. Song & Z. Yan. `Data-driven 2D stationary quantum droplets and wave propagations in the amended GP equation with two potentials via deep neural networks learning <https://arxiv.org/abs/2409.02339>`_. *arXiv preprint*, arXiv:2409.02339, 2024.
#. J\. J. Athalathil, B. Vaidya, S. Kundu, V. Upendran & M. C. M. Cheung. `Surface Flux Transport Modeling Using Physics-informed Neural Networks <https://iopscience.iop.org/article/10.3847/1538-4357/ad7d91/meta>`_. *The Astrophysical Journal*, 975(2), 258, 2024.
#. A\. A. Aghaei, M. M. Moghaddam & K. Parand. `PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems <https://arxiv.org/abs/2409.01899>`_. *arXiv preprint*, arXiv:2409.01899, 2024.
#. L\. Shang, S. Zheng, J. Wang & J. Wang. `Physics-informed neural networks incorporating energy dissipation for the phase-field model of ferroelectric microstructure evolution <https://arxiv.org/abs/2409.02959>`_. *arXiv preprint*, arXiv:2409.02959, 2024.
#. K\.-L. Lu, Y\.-M. Su, C. Qiu, Z. Bi & W\.-J. Zhang. `Solving Oscillator ODEs via Soft-constrained Physics-informed Neural Network with Small Data <https://ui.adsabs.harvard.edu/abs/2024arXiv240811077L/abstract>`_. *arXiv e-prints*, arXiv:2408, 2024.
#. Z\. Xiong, Y. Jiang, W. Lu, X. Wang & T. Tian. `Reconstructing and Forecasting Marine Dynamic Variable Fields across Space and Time Globally and Gaplessly <https://arxiv.org/abs/2408.01509>`_. *arXiv preprint*, arXiv:2408.01509, 2024.
#. J\. H. Adler, S. Hocking, X. Hu & S. Islam. `Physics-informed nonlinear vector autoregressive models for the prediction of dynamical systems <https://arxiv.org/abs/2407.18057>`_. *arXiv preprint*, arXiv:2407.18057, 2024.
#. Y\. Chen, H. Yu, C. Liu, J. Xie, J. Han & H. Dai. `Synergistic fusion of physical modeling and data-driven approaches for parameter inference to enzymatic biodiesel production system <https://doi.org/10.1016/j.apenergy.2024.123874>`_. *Applied Energy*, 373, 123874, 2024.
#. D\. Nguyen. `Advanced modeling of the childbirth system using different deep learning methods: from fetal skeleton segmentation to real-time soft tissue deformation <https://theses.hal.science/tel-04831150/>`_. PhD thesis, Centrale Lille Institut, 2024.
#. M\. Y. Hosseini & Y. Shiri. `Flow field reconstruction from sparse sensor measurements with physics-informed neural networks <https://doi.org/10.1063/5.0211680>`_. *Physics of Fluids*, 36(7), 2024.
#. H\. Lu, Q. Wang, W. Tang & H. Liu. `Physics-informed neural networks for fully non-linear free surface wave propagation <https://doi.org/10.1063/5.0210591>`_. *Physics of Fluids*, 36(6), 2024.
#. Y\. Gao, P. Xiao & Z. Li. `Physics-Informed Neural Networks for Solving Underwater Two-dimensional Sound Field <https://ieeexplore.ieee.org/abstract/document/10723708>`_. *2024 OES China Ocean Acoustics (COA)*, pp. 1-4, IEEE, 2024.
#. T\. Sahin, D. Wolff, M. von Danwitz & A. Popp. `Towards a Hybrid Digital Twin: Physics-Informed Neural Networks as Surrogate Model of a Reinforced Concrete Beam <https://arxiv.org/abs/2405.08406>`_. *arXiv preprint*, arXiv:2405.08406, 2024.
#. S\. K. Vemuri, T. Büchner, & J. Denzler. `Estimating soil hydraulic parameters for unsaturated flow using physics-informed neural networks <https://link.springer.com/chapter/10.1007/978-3-031-63759-9_37>`_. In *International Conference on Computational Science*, 338-351, Cham: Springer Nature Switzerland, 2024, June.
#. N\. A. Niewiadomska, P. Maczuga, A. Oliver-Serra, L. Siwik, P. Sepulveda-Salaz, A. Paszyńska, M. Paszyński, & K. Pingali. `Modeling tsunami waves at the coastline of Valparaiso area of Chile with physics informed neural networks <https://link.springer.com/chapter/10.1007/978-3-031-63751-3_14>`_. In *International Conference on Computational Science*, 204-218, Cham: Springer Nature Switzerland, 2024, June.
#. N\. Alzhanov, E. Y. K. Ng, & Y. Zhao. `Three-dimensional physics-informed neural network simulation in coronary artery trees <https://www.mdpi.com/2311-5521/9/7/153>`_. *Fluids*, 9(7), 2024.
#. S\. Sripada, A. U. Gaitonde, J. A. Weibel, & A. M. Marconnet. `Robust inverse parameter fitting of thermal properties from the laser-based Ångstrom method in the presence of measurement noise using physics-informed neural networks (PINNs) <https://doi.org/10.1063/5.0206247>`_. *Journal of Applied Physics*, 135(22):225106, June 2024.
#. T\. Zou, T. Yajima, & Y. Kawajiri. `A parameter estimation method for chromatographic separation process based on physics-informed neural network <https://www.sciencedirect.com/science/article/pii/S0021967324004515>`_. *Journal of Chromatography A*, 1730:465077, 2024.
#. N\. Jha & E. Mallik. `GPINN with neural tangent kernel technique for nonlinear two point boundary value problems <https://doi.org/10.1007/s11063-024-11644-7>`_. *Neural Processing Letters*, 56(3):192, May 2024.
#. H\. Zhang, L. Jiang, X. Chu, Y. Wen, L. Li, Y. Xiao, & L. Wang. `Combining physics-informed graph neural network and finite difference for solving forward and inverse spatiotemporal PDEs <https://doi.org/10.1016/j.cpc.2024.109462>`_. *Computer Physics Communications*, 308, p.109462. 2024.
#. N\. Jha & E. Mallik. `Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs <https://doi.org/10.1088/1402-4896/ad5053>`_. *Physica Scripta*, 99(7):076005, June 2024.
#. J\. H. Harmening, F. Pioch, L. Fuhrig, F.-J. Peitzmann, D. Schramm, & O. el Moctar. `Data-assisted training of a physics-informed neural network to predict the separated Reynolds-averaged turbulent flow field around an airfoil under variable angles of attack <https://doi.org/10.1007/s00521-024-09883-9>`_. *Neural Computing and Applications*, May 2024.
#. H\. Nganguia & D. Palaniappan. `Ciliary propulsion through non-uniform flows <https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/ciliary-propulsion-through-nonuniform-flows/124528EA9A3D0C1E4FC14A4B185FB949>`_. *Journal of Fluid Mechanics*, 986:A14, 2024.
#. A\. T. Deresse & T. T. Dufera. `Exploring physics-informed neural networks for the generalized nonlinear Sine-Gordon equation <https://onlinelibrary.wiley.com/doi/full/10.1155/2024/3328977>`_. *Applied Computational Intelligence and Soft Computing*, 2024(1):3328977, 2024.
#. H\. Qiumei, M. Jiaxuan, & X. Zhen. `Mass-preserving spatio-temporal adaptive PINN for Cahn-Hilliard equations with strong nonlinearity and singularity <https://arxiv.org/pdf/2404.18054>`_, 2024.
#. Z.\ Zhang, J.-H. Lee, L. Sun, & G. X. Gu. `Weak-formulated physics-informed modeling and optimization for heterogeneous digital materials <https://doi.org/10.1093/pnasnexus/pgae1869>`_. *PNAS Nexus*, 3(5):pgae186, May 2024.
#. S\. Gao, Q. Li, M. A. Gosalvez, X. Lin, Y. Xing, & Z. Zhou. `Helium focused ion beam damage in silicon: Physics-informed neural network modeling of Helium bubble nucleation and early growth <https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4795077>`_, 2024.
#. J\. Son, N. Park, H. Kwak, & J. Nam. `Optimizing a physics-informed machine learning model for pulsatile shear-thinning channel flow <https://www.jstage.jst.go.jp/article/rheology/52/2/52_113/_article/-char/ja>`_. *Journal of the Japanese Society of Rheology*, 52(2):113–122, 2024.
#. E\. Raeisi, M. Yavuz, M. Khosravifarsani, Y. Fadaei. `Mathematical modeling of interactions between colon cancer and immune system with a deep learning algorithm <https://doi.org/10.1140/epjp/s13360-024-05111-4>`_. *Eur. Phys. J. Plus*, 139(4), 345, 2024.
#. Z\. Zhang, C. Lin, & B. Wang. `Physics-informed shape optimization using coordinate projection <https://www.nature.com/articles/s41598-024-57137-4>`_. *Scientific Reports*, 14, 6537, 2024.
#. S\. Schoder & F. Kraxberger. `Feasibility study on solving the Helmholtz equation in 3D with PINNs <https://arxiv.org/abs/2403.06623>`_. *arXiv preprint arXiv:2403.06623*, 2024.
#. V\. Trávníková, D. Wolff, N. Dirkes, S. Elgeti, E. von Lieres, & M. Behr. `A model hierarchy for predicting the flow in stirred tanks with physics-informed neural networks <https://arxiv.org/abs/2403.04576>`_. *arXiv preprint arXiv:2403.04576*, 2024.
#. J\. S. Arnaud, T. Mark, & C.J. McDevitt. `A physics-constrained deep learning surrogate model of the runaway electron avalanche growth rate <https://arxiv.org/abs/2403.04948>`_. *arXiv preprint arXiv:2403.04948*, 2024.
#. Y\. D. Hu, X.H. Wang, H. Zhou, & L. Wang. `A priori knowledge-based physics-informed neural networks for electromagnetic inverse scattering <https://ieeexplore.ieee.org/abstract/document/10454009?casa_token=yr3UALpvk1wAAAAA:V7o4wHaZxCBbqqkhpj2NCrs498gBPnIkUVnzunR4rqvDwi7HKyxRHUMEaJl7ZFRgtkhrLxbqtSw>`_. *IEEE Transactions on Geoscience and Remote Sensing*, 2024.
