Research

DeepXDE has been used in

• > 230 universities, e.g., Harvard University, Massachusetts Institute of Technology, Stanford University, University of California, Berkeley, Yale University, University of California, Los Angeles, Imperial College London, Johns Hopkins University, University of Pennsylvania, Tsinghua University, University of Toronto, California Institute of Technology, Princeton University, Cornell University, National University of Singapore, Nanyang Technological University, University of California, San Diego, Peking University, University of Amsterdam, New York University Abu Dhabi, University of British Columbia, University of Copenhagen, KU Leuven, University of Pittsburgh, Zhejiang University, Shanghai Jiao Tong University, University of Texas at Austin, Leiden University, University of Minnesota, University of Manchester, University of Chinese Academy of Sciences, Georgia Institute of Technology, Boston University, University of Maryland, Universite Paris Saclay, The University of Tokyo, University of Science and Technology of China, University of Southern California, University of Wisconsin Madison, Technical University of Munich, University of California, Santa Barbara, University of Birmingham, Pennsylvania State University, University of Colorado Boulder, University of Illinois at Urbana-Champaign, University of California Irvine, Sun Yat-sen University, King Abdullah University of Science and Technology, University of Oslo, University of Florida, Aarhus University, University of Exeter, University of Southampton, University of California, Santa Cruz, University of Padua, Carnegie Mellon University, Seoul National University, University of Leeds, Sapienza University Rome, University of Alberta, University of Liverpool, Tongji University, University of Electronic Science and Technology of China, Brown University, University of Bonn, Southern University of Science and Technology, Harbin Institute of Technology, Purdue University, Kyoto University, University of Basel, Central South University, University of Massachusetts Amherst, Xi’an Jiaotong University, Tel Aviv University, Texas A&M University, Arizona State University, Beijing Institute of Technology, Southeast University, Delft University of Technology, University of Naples Federico II, University of Waterloo, Tianjin University, Xiamen University, University of Calgary, Beijing Normal University, Kapodistrian University, University of Turin, RWTH Aachen University, National Taiwan University, China University of Geosciences, Sichuan University, Rice University, Beihang University, University of Sussex, University of Bergen, KTH Royal Institute of Technology, Northwestern Polytechnical University, Tufts University, Wuhan University of Technology, Universidade do Porto, Florida State University, Karlsruhe Institute of Technology, University Duisburg-Essen, Dartmouth College, University of Western Ontario, University of Strasbourg, University of Surrey, Shanghai University, Chalmers University of Technology, Pontificia Universidad Católica de Chile, Kyushu University, Nagoya University, University of Johannesburg, University of Rome Tor Vergata, University of Kentucky, Mansoura University, Eindhoven University of Technology, Friedrich Schiller University of Jena, University of Victoria, University of Twente, University of Houston, Fuzhou University, University of Delaware, University of Mississippi, Swansea University, University of Bath, University of Trieste, University of the Basque Country, Hong Kong Baptist University, University of Hawaii Manoa, Federal University of Rio de Janeiro, George Mason University, University of Sevilla, International School for Advanced Studies, Beijing University of Technology, Nanjing Tech University, TU Wien, Beijing Jiaotong University, University of Canterbury, Universidade do Minho, Nanchang University, Carleton University, Ocean University of China, South China Normal University, Roma Tre University, AmirKabir University of Technology, Sabanci University, Indian Institute of Technology, Concordia University, Tarbiat Modares University, National University of Colombia, Clemson University, Dortmund University of Technology, University of Los Andes, University of Stuttgart, Universidad de salamanca, Harbin Engineering University, Universiti Teknologi Petronas, ITMO University, University of Nevada, Las Vegas, University of Bayreuth, Macau University of Science & Technology, Isfahan University of Technology, Rensselaer Polytechnic Institute, Missouri University of Science and Technology, AGH University of Krakow, University of Calabria, Ulster University, University of Thessaly, Kuwait University, University of Malaga, Brno University of Technology, Old Dominion University, Johannes Kepler University Linz, University of Kragujevac, California Polytechnic State University, Chung-Ang University, Graz University of Technology, Shahid Beheshti University, Shanghai Normal University, Cadi Ayyad University, Universidad Rey Juan Carlos, Zhejiang A&F University, Universidade Federal Fluminense, Pontifical Catholic University of Valparaiso, Nazarbayev University, University of A Coruña, Worcester Polytechnic Institute, Xinjiang University, LUT University, University of Las Palmas de Gran Canaria, Nanjing University of Aeronautics and Astronautics, Lahore University of Management Sciences, Hangzhou Dianzi University, London South Bank University, Taras Shevchenko National University Kiev, University of Calcutta, University of Kaiserslautern, Wuhan Textile University, San Francisco State University, Anhui University of Science and Technology, Boise State University, Necmettin Erbakan University, Shahrekord University, Shahrood University of Technology, Yangtze University, Technical University of Cartagena, Adolfo Ibáñez University, Bundeswehr University Munich, Universidad de Burgos, Dong A University, East China University of Science and Technology Shanghai, Bauhaus-Universität Weimar, Henan Institute of Economics and Trade National University of Defence Technology, University of Applied Sciences and Arts Northwestern Switzerland, University of Engineering and Management, Pontifical Catholic University of Rio de Janeiro, Fujian Agriculture and Forestry University, Central South University of Forestry and Technology, Cho Chun Shik Graduate School of Mobility, Adama Science and Technology University, The University of Waikato, Lishui University, Westphalian University, Shanghai Jian Qiao University, Tel-Aviv University,

• > 40 national labs and research institutes, e.g., Pacific Northwest National Laboratory, Sandia National Laboratories, Argonne National Laboratory, Los Alamos National Laboratory, Oak Ridge National Laboratory, Idaho National Laboratory, Institute of Applied Physics and Computational Mathematics, Institute of Computational Mathematics and Scientific/Engineering Computing, China Academy of Engineering Physics, National Key Laboratory for Remanufacturing, Laboratory of Web Science, Associate Laboratory LSRE-LCM, Center of Applied Ecology and Sustainability, NEC Lab Europe, CSIRO’s Data61, Zienkiewicz Institute for Modelling, Data and AI, Erich Schmid Institute of Materials Science, Athinoula A. Martinos Center for Biomedical Imaging, Friedrich-Alexander-Universität Erlangen-Nürnberg Research Center for Mathematics of Data, Data Observatory Foundation, Fraunhofer Institute for Integrated Systems and Device Technology IISB, ONERA, The French Aerospace Lab, Ministry of Natural Resources (China), Zhejiang Lab, United Kingdom Atomic Energy Authority, Russian Academy of Sciences, Chinese Academy of Sciences, AppliedAI Institute for Europe, Rutherford Appleton Laboratory, Mathematical Sciences Research Laboratory, German Aerospace Center, Materials Center Leoben Forschung GmbH, Ecole Polytechnique, Scuola Superiore Meridionale, Mitsubishi Electric Research Laboratories, Forschungszentrum Jülich, China Ship Scientific Research Center, Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Korea Institute of Fusion Energy, Fraunhofer Heinrich Hertz Institute, Northwest Institute of Nuclear Technology, Bay Area Environmental Research Institute, Lockheed Martin Solar and Astrophysics Laboratory, CSIRO, Space & Astronomy, Fujian Special Equipment Inspection and Research Institute, Centrale Lille Institute, Science and Technology Facilities Council Scientific Computing Department, Children’s Hospital of Philadelphia,

• > 10 industry, e.g., Anailytica, Ansys, BioME, BirenTech Research, Bosch, ExxonMobil, General Motors, Intel Corporation, RocketML, Saudi Aramco, Shell, SoftServe, Quantiph, Moldex3D,

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

1. L. Yin & X. Lv. Adapting physics-informed neural networks for bifurcation detection in ecological migration models. arXiv preprint arXiv:2409.00651, 2024.

2. 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. arXiv preprint arXiv:2408.11077, 2024.

3. H. Gangloff & N. Jouvin. jinns: a JAX library for physics-informed neural networks. arXiv preprint arXiv:2412.14132, 2024.

4. M. J. Choi. Leveraging turbulence data with physics-informed neural networks. arXiv preprint arXiv:2412.20130, 2024.

5. P. Kumar & R. Ranjan. Evaluation of physics-informed machine learning approach for computation of fluid flows. 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.

6. K. Leng, M. Shankar, & J. Thiyagalingam. Zero coordinate shift: Whetted automatic differentiation for physics-informed neural operators. Journal of Computational Physics, Volume 505, 112904, 2024.

7. R. Fang, K. Zhang, K. Song, Y. Kai, Y. Li, & B. Zheng. A deep learning method for solving thermoelastic coupling problem. Zeitschrift für Naturforschung A, 79(8), 851–871, 2024.

8. S. Schoder. Physics-informed neural networks for modal wave field predictions in 3D room acoustics. Institute of Fundamentals and Theory in Electrical Engineering, Graz University of Technology, Inffeldgasse 18/I, 8010 Graz, Austria, 2024.

9. L. Vu-Quoc & A. Humer. Partial-differential-algebraic equations of nonlinear dynamics by physics-informed neural-network: (I) Operator splitting and framework assessment. Neural Methods in Engineering, First published: 17 October, 2024.

10. A. Noorizadegan, R. Cavoretto, D.L. Young, & C.S. Chen. Stable weight updating: A key to reliable PDE solutions using deep learning. Engineering Analysis with Boundary Elements, Volume 168, 105933, 2024.

11. C. Soyarslan & M. Pradas. Physics-informed machine learning in asymptotic homogenization of elliptic equations. Computer Methods in Applied Mechanics and Engineering, Volume 427, Part 2, 117043, 2024.

12. Y. Wu, J. Guo, G. Gopalakrishna, & Z. Poulos. Deep-MacroFin: Informed equilibrium neural network for continuous time economic models. arXiv preprint arXiv:2408.10368, 2024.