#. R\. C. Sotero, J.M. Sanchez-Bornot, & I. Shaharabi-Farahani. `Parameter estimation in brain dynamics models from resting-state fMRI data using physics-informed neural networks <https://www.biorxiv.org/content/10.1101/2024.02.27.582428v1.full.pdf>`_. *bioRxiv*, 2024-02.
#. W\. Wu, M. Daneker, K.T. Turner, M.A. Jolley, & L. Lu. `Identifying heterogeneous micromechanical properties of biological tissues via physics-informed neural networks <https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11092874/pdf/nihpp-2402.10741v2.pdf>`_. *ArXiv*, 2024.
#. T\. Zhang, R. Yan, S. Zhang, D. Yang, & A. Chen. `Application of Fourier feature physics-information neural network in model of pipeline conveying fluid <https://www.sciencedirect.com/science/article/pii/S026382312400137X?casa_token=M1g8Cv35AOMAAAAA:Me31349RE7cjnIdNwDbSNkkZw-pNRpYMidqM_7G0aLMkblQUXW76ce88ef8nJq5NphrV6EqoH9I>`_. *Thin-Walled Structures*, 198, 111693, 2024.
#. S\. Alkhadhr. `Modeling a clinical acoustic information system using physics-informed machine learning <https://etda.libraries.psu.edu/files/final_submissions/29488>`_. 2024.
#. J\. Shi, K. Manjunatha, M. Behr, F. Vogt, & S. Reese. `A physics-informed deep learning framework for modeling of coronary in-stent restenosis <https://link.springer.com/article/10.1007/s10237-023-01796-1>`_. *Biomechanics and Modeling in Mechanobiology*, 23, 615-629, 2024.
#. C\. Kou, Y. Yin, Y. Zeng, S. Jia, Y. Luo, & X. Yuan. `Physics-informed neural network integrate with unclosed mechanism model for turbulent mass transfer <https://doi.org/10.1016/j.ces.2024.119752>`_. *Chemical Engineering Science*, 288, 119752, 2024.
#. B\. Jang, A. A. Kaptanoglu, R. Gaur, S. Pan, M. Landreman, & W. Dorland. `Grad–Shafranov equilibria via data-free physics informed neural networks <https://doi.org/10.1063/5.0188634>`_. *Physics of Plasmas*, 31, 3, 2024.
#. Z\. Wang, R. Keller, X. Deng, K. Hoshino, T. Tanaka, & Y. Nakahira. `Physics-informed representation and learning: Control and risk quantification <https://ojs.aaai.org/index.php/AAAI/article/view/30169>`_. In *Proceedings of the AAAI Conference on Artificial Intelligence*, 38, 19, 21699-21707, 2024, March.
#. M\. Mircea, D. Garlaschelli, & S. Semrau. `Inference of dynamical gene regulatory networks from single-cell data with physics informed neural networks <https://arxiv.org/abs/2401.07379>`_. *arXiv preprint arXiv:2401.07379*, 2024.
#. R\. Casado-Vara, M. Severt, A. Díaz-Longueira, Á. M. Rey, & J. L. Calvo-Rolle. `Dynamic Malware Mitigation Strategies for IoT Networks: A Mathematical Epidemiology Approach <https://doi.org/10.3390/math12020250>`_. *Mathematics*, 12(2), 250, 2024.
#. P\. Karnakov, S. Litvinov, & P. Koumoutsakos. `Solving inverse problems in physics by optimizing a discrete loss: Fast and accurate learning without neural networks <https://academic.oup.com/pnasnexus/article/3/1/pgae005/7516080>`_. *PNAS Nexus*, 3, pgae005, 2024.
#. J\. Seo. `Solving real-world optimization tasks using physics-informed neural computing <https://www.nature.com/articles/s41598-023-49977-3>`_. *Scientific Reports*, 14(1), 202, 2024.
#. J\. Wu, Y. Wu, G. Zhang, & Y. Zhang. `Variable linear transformation improved physics-informed neural networks to solve thin-layer flow problems <https://www.sciencedirect.com/science/article/pii/S002199912400010X>`_. *Journal of Computational Physics*, 112761, 2024.
#. Y\. Zhu, W. Kong, J. Deng, & X. Bian. `Physics-informed neural networks for incompressible flows with moving boundaries <https://pubs.aip.org/aip/pof/article-abstract/36/1/013617/3105948/Physics-informed-neural-networks-for?redirectedFrom=fulltext>`_. *Physics of Fluids*, 36, 1, 2024.
#. B\. Bhaumik, S. De, & S. Changdar. `Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow <https://www.sciencedirect.com/science/article/pii/S0378475423004469>`_. *Mathematics and Computers in Simulation*, 217, 21-36, 2024. 
#. D\. Coscia, N. Demo, & G. Rozza. `PINA: a PyTorch framework for solving differential equations by deep learning for research and production environments <https://openreview.net/forum?id=9HSPRDSkwU>`_. *ICLR 2024 Workshop on AI4DifferentialEquations In Science*, 2024.
#. S\. Liu, C. Su, J. Yao, Z. Hao, H. Su, Y. Wu, & J. Zhu. `Preconditioning for physics-informed neural networks <https://arxiv.org/abs/2402.00531>` *arXiv preprint arXiv:2402.00531*, 2024.
#. N\. Patel, A. Aykutalp, & P. Laguna. `Calculating quasi-normal modes of Schwarzschild black holes with physics informed neural networks <https://arxiv.org/abs/2401.01440>`_. *arXiv preprint arXiv:2401.01440*, 2024.
#. J\. Li, Y. Lin, & K. Zhang. `Dynamic mode decomposition of the core surface flow inverted from geomagnetic field models <https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2023GL106362>`_. *Geophysical Research Letters*, 51(1), e2023GL106362, 2024.
#. G\. Lau, A. Hemachandra, S. Ng, & B. Low. `PINNACLE: PINN adaptive colLocation and experimental points selection <https://openreview.net/forum?id=GzNaCp6Vcg>`_. *The Twelfth International Conference on Learning Representations*, 2024.
#. J\. M. Tucny, M. Durve, A. Montessori, and S. Succi. `Learning of viscosity functions in rarefied gas flows with physics-informed neural networks <https://www.sciencedirect.com/science/article/pii/S0045793023003390>`_. *Computers Fluids*, 269:106114, 2024.
#. P\. Brendel, V. Medvedev, & A. Rosskopf. `Physics-informed neural networks for magnetostatic problems on axisymmetric transformer geometries <https://ieeexplore.ieee.org/abstract/document/10373037>`_. *IEEE Journal of Emerging and Selected Topics in Industrial Electronics*, 2023.
#. T\. Zhang, D. Wang, & Y. Lu. `RheologyNet: A physics-informed neural network solution to evaluate the thixotropic properties of cementitious materials <https://doi.org/10.1016/j.cemconres.2023.107157>`_. *Cement and Concrete Research*, 168, 107157, 2023.
#. S\. C. Salas, A. O. Alvarado, F. Ortega-culaciati, & P. escapil-inchauspé. `Physics informed neural network for quasistatic fault slip forward and inverse problems <https://ci2ma.udec.cl/wonapde2024/abstract/Cobaise.pdf>`_. 2023.
#. C\. Li, & Z. Han. `Shallow water equations-fused dam-break wave propagation prediction model ensembled with a training process resampling method <https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10396666>`_. *2023 International Conference on Intelligent Computing and Next Generation Networks（ICNGN)*, 1-6. 10.1109/ICNGN59831.2023.10396666. 
#. X\. Yang, Y. Du, L. Li, Z. Zhou, & X. Zhang. `Physics-informed neural network for model prediction and dynamics parameter identification of collaborative robot joints <https://ieeexplore.ieee.org/document/10305255>`_. *IEEE Robotics and Automation Letters*, vol. 8, no. 12, pp. 8462-8469, 2023.
#. S\. H. Radbakhsh, K. Zandi, & M. Nik-bakht. `Physics-informed neural network for analyzing elastic beam behavior <https://www.dpi-proceedings.com/index.php/shm2023/article/view/36810>`_. *Structural Health Monitoring*, 2023.
#. J\. Gong, Y. Han, J. Wu, & G. Hu. `Dynamical behavior of a particle-doped multi-segment dielectric elastomer minimal energy structure <https://iopscience.iop.org/article/10.1088/1361-665X/ad126c/meta>`_. *Smart Materials and Structures*, 33(1), 015016, 2023.
#. S\. Burbulla. `Physics-informed neural networks for transformed geometries and manifolds <https://arxiv.org/abs/2311.15940>`_. *arXiv preprint arXiv:2311.15940*, 2023.
#. J\. Shi, K. Manjunatha, & S. Reese. `Deep learning-based surrogate modeling of coronary in-stent restenosis <https://onlinelibrary.wiley.com/doi/full/10.1002/pamm.202300090>`_. *Proceedings in Applied Mathematics and Mechanics*, 23, e202300090.
#. Y\. Jiang, W. Yang, Y. Zhu, & L. Hong. `Entropy structure informed learning for solving inverse problems of differential equations <https://www.sciencedirect.com/science/article/pii/S096007792300958X>`_. *Chaos, Solitons & Fractals*, Volume 175, Part 2, 2023.
#. A\. Ogueda-Oliva, & P. Seshaiyer. `Literate programming for motivating and teaching neural network-based approaches to solve differential equations <https://www.tandfonline.com/doi/full/10.1080/0020739X.2023.2249901?scroll=top&needAccess=true>`_. *International Journal of Mathematical Education in Science and Technology*, 55(2), 509–542.
#. C\. Li. `Enhancing Navier-Stokes flow learning through the level set approach <https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4641595>`_. *Available at SSRN 4641595*.
#. X\. Zhu, X. Hu, & P. Sun. `Physics-informed neural networks for solving dynamic two-phase interface problems <https://epubs.siam.org/doi/abs/10.1137/22M1517081>`_. *SIAM Journal on Scientific Computing*, 45(6), A2912-A2944, 2023.
#. H\. Patel, A. Panda, T. Nikolaienko, S. Jaso, A. Lopez, & K. Kalyanaraman. `Accurate and fast Fischer-Tropsch reaction microkinetics using PINNs <https://arxiv.org/abs/2311.10456>`_. *arXiv preprint arXiv:2311.10456*, 2023.