13. J. Kurz, B. Bowman, M. Seman, et al. A physics-informed kernel approach to learning the operator for parametric PDEs. Neural Computing and Applications, 36, 22773–22787, 2024.

14. A. Newa, A. S. Gearhart, R. A. Darragh, & M. Villafañe-Delgado. Physics-informed neural networks for scientific modeling: uses, implementations, and directions. Artificial Intelligence and Machine Learning for Multi-Domain Operations Applications VI, Vol. 13051, 130511J, 2024.

15. J. Seo. Past rewinding of fluid dynamics from noisy observation via physics-informed neural computing. Phys. Rev. E, 110(2), 025302, 2024.

16. S. Mtshali, B. A. Jacobs. Machine learning-based prediction of pharmacokinetic parameters for individualized drug dosage optimization. Int. J. Inf. Tecnol., 2024.

17. 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. An. Acad. Bras. Ciênc., 96(4), e20240262, 2024.

18. 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. Journal of Computational Physics, Volume 502, 112804, 2024.

19. L. Santos. Deep and Physics-Informed Neural Networks as a Substitute for Finite Element Analysis. ICMLT ‘24: Proceedings of the 2024 9th International Conference on Machine Learning Technologies, Pages 84–90, 2024.

20. Y. Tong, S. Xiong, X. He, et al. RoeNet: Predicting discontinuity of hyperbolic systems from continuous data. Int J Numer Methods Eng, 125(6), e7406, 2024.

21. 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. Lecture Notes in Civil Engineering, Volume 553. Springer, Singapore, 2025.

22. C. B. Ribeiro. Advanced Numerical Solution of Navier-Stokes Equations with Energy Conservation: A Physics-Informed Neural Networks Approach to Revolutionize Computational Fluid Dynamics. December 2024.

23. L. Shang, Y. Zhao, S. Zheng, J. Wang, T. Zhang, J. Wang. Quantification of gradient energy coefficients using physics-informed neural networks. International Journal of Mechanical Sciences, Volume 273, 109210, 2024.

24. 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. Physics Letters A, Volume 531, 130182, 2025.

25. D. Bonnet-Eymard, A. Persoons, M. Faes, D. Moens. Separable Physics-Informed Neural Networks for Robust Inverse Quantification in Solid Mechanics. International Symposium on Reliability Engineering and Risk Management (ISRERM), October 2024.

26. Z.-Q. Zhang, et al. Physics-Informed Neural Network Approaches in Quantum Simulations. J. Phys.: Conf. Ser., 2891, 062023, 2024.

27. J. R. Naujoks, A. Krasowski, M. Weckbecker, T. Wiegand, S. Lapuschkin, W. Samek, R. P. Klausen. PINNfluence: Influence Functions for Physics-Informed Neural Networks. arXiv preprint arXiv:2409.08958, 2024.

28. C. J. McDevitt, J. Arnaud, X. Z. Tang. An Efficient Surrogate Model of Secondary Electron Formation and Evolution. arXiv preprint arXiv:2412.13044, 2024.

29. 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. IEEE Transactions on Components, Packaging and Manufacturing Technology, 2024.

30. S. Song, H. Jin. Identifying constitutive parameters for complex hyperelastic materials using physics-informed neural networks. Soft Matter, 20(30), 5915–5926, 2024.

31. A. Ahmad, A. Khan. Pricing Rainbow Options Using Deep Learning. Preprints, 2024.

32. 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. 2024 Sensor Data Fusion: Trends, Solutions, Applications (SDF), Bonn, Germany, pp. 1–8, 2024.

33. 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. IFAC-PapersOnLine, 58(15), Pages 259–264, 2024.

34. 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. SSRN, 2024.

35. H. Mertens, F. Zhu. Comparative Analysis of Uncertainty Quantification Models in Active Learning for Efficient System Identification of Dynamical Systems. 2024 IEEE 20th International Conference on Automation Science and Engineering (CASE), Bari, Italy, pp. 1869–1876, 2024.

36. H. Zhang, L. Liu, L. Lu. Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity. arXiv preprint arXiv:2410.13141, 2024.

37. C. J. McDevitt, J. Arnaud, X.-Z. Tang. A Physics-Constrained Deep Learning Treatment of Runaway Electron Dynamics. arXiv preprint arXiv:2412.12980, 2024.

38. 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. SSRN, 2024.

39. 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. Axioms, 13(12), 872, 2024.

40. 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. arXiv preprint arXiv:2409.12712, 2024.

41. 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. Physics of Fluids, 36(12), 121902, 2024.

42. T. Sahin, M. von Danwitz, A. Popp. Solving forward and inverse problems of contact mechanics using physics-informed neural networks. Advances in Modeling and Simulation in Engineering Sciences, 11, 11, 2024.

43. 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. arXiv preprint arXiv:2409.01410, 2024.

44. J. Duan, H. Zhao, J. Song. Spatial domain decomposition-based physics-informed neural networks for practical acoustic propagation estimation under ocean dynamics. Journal of the Acoustical Society of America, 155(5), 3306–3321, 2024.

45. 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. SSRN, 2024.

46. 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. arXiv preprint arXiv:2409.19081, 2024.

47. 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. Physics of Fluids, 36(12), 121924, 2024.

48. H.-Q. Yang, C. Shi, L. Zhang. Ensemble learning of soil–water characteristic curve for unsaturated seepage using physics-informed neural networks. Soils and Foundations, 65(1), 101556, 2025.

49. M. Peng, H. Tang, Y. Kou. Adversarial and self-adaptive domain decomposition physics-informed neural networks for high-order differential equations with discontinuities. SSRN, 2024.

50. 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. SSRN Electronic Journal, 2024.

51. L. Novák, H. Sharma, M. D. Shields. Physics-informed polynomial chaos expansions. Journal of Computational Physics, Volume 506, 112926, 2024.

52. 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. arXiv preprint arXiv:2409.06402, 2024.

53. D. Sitalo, A. Ogueda-Oliva, P. Seshaiyer. Data-Driven Mathematical Modeling and Simulation of Migration Dynamics During the Russian-Ukrainian War. Spora: A Journal of Biomathematics, Vol. 10, 83–90, 2024.

54. J. Zhao, Z. Tian, X. Zhang, Z. Duan, J. Lu. Kinetics Parameter Identification of Chain Shuttling Polymerization Based on Physics-Informed Neural Networks. IFAC-PapersOnLine, 58(14), 184–191, 2024.

55. 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. arXiv preprint arXiv:2403.17607, 2024.

56. 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. J. Acoust. Soc. Am., 156(5), 3523–3533, 2024.

57. P. Xiao, M. Zheng, A. Jiao, X. Yang, L. Lu. Quantum DeepONet: Neural operators accelerated by quantum computing. arXiv preprint arXiv:2409.15683, 2024.

58. Y. Yang, P. He, X. Peng, Q. He. A number-theoretic method sampling neural network for solving partial differential equations. arXiv preprint arXiv:2411.17039, 2025.

59. J. Cho, S. Nam, H. Yang, S.-B. Yun, Y. Hong, E. Park. Separable Physics-Informed Neural Networks. Advances in Neural Information Processing Systems, 36, 23761–23788, 2023.

60. 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. arXiv preprint arXiv:2410.19759, 2024.

61. W. Hu. A new method to solve the forward and inverse problems for the spatial Solow model by using Physics Informed Neural Networks (PINNs). Engineering Analysis with Boundary Elements, 169(Part B), 106013, 2024.

62. X. Wang, C. Luo, D. Jiang, H. Wang, Z. Wang. Improved design method for gas carburizing process through data-driven and physical information. Computational Materials Science, Volume 247, 113507, 2025.

63. 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. Journal of Fluid Mechanics, 998, A46, 2024.

64. 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. Chemical Engineering Science, Volume 285, 119636, 2024.

65. C.A. Molina Catricheo, F. Lambert, J. Salomon, et al. Modeling global surface dust deposition using physics-informed neural networks. Communications Earth & Environment, 5, 778, 2024.

66. A. Deresse, T. Dufera. A deep learning approach: Physics-informed neural networks for solving the 2D nonlinear Sine–Gordon equation. Results in Applied Mathematics, 25, 2024.

67. N. Patel, A. Aykutalp, P. Laguna. Novel approach to solving Schwarzschild black hole perturbation equations via physics informed neural networks. Gen Relativ Gravit, 56, 137, 2024.

68. A. Jesser, K. Krycki, R. G. McClarren, & M. Frank. Numerical Robustness of PINNs for Multiscale Transport Equations. arXiv preprint arXiv:2412.14683, 2024.

69. H. Wu, H. Luo, Y. Ma, J. Wang, & M. Long. RoPINN: Region Optimized Physics-Informed Neural Networks. arXiv preprint arXiv:2405.14369, 2024.

70. 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. SSRN, 2024.

71. H. Wang, Y. Pu, S. Song, & G. Huang. Physics-informed Dynamics Representation Learning for Parametric PDEs. OpenReview, 2024.

72. 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. arXiv preprint, arXiv:2409.02339, 2024.

73. J. J. Athalathil, B. Vaidya, S. Kundu, V. Upendran & M. C. M. Cheung. Surface Flux Transport Modeling Using Physics-informed Neural Networks. The Astrophysical Journal, 975(2), 258, 2024.

74. A. A. Aghaei, M. M. Moghaddam & K. Parand. PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems. arXiv preprint, arXiv:2409.01899, 2024.

75. L. Shang, S. Zheng, J. Wang & J. Wang. Physics-informed neural networks incorporating energy dissipation for the phase-field model of ferroelectric microstructure evolution. arXiv preprint, arXiv:2409.02959, 2024.

76. 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. arXiv e-prints, arXiv:2408, 2024.

77. Z. Xiong, Y. Jiang, W. Lu, X. Wang & T. Tian. Reconstructing and Forecasting Marine Dynamic Variable Fields across Space and Time Globally and Gaplessly. arXiv preprint, arXiv:2408.01509, 2024.