#. J\. Plata Salas. `FísicaQuantification of gradient energy coefficients using physics-informed neural networksiciales <http://repositorioinstitucional.uaslp.mx/xmlui/handle/i/8438>`_. *Repositorio Nacional CONACYT*, 2023.
#. N\. Namaki, M. R. Eslahchi, & R. Salehi. `The use of physics-informed neural network approach to image restoration via nonlinear PDE tools <https://www.sciencedirect.com/science/article/pii/S0898122123004431>`_. *Computers & Mathematics with Applications*, 152, 355-363, 2023.
#. A\. Hvatov, D. Aminev, & N. Demyanchuk. `Easy to learn hard to master - how to solve an arbitrary equation with PINN <https://openreview.net/forum?id=0ipxdwZmFR>`_. *NeurIPS 2023 AI for Science Workshop*, 2023.
#. H\. Son, H. Cho, & H. J. Hwang. `Physics-informed neural networks for microprocessor thermal management model <https://ieeexplore.ieee.org/abstract/document/10305255>`_. *IEEE Access*, 11, 122974-122979, 2023.
#. S\. Savović, M. Ivanović, & R. Min. `A comparative study of the explicit finite difference method and physics-informed neural networks for solving the Burgers’ equation <https://www.mdpi.com/2075-1680/12/10/982>`_. *Axioms*, 12(10), 982, 2023.
#. M\. Marian, S. Tremmel. `Physics-Informed Machine Learning—An Emerging Trend in Tribology <https://www.mdpi.com/2075-4442/11/11/463>`_. *Lubricants*, 2023, 11, 463.
#. L\. S. de Oliveira, L. Kunstmann, D. Pina, D. de Oliveira, & M. Mattoso. `PINNProv: Provenance for physics-informed neural networks <https://ieeexplore.ieee.org/abstract/document/10306106>`_. *In 2023 International Symposium on Computer Architecture and High Performance Computing Workshops (SBAC-PADW) (pp. 16-23). IEEE*, 2023.
#. Z\. Wang, Z. Zhou, W. Xu, C. Sun, & R. Yan. `Physics informed neural networks for fault severity identification of axial piston pumps <https://www.sciencedirect.com/science/article/pii/S0278612523002030>`_. *Journal of Manufacturing Systems*, 71, 421-437, 2023.
#. K\. Prantikos, S. Chatzidakis, L. H. Tsoukalas, & A. Heifetz. `Physics-informed neural network with transfer learning (TL-PINN) based on domain similarity measure for prediction of nuclear reactor transients <https://www.nature.com/articles/s41598-023-43325-1>`_. *Scientific Reports*, 13(1), 16840, 2023.
#. K\. Lo, & D. Huang. `On Training Derivative-Constrained Neural Networks <https://arxiv.org/pdf/2310.01649.pdf>`_. *arXiv preprint arXiv:2310.01649*, 2023.
#. M\. Ragoza, & M. Batmanghelich. `Physics-informed neural networks for tissue elasticity reconstruction in magnetic resonance elastography <https://link.springer.com/chapter/10.1007/978-3-031-43999-5_32>`_. *In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 333-343). Cham: Springer Nature Switzerland*, 2023.
#. M\. Severt, R. Casado-Vara, & A. Martín del Rey. `A comparison of Monte Carlo-based and PINN parameter estimation methods for malware identification in IoT networks <https://www.mdpi.com/2227-7080/11/5/133>`_. *Technologies*, 11(5), 133, 2023.
#. O\. Mukhmetov, Y. Zhao, A. Mashekova, V. Zarikas, E. Y. K. Ng, & N. Aidossov. `Physics-informed neural network for fast prediction of temperature distributions in cancerous breasts as a potential efficient portable AI-based diagnostic tool <https://www.sciencedirect.com/science/article/pii/S016926072300500X>`_. *Computer Methods and Programs in Biomedicine*, 242, 107834, 2023.
#. J\. Pan, X. Xiao, L. Guo, & X. Feng. `A high resolution physics-informed neural networks for high-dimensional convection-diffusion-reaction equations <https://www.sciencedirect.com/science/article/pii/S1568494623008906>`_. *Applied Soft Computing*, 148, 110872, 2023.
#. S\. Akins, & F. Zhu. `Comparing active learning performance driven by gaussian processes or bayesian neural networks for constrained trajectory exploration <https://arxiv.org/pdf/2309.16114.pdf>`_. *arXiv preprint arXiv:2309.16114*, 2023.
#. I\. Bendaoud. `Approximation theory via deep neural networks and some applications <https://dspace.univ-ouargla.dz/jspui/bitstream/123456789/34242/1/Imane-Bendaoud.pdf>`_.
#. F\. Tangsijie, & L. Wei. `The buckling analysis of thin-walled structures based on physics-informed neural networks <https://pubs-en.cstam.org.cn/article/doi/10.6052/0459-1879-23-277>`_. *Chinese Journal of Theoretical and Applied Mechanics*, 55(11), 2539-2553, 2023.
#. J\. Ran, X. Hu, X. Yuan, A. Li, & P. Wei. `Physics-Informed neural networks based low thrust orbit transfer design for spacecraft <https://ieeexplore.ieee.org/abstract/document/10295814/authors#authors>`_. *In 2023 CAA Symposium on Fault Detection, Supervision and Safety for Technical Processes (SAFEPROCESS) (pp. 1-7). IEEE*, 2023.
#. L\. Mandl, A. Mielke, S. M. Seyedpour, & T. Ricken. `Affine transformations accelerate the training of physics-informed neural networks of a one-dimensional consolidation problem <https://www.nature.com/articles/s41598-023-42141-x>`_. *Scientific Reports*, 13(1), 15566, 2023.
#. Y\. Xu, & T. Zeng. `Multi-grade deep learning for partial differential equations with applications to the Burgers equation <https://arxiv.org/pdf/2309.07401.pdf>`_. *arXiv preprint arXiv:2309.07401*, 2023.
#. G\. Cappellini, G. Trappolini, E. Staffetti, A. Cristofaro, & M. Vendittelli. `Adaptive estimation of the Pennes' bio-heat equation-II: A NN-based implementation for real-time applications <https://css.paperplaza.net/images/temp/CDC/files/1472.pdf>`_.
#. M\. Vais. `Deep learning for the solution of differential equations <https://dspace.cuni.cz/handle/20.500.11956/185325>`_. 
#. C\. Coelho, M. F. P. Costa, & L. L. Ferrás. `The influence of the optimization algorithm in the solution of the fractional Laplacian equation by neural networks <https://pubs.aip.org/aip/acp/article/2849/1/220002/2909119>`_. *In AIP Conference Proceedings (Vol. 2849, No. 1). AIP Publishing*, 2023.
#. A\. Moreira, M. Philipps, & N. Van Riel. `Parameter estimation of a physiological diabetes model using neural networks <https://ieeexplore.ieee.org/abstract/document/10264904>`_. *In 2023 IEEE Conference on Computational Intelligence in Bioinformatics and Computational Biology (CIBCB) (pp. 1-8). IEEE*, 2023.
#. A\. G. Ogueda-Oliva, A. G. Martínez-Salinas, V. Arunachalam, & P. Seshaiyer. `Machine learning for predicting the dynamics of infectious diseases during travel through physics-informed neural networks <https://www.dl.begellhouse.com/download/article/7d502d01488ec91c/JMLMC0403(2)-47213.pdf>`_. *Journal of Machine Learning for Modeling and Computing*, 4(3), 2023.
#. S\. Y. Xu, Q. Zhou, & W. Liu. `Prediction of soliton evolution and equation parameters for NLS-MB equation based on the phPINN algorithm <https://link.springer.com/article/10.1007/s11071-023-08824-w>`_. *Nonlinear Dynamics*, 111(19), 18401-18417, 2023.
#. T\. Kapoor, A. Chandra, D. M. Tartakovsky, H. Wang, A. Nunez, & R. Dollevoet. `Neural oscillators for generalization of physics-informed machine learning <https://arxiv.org/pdf/2308.08989.pdf>`_. *arXiv preprint arXiv:2308.08989*, 2023.
#. S\. P. Moschou, E. Hicks, R. Y. Parekh, D. Mathew, S. Majumdar, & N. Vlahakis. `Physics-informed neural networks for modeling astrophysical shocks <https://iopscience.iop.org/article/10.1088/2632-2153/acf116/pdf>`_. *Machine Learning: Science and Technology*, 4(3), 035032, 2023.
#. S\. Auddy, R. Dey, N. J. Turner, & S. Basu. `GRINN: A Physics-informed neural network for solving hydrodynamic systems in the presence of self-gravity <https://arxiv.org/pdf/2308.08010.pdf>`_. *arXiv preprint arXiv:2308.08010*, 2023.
#. D\. Gazoulis, I. Gkanis, & C. G. Makridakis. `On the stability and convergence of physics informed neural networks <https://arxiv.org/pdf/2308.05423.pdf>`_. *arXiv preprint arXiv:2308.05423*, 2023.
#. Y\. D. Hu, X. H. wang, H. Zhou, L. Wang, & B. Z. Wang. `A more general electromagnetic inverse scattering method based on physics-informed neural network <https://ieeexplore.ieee.org/abstract/document/10203003>`_. *IEEE Transactions on Geoscience and Remote Sensing*, 2023.
#. H\. W. Park, & J. H. Hwang. `Predicting the early-age time-dependent behaviors of a prestressed concrete beam by using physics-informed neural network <https://www.mdpi.com/1424-8220/23/14/6649>`_. *Sensors*, 23(14), 6649, 2023.
#. D\. Bonnet-Eymard, A. Persoons, M. G. Faes, & D. Moens. `Quantifying uncertainty of physics-informed neural networks for continuum mechanics applications <https://www.researchgate.net/profile/Matthias-Faes/publication/372419185_QUANTIFYING_UNCERTAINTY_OF_PHYSICS-INFORMED_NEURAL_NETWORKS_FOR_CONTINUUM_MECHANICS_APPLICATIONS/links/64b6455c95bbbe0c6e4516a9/QUANTIFYING-UNCERTAINTY-OF-PHYSICS-INFORMED-NEURAL-NETWORKS-FOR-CONTINUUM-MECHANICS-APPLICATIONS.pdf>`_.