78. J. H. Adler, S. Hocking, X. Hu & S. Islam. Physics-informed nonlinear vector autoregressive models for the prediction of dynamical systems. arXiv preprint, arXiv:2407.18057, 2024.

79. 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. Applied Energy, 373, 123874, 2024.

80. D. Nguyen. Advanced modeling of the childbirth system using different deep learning methods: from fetal skeleton segmentation to real-time soft tissue deformation. PhD thesis, Centrale Lille Institut, 2024.

81. M. Y. Hosseini & Y. Shiri. Flow field reconstruction from sparse sensor measurements with physics-informed neural networks. Physics of Fluids, 36(7), 2024.

82. H. Lu, Q. Wang, W. Tang & H. Liu. Physics-informed neural networks for fully non-linear free surface wave propagation. Physics of Fluids, 36(6), 2024.

83. Y. Gao, P. Xiao & Z. Li. Physics-Informed Neural Networks for Solving Underwater Two-dimensional Sound Field. 2024 OES China Ocean Acoustics (COA), pp. 1-4, IEEE, 2024.

84. 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. arXiv preprint, arXiv:2405.08406, 2024.

85. S. K. Vemuri, T. Büchner, & J. Denzler. Estimating soil hydraulic parameters for unsaturated flow using physics-informed neural networks. In International Conference on Computational Science, 338-351, Cham: Springer Nature Switzerland, 2024, June.

86. 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. In International Conference on Computational Science, 204-218, Cham: Springer Nature Switzerland, 2024, June.

87. N. Alzhanov, E. Y. K. Ng, & Y. Zhao. Three-dimensional physics-informed neural network simulation in coronary artery trees. Fluids, 9(7), 2024.

88. 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). Journal of Applied Physics, 135(22):225106, June 2024.

89. T. Zou, T. Yajima, & Y. Kawajiri. A parameter estimation method for chromatographic separation process based on physics-informed neural network. Journal of Chromatography A, 1730:465077, 2024.

90. N. Jha & E. Mallik. GPINN with neural tangent kernel technique for nonlinear two point boundary value problems. Neural Processing Letters, 56(3):192, May 2024.

91. 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. Computer Physics Communications, 308, p.109462. 2024.

92. N. Jha & E. Mallik. Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs. Physica Scripta, 99(7):076005, June 2024.

93. 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. Neural Computing and Applications, May 2024.

94. H. Nganguia & D. Palaniappan. Ciliary propulsion through non-uniform flows. Journal of Fluid Mechanics, 986:A14, 2024.

95. A. T. Deresse & T. T. Dufera. Exploring physics-informed neural networks for the generalized nonlinear Sine-Gordon equation. Applied Computational Intelligence and Soft Computing, 2024(1):3328977, 2024.

96. H. Qiumei, M. Jiaxuan, & X. Zhen. Mass-preserving spatio-temporal adaptive PINN for Cahn-Hilliard equations with strong nonlinearity and singularity, 2024.

97. Z.Zhang, J.-H. Lee, L. Sun, & G. X. Gu. Weak-formulated physics-informed modeling and optimization for heterogeneous digital materials. PNAS Nexus, 3(5):pgae186, May 2024.

98. 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, 2024.

99. J. Son, N. Park, H. Kwak, & J. Nam. Optimizing a physics-informed machine learning model for pulsatile shear-thinning channel flow. Journal of the Japanese Society of Rheology, 52(2):113–122, 2024.

100. E. Raeisi, M. Yavuz, M. Khosravifarsani, Y. Fadaei. Mathematical modeling of interactions between colon cancer and immune system with a deep learning algorithm. Eur. Phys. J. Plus, 139(4), 345, 2024.

101. Z. Zhang, C. Lin, & B. Wang. Physics-informed shape optimization using coordinate projection. Scientific Reports, 14, 6537, 2024.

102. S. Schoder & F. Kraxberger. Feasibility study on solving the Helmholtz equation in 3D with PINNs. arXiv preprint arXiv:2403.06623, 2024.

103. 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. arXiv preprint arXiv:2403.04576, 2024.

104. J. S. Arnaud, T. Mark, & C.J. McDevitt. A physics-constrained deep learning surrogate model of the runaway electron avalanche growth rate. arXiv preprint arXiv:2403.04948, 2024.

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247. Y. Wang, X. Han, C. Chang, D. Zha, U. Braga-Neto, & X. Hu. Auto-PINN: Understanding and optimizing physics-informed neural architecture. arXiv preprint arXiv:2205.13748, 2022.

248. B. Dalen. Characterization of Cardiac cellular dynamics using physics-informed neural networks. 2022.

249. 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. Computer Aided Geometric Design, 95, p.102087, 2022.

250. A. Ncube. Investigating new computational approaches for solving black hole perturbation equations. Doctoral dissertation, University of Johannesburg, 2022.

251. C. Garcıa-Cervera, M. Kessler, & F. Periago. A first step towards controllability of partial differential equations via physics-informed neural networks. 2022.

252. 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. arXiv preprint arXiv:2203.08501, 2022.

253. P. Escapil-Inchauspé, & G. A. Ruz. Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems. Neurocomputing, 126826, 2023.

254. P. Escapil-Inchauspé, & G. Ruz. Physics-informed neural networks for operator equations with stochastic data. arXiv preprint arXiv:2211.10344, 2022.

255. H. Xie, C. Zhai, L. Liu, & H. Yong. A weighted first-order formulation for solving anisotropic diffusion equations with deep neural networks. arXiv preprint arXiv:2205.06658, 2022.

256. Y. Lu, G. Mei, & F. Piccialli. A deep learning approach for predicting two-dimensional soil consolidation using physics-informed neural networks (PINN). arXiv preprint arXiv:2205.05710, 2022.

257. J. Yu, L. Lu, X. Meng, & G. Karniadakis. Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Computer Methods in Applied Mechanics and Engineering, 393, 114823, 2022.

258. A. Sacchetti, B. Bachmann, K. Löffel, U. Künzi, & B. Paoli. Neural networks to solve partial differential equations: A comparison with finite elements. IEEE Access, 10, 32271-32279, 2022.

259. Y. Xue, Y. Li, K. Zhang, & F. Yang. A physics-inspired neural network to solve partial differential equations - application in diffusion-induced stress. Physical Chemistry Chemical Physics, 24(13), 7937-7949, 2022.

260. 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. ChemEngineering, 6.2 (2022): 21, 2022.

261. M. Daneker, Z. Zhang, G. Karniadakis, & L. Lu. Systems biology: Identifiability analysis and parameter identification via systems-biology-informed neural networks. Computational Modeling of Signaling Networks, Springer, 87–105, 2023.

262. C. Martin, A. Oved, R. Chowdhury, E. Ullmann, N. Peters, A. Bharath, & M. Varela. EP-PINNs: Cardiac electrophysiology characterisation using physics-informed neural networks. Frontiers in cardiovascular medicine, 2179, 2022.

263. V. Schäfer. Generalization of physics-informed neural networks for various boundary and initial conditions. Doctoral dissertation, Technische Universität Kaiserslautern, 2022.

264. S. Alkhadhr, & M. Almekkawy. A combination of deep neural networks and physics to solve the inverse problem of Burger’s equation. 43rd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC), 2021.

265. K. Iversen. Physics informed neural networks for inverse advection-diffusion problems. The University of Bergen, 2021.

266. S. Markidis. The old and the new: Can physics-informed deep-learning replace traditional linear solvers?. Frontiers in Big Data, 4:669097, 2021.

267. S. Alkhadhr, X. Liu, & M. Almekkawy. Modeling of the forward wave propagation using physics-informed neural networks. 2021 IEEE International Ultrasonics Symposium (IUS), pp. 1–4, 2021.

268. L. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo, & S. Johnson. Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing, 43(6), B1105–B1132, 2021.

269. K. Goswami, A. Sharma, M. Pruthi, & R. Gupta. Study of drug assimilation in human system using physics informed neural networks. arXiv preprint arXiv:2110.05531, 2021.

270. C. Hennigan. The primal Hamiltonian: A new global approach to monetary policy. 2021.

271. S. Lee, & T. Kadeethum. Physics-informed neural networks for solving coupled flow and transport system. 2021.

272. Y. Chen, & L. Dal Negro. Physics-informed neural networks for imaging and parameter retrieval of photonic nanostructures from near-field data. arXiv preprint arXiv:2109.12754, 2021.

273. A. Ncube, G. Harmsen, & A. Cornell. Investigating a new approach to quasinormal modes: Physics-informed neural networks. arXiv preprint arXiv:2108.05867, 2021.

274. M. Almajid, & M. Abu-Alsaud. Prediction of porous media fluid flow using physics informed neural networks. Journal of Petroleum Science and Engineering, 109205, 2021.

275. J. Kuhlmann. Development of a physics-informed machine learning method for aerodynamic and fluids simulation. 2021.

276. E. Whalen. Enhancing surrogate models of engineering structures with graph-based and physics-informed learning. PhD dissertation, Massachusetts Institute of Technology, 2021.

277. M. Merkle. Boosting the training of physics-informed neural networks with transfer learning. 2021.

278. A. Warey, T. Han, & S. Kaushik. Investigation of numerical diffusion in aerodynamic flow simulations with physics informed neural networks. arXiv preprint arXiv:2103.03115, 2021.

279. L. Lu, X. Meng, Z. Mao, & G. Karniadakis. DeepXDE: A deep learning library for solving differential equations. SIAM Review, 63(1), 208–228, 2021.

280. V. Liu, & H. Yoon. Prediction of advection and diffusion transport using physics informed neural networks. 2020 AGU Fall Meeting, 2020.

281. A. Yazdani, L. Lu, M. Raissi, & G. Karniadakis. Systems biology informed deep learning for inferring parameters and hidden dynamics. PLoS Computational Biology, 16(11), e1007575, 2020.