#. M\. Z. Asadzadeh, K. Roppert, & P. Raninger. `Material data identification in an induction hardening test rig with physics-informed neural networks <https://www.mdpi.com/1996-1944/16/14/5013>`_. *Materials*, 16(14), 5013, 2023.
#. W\. Xuan, H. Lou, S. Fu, Z. Zhang, & N. Ding. `Physics-informed deep learning method for the refrigerant filling mass flow metering <https://www.sciencedirect.com/science/article/pii/S0955598623001140>`_. *Flow Measurement and Instrumentation*, 93, 102418, 2023.
#. S\. Alkhadhr and M. Almekkawy. `Wave equation modeling via physics-informed neural networks: Models of soft and hard constraints for initial and boundary conditions <https://www.mdpi.com/1424-8220/23/5/2792>`_. *Sensors*, 23(5), 2023.
#. M\. Bazmara, M. Mianroodi, and M. Silani. `Application of physics-informed neural networks for nonlinear buckling analysis of beams <https://link.springer.com/article/10.1007/s10409-023-22438-x>`_. *Acta Mechanica Sinica*, 39(6):422438, 2023.
#. M\. Bazmara, M. Silani, M. Mianroodi, and M. sheibanian. `Physics-informed neural networks for nonlinear bending of 3D functionally graded beam <https://www.sciencedirect.com/science/article/pii/S2352012423001169?casa_token=SthbqcM_IV0AAAAA:W-mYynSBMYLx6bdlXk7g_wsp15CjzX2OZcvBg_M_az_SgfJH9gPxdaxC259-GVANqw5V0U3MjA>`_. *Structures*, 49:152-162, 2023.
#. J\. Duan and H. Zhao. `PINNs for sound propagation and sound speed field estimation simultaneously <https://ieeexplore.ieee.org/abstract/document/10244314>`_. In *OCEANS 2023 - Limerick*, p. 1-5, 2023.
#. A\. Fallah and M. M. Aghdam. `Physics-informed neural network for bending and free vibration analysis of three-dimensional functionally graded porous beam resting on elastic foundation <https://link.springer.com/article/10.1007/s00366-023-01799-7>`_. *Engineering with Computers*, 2023.
#. F\. Fonseca. `A solution of a 3D cartesian poisson-boltzmann equation <https://www.m-hikari.com/ces/ces2023/ces1-2023/p/fonsecaCES1-2023.pdf>`_. *Contemporary Engineering Sciences*, 16(1):1-10, 2023.
#. L\. Fritschi and K. Lenk. `Parameter inference for an astrocyte model using machine learning approaches <https://www.biorxiv.org/content/10.1101/2023.05.16.540982v1.abstract>`_. *bioRxiv*, p. 2023-05, 2023.
#. Z\. Gong, Y. Chu, and S. Yang. `Physics-informed neural networks for solving 2-D magnetostatic fields <https://ieeexplore.ieee.org/abstract/document/10141630?casa_token=1fiPdhDF_ewAAAAA:RH2pE-IZj1prPA4IMx6Dn3fLLkTu9jkp09g-CKPC7WbW6xN9iwT-pYJNAJcXp0d4eOj8f0ILmg>`_. *IEEE Transactions on Magnetics*, 59(11):1-5, 2023.
#. M\. A. Haddou. `Quasi-normal modes of near-extremal black holes in dRGT massive gravity using physics-informed neural networks (PINNs) <https://arxiv.org/abs/2303.02395>`_. 2023.
#. Z\. Hao, J. Yao, C. Su, H. Su, Z. Wang, F. Lu, Z. Xia, Y. Zhang, S. Liu, L. Lu, & J. Zhu. `PINNacle: A comprehensive benchmark of physics-informed neural networks for solving PDEs <https://arxiv.org/abs/2306.08827>`_. *arXiv preprint arXiv:2306.08827*, 2023.
#. H\. Huang, Y. Li, Y. Xue, K. Zhang, and F. Yang. `A deep learning approach for solving diffusion-induced stress in large-deformed thin film electrodes <https://www.sciencedirect.com/science/article/pii/S2352152X23004346?casa_token=ARNFtXjuW4EAAAAA:05XIn4QUMFNaXXASRd_mjk4ETNedF0KsCNI4Cf-gT3n9vWJuRHWLg0_pdqg0zp8L65_r8-359w>`_. *Journal of Energy Storage*, 63:107037, 2023.
#. Y\. Huang, Z. Xu, C. Qian, & L. Liu. `Solving free-surface problems for non-shallow water using boundary and initial conditions-free physics-informed neural network (bif-PINN) <https://www.sciencedirect.com/science/article/pii/S0021999123000980?casa_token=8BLfjYZRFEIAAAAA:SGSFk_XdumSVBV83RReAstGdGtIEY5Z5TLZzYKXYX_vp8b0OqksWmhvJclVEctAI8wH3hcVDuZc>`_. *Journal of Computational Physics*, p.112003, 2023.
#. H\. Jung, J. Gupta, B. Jayaprakash, M. Eagon, H. P. Selvam,C. Molnar, W. Northrop, and S. Shekhar. `A survey on solving and discovering differential equations using deep neural networks <https://arxiv.org/abs/2304.13807>`_. 2023.
#. N\. V. Jagtap, M. Mudunuru, and K. Nakshatrala. `CoolPINNs: A physics-informed neural network modeling of active cooling in vascular systems <https://www.sciencedirect.com/science/article/pii/S0307904X23001725?casa_token=ghG1n0pXaUoAAAAA:LcHDc1asGmreWK3OWe8k-83CBilTqjf05vzX5pD6rUbicE8BQfQeLebUQlAu4tR7O4fRye3yCQ>`_. *Applied Mathematical Modelling*, 122:265-287, 2023.
#. Q\. Jiang, X. Wang, M. Yu, M. Tang, B. Zhan, and S. Dong. `Study on pile driving and sound propagation in shallow water using physics-informed neural network <https://www.sciencedirect.com/science/article/pii/S0029801823010685?casa_token=ipvWqqWrzjgAAAAA:nFwBdP8DinBH9Xu4OCtLNxiztID_91YnWAVGzk4CTLFUAqx_PdKxy1e6DlERhXSZ_5slpoPrbw>`_. *Ocean Engineering*, 281:114684, 2023.
#. G\. Lei, N. Ma, B. Sun, K. Mao, B. Chen, and Y. Zhai. `Physics-informed neural networks for solving nonlinear Bloch equations in atomic magnetometry <https://iopscience.iop.org/article/10.1088/1402-4896/ace290/meta?casa_token=cH6Su_4_ui0AAAAA:15B5F5gx8ck_PqH92RQQIWTyojxKKSWYhOh8NJ1i-rEoTQiYu3TwBVYjLBJIAex3k8yZArtlp_vSWATjg_7XW7c42g>`_. *Physica Scripta*, 98(8):085010, 2023.
#. C\. Li, Z. Han, Y. Li, M. Li, W. Wang, J. Dou, L. Xu, and G. Chen. `Physical information-fused deep learning model ensembled with a subregion-specific sampling method for predicting flood dynamics <https://www.sciencedirect.com/science/article/pii/S0022169423004079?casa_token=Ysw-awi8xM4AAAAA:QyCKJYA5w1WdvotsegNKYt87EEPpbDH3MxEKiSQmIuT-LGW34S6FvxmFiN2GuPvND0hH2BBaAg>`_. *Journal of Hydrology*, 620:129465, 2023.
#. S\. Li, G. Wang, Y. Di, L. Wang, H. Wang, and Q. Zhou. `A physics-informed neural network framework to predict 3D temperature field without labeled data in process of laser metal deposition <https://www.sciencedirect.com/science/article/pii/S0952197623000921?casa_token=wKrzmLM53moAAAAA:Dj0ztY7QBjSZLE0-_GuF5F6Td1yEygz2jZN5rJ94cL9Vzz7QXkiGmoLJL5WMpH-crx9RLqI87A>`_. *Engineering Applications of Artificial Intelligence*, 120:105908, 2023.
#. R\. Liang, W. Liu, L. Xu, X. Qu, and S. Kaewunruen. `Solving elastodynamics via physics-informed neural network frequency domain method <https://www.sciencedirect.com/science/article/pii/S0020740323004770?casa_token=ph_v6mL_Z1gAAAAA:j295td9FH7YZcVTyGNzQnFsOiWKtVKFtWeP6DNsEYkrTZapT9b88TwW7YlJOYo4_nRIOAY6g>`_. *International Journal of Mechanical Sciences*, 258:108575, 2023.
#. H\. Liu, C. Hou, H. Qu, and Y. Hou. `Learning mean curvature-based regularization to solve the inverse variational problems from noisy data <https://link.springer.com/article/10.1007/s11760-023-02544-9>`_. *Signal, Image and Video Processing*, 17(6):3193-3200, 2023.
#. M\. L. Mamud, M. K. Mudunuru, S. Karra, and B. Ahmmed. `Do physics-informed neural networks satisfy local and global mass balance <https://arxiv.org/abs/2305.13920>`_? 2023.
#. C\. McDevitt. `A physics-informed deep learning model of the hot tail runaway electron seed <https://arxiv.org/abs/2306.13224>`_. 2023.
#. P\. P. Nagrani, R. V. Kulkarni, P. U. Kelkar, R. D. Corder, K. A. Erk, A. M. Marconnet, and I. C. Christov. `Data-driven rheological characterization of stress buildup and relaxation in thermal greases <https://arxiv.org/abs/2304.12104>`_. *Journal of Rheology*, 67(6):1129-1140, 2023.
#. Y\. Patel, V. Mons, O. Marquet, and G. Rigas. `Turbulence model augmented physics informed neural networks for mean flow reconstruction <https://arxiv.org/abs/2306.01065>`_. 2023.
#. F\. Pioch, J. H. Harmening, A. M. Müller, F. Peitzmann, D. Schramm, and O. el Moctar. `Turbulence modeling for physics-informed neural networks: Comparison of different RANS models for the backward-facing step flow <https://www.mdpi.com/2311-5521/8/2/43>`_. *Fluids*, 8(2), 2023.
#. P\. Sharma, L. Evans, M. Tindall, and P. Nithiarasu. `Stiff-PDEs and physics-informed neural networks <https://link.springer.com/article/10.1007/s11831-023-09890-4>`_. *Archives of Computational Methods in Engineering*, p. 1-30, 2023.
#. C\. Soyarslan and M. Pradas. `Physics-informed machine learning in the determination of effective thermomechanical properties <https://www.mrforum.com/product/9781644902479-175>`_. *Material Forming - The 26th International ESAFORM Conference on Material Forming - ESAFORM 2023*, Materials Research Proceedings, p. 1621-1630, 2023.
#. Z\. Wang and Y. Nakahira. `A generalizable physics-informed learning framework for risk probability estimation <https://proceedings.mlr.press/v211/wang23a.html>`_. *Proceedings of The 5th Annual Learning for Dynamics and Control Conference*, Vol. 211 of *Proceedings of Machine Learning Research*, p. 358-370. PMLR, 15-16, 2023.
#. W\. Xuan, H. Lou, S. Fu, Z. Zhang, and N. Ding. `Physics-informed deep learning method for the refrigerant filling mass flow metering <https://www.sciencedirect.com/science/article/pii/S0955598623001140?casa_token=xhB__ZuLF_wAAAAA:V_02cWSKx8r7JwEHeQzHJwpSOBCukhwDGTdhiaP0gbsLAg1Zj-5X5GsOOZg1nxBxaUEw2o-j>`_. *Flow Measurement and Instrumentation*, 93:102418, 2023.
#. J\. Yao, C. Su, Z. Hao, S. Liu, H. Su, and J. Zhu. `MultiAdam: Parameter-wise scale-invariant optimizer for multiscale training of physics-informed neural networks <https://arxiv.org/abs/2306.02816>`_. 2023.
#. X\. Zeng, S. Zhang, C. Ren, and T. Shao. `Physics informed neural networks for electric field distribution characteristics analysis <https://iopscience.iop.org/article/10.1088/1361-6463/acbec3/meta?casa_token=kSR8jrv_CucAAAAA:4pet2OAmsFh9uQLzJWzE0-c_QDBeJ3_jqeq0iTM4P_wI5VwEfpEI-c-j1HGnpoQgwfKtOGVPh8IDOPuT1dFy3GNJK2E>`_. *Journal of Physics D: Applied Physics*, 56(16):165202, 2023.
#. Z\. Zhang. `Modeling and control for renal anemia treatment with erythropoietin using physics-informed neural network <https://era.library.ualberta.ca/items/3fe4c633-84ae-4c42-ad70-aa74a5e03b6e>`_. 2023.
#. Z\. Zhang and Z. Li. `Haemoglobin response modelling under erythropoietin treatment: Physiological model-informed machine learning method <https://onlinelibrary.wiley.com/doi/full/10.1002/cjce.24922?casa_token=7w5PAgcDaaEAAAAA%3AMU8thygdlKxIV6JwR3FzffNNOm-g80wSIoIKCNFTs8tDJ56fBlrjdNEMIAKJhQW0CDnKe8Tj3WmKFDg>`_. *The Canadian Journal of Chemical Engineering*, 2023.
#. M\. Zhou and G. Mei. `Transfer learning-based coupling of smoothed finite element method and physics-informed neural network for solving elastoplastic inverse problems <https://www.mdpi.com/2227-7390/11/11/2529>`_. *Mathematics*, 11(11), 2023.
#. V\. Medvedev, A. Erdmann, & A. Rosskopf. `Modeling of near- and far-field diffraction from EUV absorbers using physics-informed neural networks <https://ieeexplore.ieee.org/abstract/document/10221390>`_. *Photonics & Electromagnetics Research Symposium (PIERS)*, 297-305, 2023.
#. B\. Fan, E. Qiao, A. Jiao, Z. Gu, W. Li, & L. Lu. `Deep learning for solving and estimating dynamic macro-finance models <https://doi.org/10.48550/arXiv.2305.09783>`_. *arXiv preprint arXiv:2305.09783*, 2023.
#. T\. Grossmann, U. Komorowska, J. Latz, & C. Schönlieb. `Can physics-informed neural networks beat the finite element method <https://arxiv.org/pdf/2302.04107.pdf>`_? *arXiv preprint arXiv:2302.04107*, 2023.
#. L\. Sliwinski, & G. Rigas. `Mean flow reconstruction of unsteady flows using physics-informed neural networks <https://www.cambridge.org/core/journals/data-centric-engineering/article/mean-flow-reconstruction-of-unsteady-flows-using-physicsinformed-neural-networks/FA2A09B976B0ACE4C8C2CEA9205C540D>`_. *Data-Centric Engineering*, 4, p.e4, 2023.
#. E\. Lorin, & X. Yang. `Schwarz waveform relaxation-learning for advection-diffusion-reaction equations <https://www.sciencedirect.com/science/article/pii/S0021999122007203?casa_token=hZsXqzdjoXoAAAAA:czw_GhUSII7WfE0UehrS6UXadNdldXqO_pN-zRU4r480Wn_fB_PswseUR_fnd4WmPPYk-Tx2YVU>`_. *Journal of Computational Physics*, 473, p.111657, 2023.
#. C\. Wu, M. Zhu, Q. Tan, Y. Kartha, & L. Lu. `A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks <https://doi.org/10.1016/j.cma.2022.115671>`_. *Computer Methods in Applied Mechanics and Engineering*, 403, 115671, 2023.
#. S\. Carney, A. Gangal, & L. Kim. `Physics informed neural networks for elliptic equations with oscillatory differential operators <https://arxiv.org/pdf/2212.13531.pdf>`_. *arXiv preprint arXiv:2212.13531*, 2022.
#. R\. Usman, & D. Amato. `ML-Ops pipeline for improved physics-informed ODE modeling <https://www.researchgate.net/profile/Raihaan_Usman/publication/366250867_ML-Ops_Pipeline_for_Improved_Physics-Informed_ODE_Modelling/links/6399c269095a6a777429ffeb/ML-Ops-Pipeline-for-Improved-Physics-Informed-ODE-Modelling.pdf>`_. 2022.
#. S\. Saqlain, W. Zhu, E. Charalampidis, & P. Kevrekidis. `Discovering governing equations in discrete systems using PINNs <https://arxiv.org/pdf/2212.00971.pdf>`_. *arXiv preprint arXiv:2212.00971*, 2022.
#. W\. Wu, M. Daneker, M. Jolley, K. Turner, & L. Lu. `Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics <https://link.springer.com/article/10.1007/s10483-023-2995-8>`_. *Applied Mathematics and Mechanics*, 44(7), 1039-1068, 2023.
#. C\. McDevitt, E. Fowler, & S. Roy. `Physics-constrained deep learning of incompressible cavity flows <https://arxiv.org/pdf/2211.06375.pdf>`_. *arXiv preprint arXiv:2211.06375*, 2022.
#. E\. Lorin, & X. Yang. `Time-dependent Dirac equation with physics-informed neural networks: Computation and properties <https://www.sciencedirect.com/science/article/pii/S001046552200193X>`_. *Computer Physics Communications*, 280, p.108474, 2022.
#. Y\. Ji. `Solving singular Liouville equations using deep learning <https://openreview.net/pdf?id=dCZsSsvraQ>`_. *The Symbiosis of Deep Learning and Differential Equations II*, 2022.
#. A\. Serebrennikova, R. Teubler, L. Hoffellner, E. Leitner, U. Hirn, & K. Zojer. `Transport of organic volatiles through paper: Physics-informed neural networks for solving inverse and forward problems <https://link.springer.com/article/10.1007/s11242-022-01864-7>`_. *Transport in Porous Media*, 1-24, 2022.
#. A\. Cornell, A. Ncube, & G. Harmsen. `Determining QNMs using PINNs <https://arxiv.org/pdf/2205.08284.pdf>`_. *arXiv preprint arXiv:2205.08284*, 2022.
#. M\. Mukhametzhanov. `High precision differentiation techniques for data-driven solution of nonlinear PDEs by physics-informed neural networks <https://arxiv.org/pdf/2210.00518.pdf>`_. *arXiv preprint arXiv:2210.00518*, 2022.
#. A\. New, B. Eng, A. Timm, & A. Gearhart. `Tunable complexity benchmarks for evaluating physics-informed neural networks on coupled ordinary differential equations <https://arxiv.org/pdf/2210.07880.pdf>`_. *arXiv preprint arXiv:2210.07880*, 2022.
#. N\. Dhamirah Mohamad, A. Yousif, N. Shaari, H. Mustafa, S. Abdul Karim, A. Shafie, & M. Izzatullah. `Heat transfer modeling with physics-informed neural network (PINN) <https://link.springer.com/chapter/10.1007/978-3-031-04028-3_3>`_. *Intelligent Systems Modeling and Simulation II: Machine Learning, Neural Networks, Efficient Numerical Algorithm and Statistical Methods*, pp. 25-35, Cham: Springer International Publishing, 2022.
#. K\. Prantikos, L. Tsoukalas, & A. Heifetz. `Physics-informed neural network solution of point kinetics equations for a nuclear reactor digital twin <https://doi.org/10.3390/en15207697>`_. *Energies*, 15(20), 7697, 2022.
#. A\. Zhu. `Accelerating parameter inference in diffusion-reaction models of glioblastoma using physics-informed neural networks <https://www.siam.org/Portals/0/Publications/SIURO/Vol15/S147281R.pdf?ver=2022-10-11-122057-963>`_. 2022.
#. Y\. Wang, J. Xing, K. Luo, H. Wang, & J. Fan. `Solving combustion chemical differential equations via physics-informed neural network <https://www.zjujournals.com/eng/article/2022/1008-973X/202210020.shtml>`_. *Journal of Zhejiang University(Engineering Science)*, 2022.
#. Y\. Zhou, M. Dan, Y. Shao, & Y. Zhang. `Deep-neural-network solution of piezo-phototronic transistor based on GaN/AlN quantum wells <https://www.sciencedirect.com/science/article/pii/S2211285522006644?casa_token=X4ir1WGO9EAAAAAA:HJj8gjGsU53HU7LOtY3a4Kr_vvlegAyz8xFnePT9panY5HR1f4xOAKbizXSFMBtAtdgUCHR_gbI>`_. *Nano Energy*, 101, p.107586, 2022.