282. A. Kapetanović, A. Šušnjara, & D. Poljak. Numerical solution and uncertainty quantification of bioheat transfer equation using neural network approach. 2020 5th International Conference on Smart and Sustainable Technologies (SpliTech)*, 2020.

283. Q. Zhang, Y. Chen, & Z. Yang. Data driven solutions and discoveries in mechanics using physics informed neural network. Preprints, 2020060258, 2020.

284. W. Peng, W. Zhou, J. Zhang, & W. Yao. Accelerating physics-informed neural network training with prior dictionaries. arXiv preprint arXiv:2004.08151, 2020.

285. Y. Chen, L. Lu, G. Karniadakis, & L. Negro. Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Optics Express, 28(8), 11618–11633, 2020.

286. G. Pang, L. Lu, & G. Karniadakis. fPINNs: Fractional physics-informed neural networks. SIAM Journal on Scientific Computing, 41(4), A2603–A2626, 2019.

287. D. Zhang, L. Lu, L. Guo, & G. Karniadakis. Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. Journal of Computational Physics, 397, 108850, 2019.

Deep neural operators

1. 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. npj Artificial Intelligence, 1, 35, 2025.

2. A. Jiao, Q. Yan, J. Harlim, & L. Lu. Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators. arXiv preprint arXiv:2407.05477, 2024.

3. J. Park, & N. Kang. Point-DeepONet: A Deep Operator Network Integrating PointNet for Nonlinear Analysis of Non-Parametric 3D Geometries and Load Conditions. arXiv preprint arXiv:2412.18362, 2024.

4. K. Lv, J. Wang, Y. Zhang, & H. Yu. Neural Operators for Adaptive Control of Freeway Traffic. arXiv preprint arXiv:2410.20708, 2024.

5. 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. Association for Computing Machinery, 1(3), September 2024.

6. C. García-Cervera, M. Kessler, P. Pedregal, & F. Periago. Universal approximation of set-valued maps and DeepONet approximation of the controllability map. ResearchGate, December 2024.

7. 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. Computer Methods in Applied Mechanics and Engineering, Volume 429, 117130, 2024.

8. 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. Additive Manufacturing, Volume 88, 104266, 2024.

9. O. Ovadia, A. Kahana, P. Stinis, E. Turkel, D. Givoli, & G. E. Karniadakis. ViTO: Vision Transformer-Operator. Computer Methods in Applied Mechanics and Engineering, Volume 428, 117109, 2024.

10. Z. Jiang, M. Zhu, & L. Lu. Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration. Reliability Engineering & System Safety, Volume 251, 110392, 2024.

11. 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. arXiv preprint arXiv:2409.05508, 2024.

12. K. Lv, J. Wang, & Y. Cao. Neural Operator Approximations for Boundary Stabilization of Cascaded Parabolic PDEs. International Journal of Adaptive Control and Signal Processing, Wiley Online Library, 2024.

13. 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. arXiv preprint arXiv:2409.00994, 2024.

14. P. Gao, G. E. Karniadakis, & P. Stinis. Multiscale modeling framework of a constrained fluid with complex boundaries using twin neural networks. arXiv preprint arXiv:2408.03263, 2024.

15. L. Xiao, G. Mei, & N. Xu. Knowledge-integrated deep learning for predicting stochastic thermal regime of embankment in permafrost region. Journal of Rock Mechanics and Geotechnical Engineering, Elsevier, 2024.

16. G. Fabiani, I. G. Kevrekidis, C. Siettos, & A. N. Yannacopoulos. RandONet: Shallow-networks with random projections for learning linear and nonlinear operators. Computer Methods in Applied Mechanics and Engineering, 429:117130, 2024.

17. A. Jiao, H. He, R. Ranade, J. Pathak, & L. Lu. One-shot learning for solution operators of partial differential equations. arXiv preprint arXiv:2104.05512, 2024.

18. S. Zampini, U. Zerbinati, G. Turkyyiah, & D. Keyes. PETScML: Second-order solvers for training regression problems in Scientific Machine Learning. In Proceedings of the Platform for Advanced Scientific Computing Conference, 1-12, 2024, June.

19. L. Branca & A. Pallottini. Emulating the interstellar medium chemistry with neural operators. Astronomy & Astrophysics, 684, A203, 2024.

20. J. Hayford, J. Goldman-Wetzler, E. Wang, & L. Lu. Speeding up and reducing memory usage for scientific machine learning via mixed precision. Computer Methods in Applied Mechanics and Engineering, 428, 117093, 2024.

21. K. Kobayashi, J. Daniell, & S.B. Alam. Improved generalization with deep neural operators for engineering systems: Path towards digital twin. Engineering Applications of Artificial Intelligence, 131, 107844, 2024.

22. K. Kobayashi & S.B. Alam. Deep neural operator-driven real-time inference to enable digital twin solutions for nuclear energy systems. Scientific Reports, 14, 2101, 2024.

23. H. Liu, B. Dahal, R. Lai, & W. Liao. Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning. arXiv preprint arXiv:2401.10490, 2024.

24. 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. JOM, 1-11, 2024.