#. M\. Ferrante, A. Duggento, & N. Toschi. `Physically constrained neural networks to solve the inverse problem for neuron models <https://arxiv.org/pdf/2209.11998.pdf>`_. *arXiv preprint arXiv:2209.11998*, 2022.
#. R\. Hu, Q. Lin, A. Raydan, & S. Tang. `Higher-order error estimates for physics-informed neural networks approximating the primitive equations <https://arxiv.org/pdf/2209.11929.pdf>`_. *arXiv preprint arXiv:2209.11929*, 2022.
#. D\. Sana. `Approximating the wave equation via physics informed neural networks: Various forward and inverse problems <https://dcn.nat.fau.eu/wp-content/uploads/FAUMoD_DaniaSana-InternReport_PINN.pdf>`_. 2022.
#. C\. Garcia-Cervera, M. Kessler, & F. Periago. `Control of partial differential equations via physics-informed neural networks <https://link.springer.com/article/10.1007/s10957-022-02100-4>`_. *Journal of Optimization Theory and Applications*, 1-24, 2022.
#. M\. Takamoto, T. Praditia, R. Leiteritz, D. MacKinlay, F. Alesiani, D. Pflüger, & M. Niepert. `PDEBENCH: An extensive benchmark for scientific machine learning <https://arxiv.org/abs/2210.07182>`_. *arXiv preprint arXiv:2210.07182*, 2022.
#. E\. Pickering,  & T. Sapsis. `Information FOMO: The unhealthy fear of missing out on information. A method for removing misleading data for healthier models <https://arxiv.org/pdf/2208.13080.pdf>`_. *arXiv preprint arXiv:2208.13080*, 2022.
#. I\. Nodozi, J. O'Leary, A. Mesbah, & A. Halder. `A physics-informed deep learning approach for minimum effort stochastic control of colloidal self-assembly <https://arxiv.org/pdf/2208.09182.pdf>`_. *arXiv preprint arXiv:2208.09182*, 2022.
#. Y\. Yang, & G. Mei. `A deep learning-based approach for a numerical investigation of soil–water vertical infiltration with physics-informed neural networks <https://www.mdpi.com/2227-7390/10/16/2945>`_. *Mathematics*, 10(16), p.2945, 2022.
#. L\. Jiang, L. Wang, X. Chu, Y. Xiao, & H. Zhang. `PhyGNNet: Solving spatiotemporal PDEs with physics-informed graph neural network <https://arxiv.org/pdf/2208.04319.pdf>`_. *arXiv preprint arXiv:2208.04319*, 2022.
#. J\. Yu. `Indifference computer experiment for mathematical identification of two variables <https://www.hindawi.com/journals/wcmc/2022/9181840>`_. *Wireless Communications and Mobile Computing*, 2022.
#. C\. Trost, S. Zak, S. Schaffer, C. Saringer, L. Exl, & M. Cordill. `Bridging fidelities to predict nanoindentation tip radii using interpretable deep learning models <https://link.springer.com/article/10.1007/s11837-022-05233-z>`_. *JOM*, 74(6), pp.2195-2205, 2022.
#. F\. Torres, M. Negri, M. Nagy-Huber, M. Samarin, & V. Roth. `Mesh-free Eulerian physics-informed neural networks <https://arxiv.org/pdf/2206.01545.pdf>`_. *arXiv preprint arXiv:2206.01545*, 2022.
#. R\. Anelli. `Physics-informed neural networks for shallow water equations <https://www.politesi.polimi.it/handle/10589/195179>`_. 2022.
#. A\. Konradsson. `Physics-informed neural networks for charge dynamics in air <https://odr.chalmers.se/server/api/core/bitstreams/1c6854a9-f547-4af4-a7b1-dcb2b711b74d/content>`_. *Master’s thesis in Complex Adaptive Systems*, 2022.
#. X\. Wang, J. Li, & J. Li. `A deep learning based numerical PDE method for option pricing <https://link.springer.com/article/10.1007/s10614-022-10279-x>`_. *Computational Economics*, 1-16, 2022.
#. Y\. Wang, X. Han, C. Chang, D. Zha, U. Braga-Neto, & X. Hu. `Auto-PINN: Understanding and optimizing physics-informed neural architecture <https://arxiv.org/abs/2205.13748>`_. *arXiv preprint arXiv:2205.13748*, 2022.
#. B\. Dalen. `Characterization of Cardiac cellular dynamics using physics-informed neural networks <https://www.duo.uio.no/handle/10852/95723>`_. 2022.
#. D\. Wang, J. Xu, F. Gao, C. Wang, R. Gu, F. Lin, T. Rabczuk, & G. Xu. `IGA-Reuse-NET: A deep-learning-based isogeometric analysis-reuse approach with topology-consistent parameterization <https://www.sciencedirect.com/science/article/pii/S0167839622000231?casa_token=9eLzbjE8T2cAAAAA:hTPRjKVgdvubCTtVAzzxgtvB1pM8dBaj0NzSJM3YF0tqR8Fvy2QMRezghUFbbKHzDEWvSklVJJc>`_. *Computer Aided Geometric Design*, 95, p.102087, 2022.
#. A\. Ncube. `Investigating new computational approaches for solving black hole perturbation equations <https://ujcontent.uj.ac.za/view/pdfCoverPage?instCode=27UOJ_INST&filePid=136787410007691&download=true>`_. *Doctoral dissertation, University of Johannesburg*, 2022.
#. C\. Garcıa-Cervera, M. Kessler, & F. Periago. `A first step towards controllability of partial differential equations via physics-informed neural networks <https://www.researchgate.net/profile/Francisco-Periago/publication/359524458_A_first_step_towards_controllability_of_partial_differential_equations_via_physics-informed_neural_networks/links/6242e24c8068956f3c56d679/A-first-step-towards-controllability-of-partial-differential-equations-via-physics-informed-neural-networks.pdf>`_. 2022.
#. L\. Guo, H. Wu, X. Yu, & T. Zhou. `Monte Carlo PINNs: Deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations <https://arxiv.org/pdf/2203.08501.pdf>`_. *arXiv preprint arXiv:2203.08501*, 2022.
#. P\. Escapil-Inchauspé, & G. A. Ruz. `Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems <https://doi.org/10.1016/j.neucom.2023.126826>`_. *Neurocomputing*, 126826, 2023.
#. P\. Escapil-Inchauspé, & G. Ruz. `Physics-informed neural networks for operator equations with stochastic data <https://arxiv.org/abs/2211.10344>`_. *arXiv preprint arXiv:2211.10344*, 2022.
#. H\. Xie, C. Zhai, L. Liu, & H. Yong. `A weighted first-order formulation for solving anisotropic diffusion equations with deep neural networks <https://arxiv.org/pdf/2205.06658.pdf>`_. *arXiv preprint arXiv:2205.06658*, 2022.
#. Y\. Lu, G. Mei, & F. Piccialli.  `A deep learning approach for predicting two-dimensional soil consolidation using physics-informed neural networks (PINN) <https://arxiv.org/pdf/2205.05710.pdf>`_. *arXiv preprint arXiv:2205.05710*, 2022.
#. J\. Yu, L. Lu, X. Meng, & G. Karniadakis. `Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems <https://doi.org/10.1016/j.cma.2022.114823>`_. *Computer Methods in Applied Mechanics and Engineering*, 393, 114823, 2022.
#. A\. Sacchetti, B. Bachmann, K. Löffel, U. Künzi, & B. Paoli. `Neural networks to solve partial differential equations: A comparison with finite elements <https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=9737092>`_. *IEEE Access*, 10, 32271-32279, 2022.
#. Y\. Xue, Y. Li, K. Zhang, & F. Yang. `A physics-inspired neural network to solve partial differential equations - application in diffusion-induced stress <https://pubs.rsc.org/en/content/articlehtml/2022/cp/d1cp04893g>`_. *Physical Chemistry Chemical Physics*, 24(13), 7937-7949, 2022.
#. V\. Santana, M. Gama, J. Loureiro, A. Rodrigues, A. Ribeiro, F. Tavares, A. Barreto Jr, I. Nogueira. `A first approach towards adsorption-oriented physics-informed neural networks: Monoclonal antibody adsorption performance on an ion-exchange column as a case study <https://www.mdpi.com/2305-7084/6/2/21>`_. *ChemEngineering*, 6.2 (2022): 21, 2022.
#. M\. Daneker, Z. Zhang, G. Karniadakis, & L. Lu. `Systems biology: Identifiability analysis and parameter identification via systems-biology-informed neural networks <https://link.springer.com/protocol/10.1007/978-1-0716-3008-2_4>`_. *Computational Modeling of Signaling Networks*, Springer, 87–105, 2023.
#. C\. Martin, A. Oved, R. Chowdhury, E. Ullmann, N. Peters, A. Bharath, & M. Varela. `EP-PINNs: Cardiac electrophysiology characterisation using physics-informed neural networks <https://www.frontiersin.org/articles/10.3389/fcvm.2021.768419/full?&utm_source=Email_to_authors_&utm_medium=Email&utm_content=T1_11.5e1_author&utm_campaign=Email_publication&field=&journalName=Frontiers_in_Cardiovascular_Medicine&id=768419>`_. *Frontiers in cardiovascular medicine*, 2179, 2022.
#. V\. Schäfer. `Generalization of physics-informed neural networks for various boundary and initial conditions <https://elib.dlr.de/185457/1/master_thesis.pdf>`_. *Doctoral dissertation, Technische Universität Kaiserslautern*, 2022.
#. S\. Alkhadhr, & M. Almekkawy.  `A combination of deep neural networks and physics to solve the inverse problem of Burger's equation <https://ieeexplore.ieee.org/document/9630259>`_. *43rd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC)*, 2021.
#. K\. Iversen. `Physics informed neural networks for inverse advection-diffusion problems <https://bora.uib.no/bora-xmlui/handle/11250/2835305>`_. *The University of Bergen*, 2021.
#. S\. Markidis. `The old and the new: Can physics-informed deep-learning replace traditional linear solvers? <https://www.frontiersin.org/articles/10.3389/fdata.2021.669097/full>`_. *Frontiers in Big Data*, 4:669097, 2021.
#. S\. Alkhadhr, X. Liu, & M. Almekkawy. `Modeling of the forward wave propagation using physics-informed neural networks <https://doi.org/10.1109/IUS52206.2021.9593574>`_. *2021 IEEE International Ultrasonics Symposium (IUS)*, pp. 1--4, 2021.
#. L\. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo, & S. Johnson. `Physics-informed neural networks with hard constraints for inverse design <https://doi.org/10.1137/21M1397908>`_. *SIAM Journal on Scientific Computing*, 43(6), B1105--B1132, 2021.
#. K\. Goswami, A. Sharma, M. Pruthi, & R. Gupta. `Study of drug assimilation in human system using physics informed neural networks <https://arxiv.org/abs/2110.05531>`_. *arXiv preprint arXiv:2110.05531*, 2021.
#. C\. Hennigan. `The primal Hamiltonian: A new global approach to monetary policy <https://www.colorado.edu/economics/sites/default/files/attached-files/21-02_-_hennigan.pdf>`_. 2021.
#. S\. Lee, & T. Kadeethum. `Physics-informed neural networks for solving coupled flow and transport system <http://ceur-ws.org/Vol-2964/article_197.pdf>`_. 2021.
#. Y\. Chen, & L. Dal Negro. `Physics-informed neural networks for imaging and parameter retrieval of photonic nanostructures from near-field data <https://arxiv.org/abs/2109.12754>`_. *arXiv preprint arXiv:2109.12754*, 2021.
#. A\. Ncube, G. Harmsen, & A. Cornell. `Investigating a new approach to quasinormal modes: Physics-informed neural networks <https://arxiv.org/abs/2108.05867>`_. *arXiv preprint arXiv:2108.05867*, 2021.
#. M\. Almajid, & M. Abu-Alsaud. `Prediction of porous media fluid flow using physics informed neural networks <https://doi.org/10.1016/j.petrol.2021.109205>`_. *Journal of Petroleum Science and Engineering*, 109205, 2021.
#. J\. Kuhlmann. `Development of a physics-informed machine learning method for aerodynamic and fluids simulation <https://repositorio.uniandes.edu.co/bitstream/handle/1992/53423/24382.pdf?sequence=1&isAllowed=y>`_. 2021.
#. E\. Whalen. `Enhancing surrogate models of engineering structures with graph-based and physics-informed learning <https://dspace.mit.edu/handle/1721.1/139609>`_. *PhD dissertation, Massachusetts Institute of Technology*, 2021.
#. M\. Merkle. `Boosting the training of physics-informed neural networks with transfer learning <https://github.com/mariusmerkle/TL-PINNs/blob/main/Bachelor%20Thesis.pdf>`_. 2021.
#. A\. Warey, T. Han, & S. Kaushik. `Investigation of numerical diffusion in aerodynamic flow simulations with physics informed neural networks <https://arxiv.org/abs/2103.03115>`_. *arXiv preprint arXiv:2103.03115*, 2021.
#. L\. Lu, X. Meng, Z. Mao, & G. Karniadakis. `DeepXDE: A deep learning library for solving differential equations <https://doi.org/10.1137/19M1274067>`_. *SIAM Review*, 63(1), 208--228, 2021.
#. V\. Liu, & H. Yoon. `Prediction of advection and diffusion transport using physics informed neural networks <https://www.osti.gov/servlets/purl/1833169>`_. *2020 AGU Fall Meeting*, 2020. 
#. A\. Yazdani, L. Lu, M. Raissi, & G. Karniadakis. `Systems biology informed deep learning for inferring parameters and hidden dynamics <https://doi.org/10.1371/journal.pcbi.1007575>`_. *PLoS Computational Biology*, 16(11), e1007575, 2020.
#. A\. Kapetanović, A. Šušnjara, & D. Poljak. `Numerical solution and uncertainty quantification of bioheat transfer equation using neural network approach <https://ieeexplore.ieee.org/abstract/document/9243733>`_. 2020 5th International Conference on Smart and Sustainable Technologies (SpliTech)*, 2020.
#. Q\. Zhang, Y. Chen, & Z. Yang. `Data driven solutions and discoveries in mechanics using physics informed neural network <https://www.preprints.org/manuscript/202006.0258>`_. *Preprints*, 2020060258, 2020.
#. W\. Peng, W. Zhou, J. Zhang, & W. Yao. `Accelerating physics-informed neural network training with prior dictionaries <https://arxiv.org/abs/2004.08151>`_. *arXiv preprint arXiv:2004.08151*, 2020.
#. Y\. Chen, L. Lu, G. Karniadakis, & L. Negro. `Physics-informed neural networks for inverse problems in nano-optics and metamaterials <https://doi.org/10.1364/OE.384875>`_. *Optics Express*, 28(8), 11618--11633, 2020.
#. G\. Pang, L. Lu, & G. Karniadakis. `fPINNs: Fractional physics-informed neural networks <https://doi.org/10.1137/18M1229845>`_. *SIAM Journal on Scientific Computing*, 41(4), A2603--A2626, 2019.
#. D\. Zhang, L. Lu, L. Guo, & G. Karniadakis. `Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems <https://doi.org/10.1016/j.jcp.2019.07.048>`_. *Journal of Computational Physics*, 397, 108850, 2019.

Deep neural operators
--------

#. A\. Velikorodny, L. Lu, V. Dudenkov, V. Glanz, B. Chernyavsky, A. Neylon, & P. C. Smits. `Deep operator learning for blood flow modelling in stenosed vessels <https://doi.org/10.1038/s44387-025-00035-5>`_. *npj Artificial Intelligence*, 1, 35, 2025.
#. A\. Jiao, Q. Yan, J. Harlim, & L. Lu. `Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators <https://arxiv.org/abs/2407.05477>`_. *arXiv preprint arXiv:2407.05477*, 2024.
#. J\. Park, & N. Kang. `Point-DeepONet: A Deep Operator Network Integrating PointNet for Nonlinear Analysis of Non-Parametric 3D Geometries and Load Conditions <https://arxiv.org/abs/2412.18362>`_. *arXiv preprint arXiv:2412.18362*, 2024.
#. K\. Lv, J. Wang, Y. Zhang, & H. Yu. `Neural Operators for Adaptive Control of Freeway Traffic <https://arxiv.org/abs/2410.20708>`_. *arXiv preprint arXiv:2410.20708*, 2024.
#. Z\. Li, H. Zheng, N. Kovachki, D. Jin, H. Chen, B. Liu, K. Azizzadenesheli, & A. Anandkumar. `Physics-Informed Neural Operator for Learning Partial Differential Equations <https://doi.org/10.1145/3648506>`_. *Association for Computing Machinery*, 1(3), September 2024.
#. C\. García-Cervera, M. Kessler, P. Pedregal, & F. Periago. `Universal approximation of set-valued maps and DeepONet approximation of the controllability map <https://www.researchgate.net/publication/387067822>`_. *ResearchGate*, December 2024.
#. J\. He, S. Koric, D. Abueidda, A. Najafi, & I. Jasiuk. `Geom-DeepONet: A point-cloud-based deep operator network for field predictions on 3D parameterized geometries <https://doi.org/10.1016/j.cma.2024.117130>`_. *Computer Methods in Applied Mechanics and Engineering*, Volume 429, 117130, 2024.
#. S\. Kushwaha, J. Park, S. Koric, J. He, I. Jasiuk, & D. Abueidda. `Advanced deep operator networks to predict multiphysics solution fields in materials processing and additive manufacturing <https://doi.org/10.1016/j.addma.2024.104266>`_. *Additive Manufacturing*, Volume 88, 104266, 2024.
#. O\. Ovadia, A. Kahana, P. Stinis, E. Turkel, D. Givoli, & G. E. Karniadakis. `ViTO: Vision Transformer-Operator <https://doi.org/10.1016/j.cma.2024.117109>`_. *Computer Methods in Applied Mechanics and Engineering*, Volume 428, 117109, 2024.
#. Z\. Jiang, M. Zhu, & L. Lu. `Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration <https://doi.org/10.1016/j.ress.2024.110392>`_. *Reliability Engineering & System Safety*, Volume 251, 110392, 2024.
#. Q\. Meng, Y. Li, Z. Deng, X. Liu, G. Chen, Q. Wu, C. Liu, & X. Hao. `A general reduced-order neural operator for spatio-temporal predictive learning on complex spatial domains <https://arxiv.org/abs/2409.05508>`_. *arXiv preprint arXiv:2409.05508*, 2024.
#. K\. Lv, J. Wang, & Y. Cao. `Neural Operator Approximations for Boundary Stabilization of Cascaded Parabolic PDEs <https://doi.org/10.1002/acs.3902>`_. *International Journal of Adaptive Control and Signal Processing*, Wiley Online Library, 2024.
#. B\. Ahmed, Y. Qiu, D. W. Abueidda, W. El-Sekelly, B. G. de Soto, T. Abdoun, & M. E. Mobasher. `Physics-informed DeepONet with stiffness-based loss functions for structural response prediction <https://arxiv.org/abs/2409.00994>`_. *arXiv preprint arXiv:2409.00994*, 2024.
#. P\. Gao, G. E. Karniadakis, & P. Stinis. `Multiscale modeling framework of a constrained fluid with complex boundaries using twin neural networks <https://arxiv.org/abs/2408.03263>`_. *arXiv preprint arXiv:2408.03263*, 2024.
#. L\. Xiao, G. Mei, & N. Xu. `Knowledge-integrated deep learning for predicting stochastic thermal regime of embankment in permafrost region <https://doi.org/10.1016/j.jrmge.2024.05.005>`_. *Journal of Rock Mechanics and Geotechnical Engineering*, Elsevier, 2024.
#. G\. Fabiani, I. G. Kevrekidis, C. Siettos, & A. N. Yannacopoulos. `RandONet: Shallow-networks with random projections for learning linear and nonlinear operators <https://arxiv.org/abs/2406.05470>`_. *Computer Methods in Applied Mechanics and Engineering*, 429:117130, 2024.
#. A\. Jiao, H. He, R. Ranade, J. Pathak, & L. Lu. `One-shot learning for solution operators of partial differential equations <https://arxiv.org/abs/2104.05512>`_. *arXiv preprint arXiv:2104.05512*, 2024.
#. S\. Zampini, U. Zerbinati, G. Turkyyiah, & D. Keyes. `PETScML: Second-order solvers for training regression problems in Scientific Machine Learning <https://dl.acm.org/doi/pdf/10.1145/3659914.3659931>`_. In *Proceedings of the Platform for Advanced Scientific Computing Conference*, 1-12, 2024, June.