25. M. Lamarque, L. Bhan, Y. Shi, & M. Krstic. Adaptive neural-operator backstepping control of a benchmark hyperbolic PDE. arXiv preprint arXiv:2401.07862, 2024.

26. K. Leng, M. Shankar, & J. Thiyagalingam. Zero coordinate shift: Whetted automatic differentiation for physics-informed operator learning. Journal of Computational Physics, 505, 112904, 2024.

27. M. Lamarque, L. Bhan, R. Vazquez, & M. Krstic. Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation. arXiv preprint arXiv:2401.02511, 2024.

28. A. Xavier. Solving Heat Conduction Problems with DeepONets. 2023.

29. L. Xu, H. Zhang, & M. Zhang. Training a deep operator network as a surrogate solver for two-dimensional parabolic-equation models. The Journal of the Acoustical Society of America, 154(5), 3276-3284, 2023.

30. N. Ford, V. J. Leon, H. Merman, J. Gilbert, & A. New. Data-efficient operator learning for solving high Mach number fluid flow problems. arXiv preprint arXiv:2311.16860, 2023.

31. B. Chen, C. Wang, W. Li, & H. Fu. A hybrid Decoder-DeepONet operator regression framework for unaligned observation data. arXiv preprint arXiv:2308.09274, 2023.

32. K. Kobayashi, & S. B. Alam. Potential of deep operator networks in digital twin-enabling technology for nuclear system. arXiv preprint arXiv:2308.07523, 2023.

33. 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. Engineering Applications of Artificial Intelligence, 127:107258, 2024.

34. E. L. Bolager, I. Burak, C. Datar, Q. Sun, & F. Dietrich. Sampling weights of deep neural networks. 2023.

35. V. Fanaskov, T. Yu, A. Rudikov, & I. Oseledets. General covariance data augmentation for neural PDE solvers. 2023.

36. 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. Computer Methods in Applied Mechanics and Engineering, 415:116277, 2023.

37. K. Kobayashi, J. Daniell, & S. B. Alam. Operator learning framework for digital twin and complex engineering systems. 2023.

38. 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. Computer Methods in Applied Mechanics and Engineering, 416, 116300, 2023.

39. 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. The Astrophysical Journal Letters, 950(2), L12, 2023.

40. S. Wang, & P. Perdikaris. Long-time integration of parametric evolution equations with physics-informed deeponets. Journal of Computational Physics, 475, p.111855, 2023.

41. E. Pickering, S. Guth, G. Karniadakis, & T. Sapsis. Discovering and forecasting extreme events via active learning in neural operators. Nature Computational Science, 2(12), pp.823-833, 2022.

42. S. Dhulipala, & R. Hruska. Efficient interdependent systems recovery modeling with DeepONets. 2022 Resilience Week (RWS), pp. 1-6. IEEE, 2022.

43. M. Zhu, H. Zhang, A. Jiao, G. Karniadakis, & L. Lu. Reliable extrapolation of deep neural operators informed by physics or sparse observations. Computer Methods in Applied Mechanics and Engineering, 412, 116064, 2023.

44. P. Clark Di Leoni, L. Lu, C. Meneveau, G. Karniadakis, & T. Zaki. Neural operator prediction of linear instability waves in high-speed boundary layers. Journal of Computational Physics, 474, 111793, 2023.

45. P. Jin, S. Meng, & L. Lu. MIONet: Learning multiple-input operators via tensor product. SIAM Journal on Scientific Computing, 44(6), A3490–A3514, 2022.

46. 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. Physical Review Research, 4(2), 023210, 2022.

47. 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. Computer Methods in Applied Mechanics and Engineering, 393, 114778, 2022.

48. L. Tan, & L. Chen. Enhanced DeepONet for modeling partial differential operators considering multiple input functions. arXiv preprint arXiv:2202.08942, 2022.

49. C. Lin, M. Maxey, Z. Li, & G. Karniadakis. A seamless multiscale operator neural network for inferring bubble dynamics. Journal of Fluid Mechanics, 929, A18, 2021.

50. 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. Journal of Computational Physics, 447, 110698, 2021.

51. S. Cai, Z. Wang, L. Lu, T. Zaki, & G. Karniadakis. DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks. Journal of Computational Physics, 436, 110296, 2021.

52. L. Lu, P. Jin, G. Pang, Z. Zhang, & G. Karniadakis. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3, 218–229, 2021.

53. C. Lin, Z. Li, L. Lu, S. Cai, M. Maxey, & G. Karniadakis. Operator learning for predicting multiscale bubble growth dynamics. The Journal of Chemical Physics, 154(10), 104118, 2021.

Multi-fidelity NN

1. L. Lu, M. Dao, P. Kumar, U. Ramamurty, G. Karniadakis, & S. Suresh. Extraction of mechanical properties of materials through deep learning from instrumented indentation. Proceedings of the National Academy of Sciences, 117(13), 7052–7062, 2020.

2. X. Meng, & G. Karniadakis. A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems. Journal of Computational Physics, 401, 109020, 2020.

Tutorial

1. An introductory course to PINNs and DeepONet