#. L\. Branca & A. Pallottini. `Emulating the interstellar medium chemistry with neural operators <https://www.aanda.org/articles/aa/full_html/2024/04/aa49193-24/aa49193-24.html>`_. *Astronomy & Astrophysics*, 684, A203, 2024.
#. J\. Hayford, J. Goldman-Wetzler, E. Wang, & L. Lu. `Speeding up and reducing memory usage for scientific machine learning via mixed precision <https://doi.org/10.1016/j.cma.2024.117093>`_. *Computer Methods in Applied Mechanics and Engineering*, 428, 117093, 2024.
#. K\. Kobayashi, J. Daniell, & S.B. Alam. `Improved generalization with deep neural operators for engineering systems: Path towards digital twin <https://www.sciencedirect.com/science/article/pii/S0952197624000022#d1e4572>`_. *Engineering Applications of Artificial Intelligence*, 131, 107844, 2024.
#. K\. Kobayashi & S.B. Alam. `Deep neural operator-driven real-time inference to enable digital twin solutions for nuclear energy systems <https://www.nature.com/articles/s41598-024-51984-x#Abs1>`_. *Scientific Reports*, 14, 2101, 2024.
#. H\. Liu, B. Dahal, R. Lai, & W. Liao. `Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning <https://arxiv.org/abs/2401.10490>`_. *arXiv preprint arXiv:2401.10490*, 2024.
#. J\. He, D. Pal, A. Najafi, D. Abueidda, S. Koric, & I. Jasiuk. `Material-response-informed DeepONet and its application to polycrystal stress–strain prediction in crystal plasticity <https://link.springer.com/article/10.1007/s11837-024-06681-5>`_. *JOM*, 1-11, 2024.
#. M\. Lamarque, L. Bhan, Y. Shi, & M. Krstic. `Adaptive neural-operator backstepping control of a benchmark hyperbolic PDE <https://arxiv.org/abs/2401.07862>`_. *arXiv preprint arXiv:2401.07862*, 2024.
#. K\. Leng, M. Shankar, & J. Thiyagalingam. `Zero coordinate shift: Whetted automatic differentiation for physics-informed operator learning <https://doi.org/10.1016/j.jcp.2024.112904>`_. *Journal of Computational Physics*, 505, 112904, 2024.
#. M\. Lamarque, L. Bhan, R. Vazquez, & M. Krstic. `Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation <https://arxiv.org/abs/2401.02511>`_. *arXiv preprint arXiv:2401.02511*, 2024.
#. A\. Xavier. `Solving Heat Conduction Problems with DeepONets <https://mediatum.ub.tum.de/doc/1728141/document.pdf>`_. 2023.
#. L\. Xu, H. Zhang, & M. Zhang. `Training a deep operator network as a surrogate solver for two-dimensional parabolic-equation models <https://pubs.aip.org/asa/jasa/article/154/5/3276/2922093>`_. *The Journal of the Acoustical Society of America*, 154(5), 3276-3284, 2023.
#. N\. Ford, V. J. Leon, H. Merman, J. Gilbert, & A. New. `Data-efficient operator learning for solving high Mach number fluid flow problems <https://arxiv.org/abs/2311.16860>`_. *arXiv preprint arXiv:2311.16860*, 2023.
#. B\. Chen, C. Wang, W. Li, & H. Fu. `A hybrid Decoder-DeepONet operator regression framework for unaligned observation data <https://arxiv.org/pdf/2308.09274.pdf>`_. *arXiv preprint arXiv:2308.09274*, 2023.
#. K\. Kobayashi, & S. B. Alam. `Potential of deep operator networks in digital twin-enabling technology for nuclear system <https://arxiv.org/pdf/2308.07523.pdf>`_. *arXiv preprint arXiv:2308.07523*, 2023.
#. J\. He, S. Kushwaha, J. Park, S. Koric, D. Abueidda, & I. Jasiuk. `Sequential deep operator networks (S-DeepONet) for predicting full-field solutions under time-dependent loads <https://arxiv.org/abs/2306.08218>`_. *Engineering Applications of Artificial Intelligence*, 127:107258, 2024.
#. E\. L. Bolager, I. Burak, C. Datar, Q. Sun, & F. Dietrich. `Sampling weights of deep neural networks <https://arxiv.org/pdf/2306.16830>`_. 2023.
#. V\. Fanaskov, T. Yu, A. Rudikov, & I. Oseledets. `General covariance data augmentation for neural PDE solvers <https://arxiv.org/abs/2301.12730>`_. 2023.
#. J\. He, S. Koric, S. Kushwaha, J. Park, D. Abueidda, & I. Jasiuk. `Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads <https://arxiv.org/abs/2306.03645>`_. *Computer Methods in Applied Mechanics and Engineering*, 415:116277, 2023.
#. K\. Kobayashi, J. Daniell, & S. B. Alam. `Operator learning framework for digital twin and complex engineering systems <https://arxiv.org/abs/2301.06701>`_. 2023.
#. M\. Zhu, S. Feng, Y. Lin, & L. Lu. `Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness <https://doi.org/10.1016/j.cma.2023.116300>`_. *Computer Methods in Applied Mechanics and Engineering*, 416, 116300, 2023.
#. S\. Mao, R. Dong, L. Lu, K. M. Yi, S. Wang, & P. Perdikaris. `PPDONet: Deep operator networks for fast prediction of steady-state solutions in disk-planet systems <https://iopscience.iop.org/article/10.3847/2041-8213/acd77f>`_. *The Astrophysical Journal Letters*, 950(2), L12, 2023.
#. S\. Wang, & P. Perdikaris. `Long-time integration of parametric evolution equations with physics-informed deeponets <https://www.sciencedirect.com/science/article/pii/S0021999122009184>`_. *Journal of Computational Physics*, 475, p.111855, 2023.
#. E\. Pickering, S. Guth, G. Karniadakis, & T. Sapsis. `Discovering and forecasting extreme events via active learning in neural operators <https://www.nature.com/articles/s43588-022-00376-0>`_. *Nature Computational Science*, 2(12), pp.823-833, 2022.
#. S\. Dhulipala, & R. Hruska. `Efficient interdependent systems recovery modeling with DeepONets <https://ieeexplore.ieee.org/abstract/document/9984029?casa_token=YzcUx1VcodwAAAAA:YnIaTDR230p0yeoXKAlfu_DclexyyIs4i5hg42Y9YIGkCMWBzk0QwWT-SBaHWeD2-GNi-znQsSQ>`_. *2022 Resilience Week (RWS)*, pp. 1-6. IEEE, 2022.
#. M\. Zhu, H. Zhang, A. Jiao, G. Karniadakis, & L. Lu. `Reliable extrapolation of deep neural operators informed by physics or sparse observations <https://doi.org/10.1016/j.cma.2023.116064>`_. *Computer Methods in Applied Mechanics and Engineering*, 412, 116064, 2023.
#. P\. Clark Di Leoni, L. Lu, C. Meneveau, G. Karniadakis, & T. Zaki. `Neural operator prediction of linear instability waves in high-speed boundary layers <https://doi.org/10.1016/j.jcp.2022.111793>`_. *Journal of Computational Physics*, 474, 111793, 2023.
#. P\. Jin, S. Meng, & L. Lu. `MIONet: Learning multiple-input operators via tensor product <https://doi.org/10.1137/22M1477751>`_. *SIAM Journal on Scientific Computing*, 44(6), A3490--A3514, 2022.
#. L\. Lu, R. Pestourie, S. Johnson, & G. Romano. `Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport <https://doi.org/10.1103/PhysRevResearch.4.023210>`_. *Physical Review Research*, 4(2), 023210, 2022.
#. L\. Lu, X. Meng, S. Cai, Z. Mao, S. Goswami, Z. Zhang, & G. Karniadakis. `A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data <https://doi.org/10.1016/j.cma.2022.114778>`_. *Computer Methods in Applied Mechanics and Engineering*, 393, 114778, 2022.
#. L\. Tan, & L. Chen. `Enhanced DeepONet for modeling partial differential operators considering multiple input functions <https://arxiv.org/pdf/2202.08942.pdf>`_. *arXiv preprint arXiv:2202.08942*, 2022.
#. C\. Lin, M. Maxey, Z. Li, & G. Karniadakis. `A seamless multiscale operator neural network for inferring bubble dynamics <https://doi.org/10.1017/jfm.2021.866>`_. *Journal of Fluid Mechanics*, 929, A18, 2021.
#. Z\. Mao, L. Lu, O. Marxen, T. Zaki, & G. Karniadakis. `DeepM&Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators <https://doi.org/10.1016/j.jcp.2021.110698>`_. *Journal of Computational Physics*, 447, 110698, 2021.
#. S\. Cai, Z. Wang, L. Lu, T. Zaki, & G. Karniadakis. `DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks <https://doi.org/10.1016/j.jcp.2021.110296>`_. *Journal of Computational Physics*, 436, 110296, 2021.
#. L\. Lu, P. Jin, G. Pang, Z. Zhang, & G. Karniadakis. `Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators <https://doi.org/10.1038/s42256-021-00302-5>`_. *Nature Machine Intelligence*, 3, 218--229, 2021.
#. C\. Lin, Z. Li, L. Lu, S. Cai, M. Maxey, & G. Karniadakis. `Operator learning for predicting multiscale bubble growth dynamics <https://doi.org/10.1063/5.0041203>`_. *The Journal of Chemical Physics*, 154(10), 104118, 2021.

Multi-fidelity NN
-----------------

#. L\. Lu, M. Dao, P. Kumar, U. Ramamurty, G. Karniadakis, & S. Suresh. `Extraction of mechanical properties of materials through deep learning from instrumented indentation <https://doi.org/10.1073/pnas.1922210117>`_. *Proceedings of the National Academy of Sciences*, 117(13), 7052--7062, 2020.
#. X\. Meng, & G. Karniadakis. `A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems <https://doi.org/10.1016/j.jcp.2019.109020>`_. *Journal of Computational Physics*, 401, 109020, 2020.

Tutorial
--------

#. `An introductory course to PINNs and DeepONet <https://github.com/fperiago/pinn_deeponet_for_beginners>`_