# Research

DeepXDE has been used in

> 150 universities, e.g., Harvard University, Massachusetts Institute of Technology, Stanford University, California Institute of Technology, University of California, Berkeley, Johns Hopkins University, University of Pennsylvania, Princeton University, Imperial College London, University of California, Los Angeles, Cornell University, Nanyang Technological University, New York University Abu Dhabi, University of British Columbia, Tsinghua University, Peking University, University of Texas at Austin, University of Pittsburgh, KU Leuven, Georgia Institute of Technology, University of Minnesota, University of Manchester, University of Wisconsin Madison, University of California, Santa Barbara, Boston University, University of Colorado Boulder, University of Illinois at Urbana-Champaign, Technical University of Munich, University of Southern California, Pennsylvania State University, University of California Irvine, University of Birmingham, University of Oslo, Zhejiang University, University of Florida, University of Southampton, University of California, Santa Cruz, Aarhus University, University of Chinese Academy of Sciences, Carnegie Mellon University, Sapienza University Rome, Brown University, University of Alberta, University of Bonn, Purdue University, Kyoto University, University of Liverpool, Texas A&M University, University of Basel, Arizona State University, University of Massachusetts Amherst, Delft University of Technology, Tongji University, University of Sussex, Tel Aviv University, Rice University, University of Bergen, RWTH Aachen University, University of Naples Federico II, Central South University, Southern University of Science and Technology, Xi’an Jiaotong University, University of Electronic Science and Technology of China, Southeast University, KTH Royal Institute of Technology, Florida State University, Xiamen University, Tufts University, Universidade do Porto, Kapodistrian University, Beihang University, University of Strasbourg, Tianjing University, Beijing Normal University, China University of Geosciences, Beijing Institute of Technology, University of Western Ontario, University Duisburg-Essen, University of Rome Tor Vergata, Chalmers University of Technology, University of Victoria, Northwestern Polytechnical University, Wuhan University of Technology, Eindhoven University of Technology, Kyushu University, University of Delaware, University of Kentucky, University of Surrey, University of Mississippi, University of Hawaii Manoa, University of Twente, Federal University of Rio de Janeiro, Shanghai University, University of Johannesburg, University of Houston, Carleton University, George Mason University, Universidade do Minho, University of Seville, Università degli Studi Roma Tre, Beijing Jiaotong University, Dortmund University of Technology, University of Stuttgart, AmirKabir University of Technology, National University of Colombia, University of Calabria, Clemson University, Rensselaer Polytechnic Institute, Tarbiat Modares University, Isfahan University of Technology, Nanchang University, South China Normal University, Universidad de salamanca, Harbin Engineering University, Graz University of Technology, Missouri University of Science and Technology, California Polytechnic State University, University of Nevada, Las Vegas, Universiti Teknologi PETRONAS, Sabanci University, University of Thessaly, University of Kragujevac, Old Dominion University, Macau University of Science & Technology, Universidade Federal Fluminense, Chung-Ang University, Brno University of Technology, Cadi Ayyad University, Zhejiang A&F University, Hangzhou Dianzi University, University of Kaiserslautern, Universidad Rey Juan Carlos, Shanghai Normal University, San Francisco State University, Taras Shevchenko National University Kiev, Worcester Polytechnic Institute, Nazarbayev University, Xinjiang University, Technical University of Cartagena, Dong A University, Universidad de Burgos, Bundeswehr University Munich, Adolfo Ibáñez University, Bauhaus-Universität Weimar, Henan Institute of Economics and Trade Swansea University, National University of Defence Technology, University of Applied Sciences and Arts Northwestern Switzerland, University of Los Andes, Kuwait University

> 30 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

> 10 industry, e.g., Anailytica, Ansys, BirenTech Research, ExxonMobil, General Motors, RocketML, Saudi Aramco, Shell, SoftServe, Fraunhofer Institute for Integrated Systems and Device Technology IISB, Quantiph

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

J. Seo. Solving real-world optimization tasks using physics-informed neural computing.

*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.

*Journal of Computational Physics*, 112761, 2024.N. Patel, A. Aykutalp, & P. Laguna. Calculating quasi-normal modes of Schwarzschild black holes with physics informed neural networks.

*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.

*Geophysical Research Letters*, 51(1), e2023GL106362, 2024.J. M. Tucny, M. Durve, A. Montessori, and S. Succi. Learning of viscosity functions in rarefied gas flows with physics-informed neural networks.

*Computers Fluids*, 269:106114, 2024.P. Brendel, V. Medvedev, & A. Rosskopf. Physics-informed neural networks for magnetostatic problems on axisymmetric transformer geometries.

*IEEE Journal of Emerging and Selected Topics in Industrial Electronics*, 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. 2023.

Z. Wang, R. Keller, X. Deng, K. Hoshino, T. Tanaka, & Y. Nakahira. Physics-informed representation and learning: Control and risk quantification.

*arXiv preprint arXiv:2312.10594*, 2023.S. H. Radbakhsh, K. Zandi, & M. Nik-bakht. Physics-informed neural network for analyzing elastic beam behavior.

*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.

*Smart Materials and Structures*, 33(1), 015016, 2023.S. Burbulla. Physics-informed neural networks for transformed geometries and manifolds.

*arXiv preprint arXiv:2311.15940*, 2023.B. Jang, A. A. Kaptanoglu, R. Gaur, S. Pan, M. Landreman, & W. Dorland. Grad-Shafranov equilibria via data-free physics informed neural networks.

*arXiv preprint arXiv:2311.13491*, 2023.C. Li. Enhancing Navier-Stokes flow learning through the level set approach.

*Available at SSRN 4641595*.X. Zhu, X. Hu, & P. Sun. Physics-informed neural networks for solving dynamic two-phase interface problems.

*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.

*arXiv preprint arXiv:2311.10456*, 2023.J. Plata Salas. Física asistida por redes neuronales artificiales.

*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.

*Computers & Mathematics with Applications*, 152, 355-363, 2023.H. Son, H. Cho, & H. J. Hwang. Physics-informed neural networks for microprocessor thermal management model.

*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.

*Axioms*, 12(10), 982, 2023.L. S. de Oliveira, L. Kunstmann, D. Pina, D. de Oliveira, & M. Mattoso. PINNProv: Provenance for physics-informed neural networks.

*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.

*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.

*Scientific Reports*, 13(1), 16840, 2023.K. Lo, & D. Huang. On Training Derivative-Constrained Neural Networks.

*arXiv preprint arXiv:2310.01649*, 2023.M. Ragoza, & M. Batmanghelich. Physics-informed neural networks for tissue elasticity reconstruction in magnetic resonance elastography.

*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.

*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.

*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.

*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.

*arXiv preprint arXiv:2309.16114*, 2023.I. Bendaoud. Approximation theory via deep neural networks and some applications.

J. Shi, K. Manjunatha, & S. Reese. Deep learning-based surrogate modeling of coronary in-stent restenosis.

*PAMM*, e202300090, 2023.F. Tangsijie, & L. Wei. The buckling analysis of thin-walled structures based on physics-informed neural networks.

*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.

*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.

*Scientific Reports*, 13(1), 15566, 2023.Y. Xu, & T. Zeng. Multi-grade deep learning for partial differential equations with applications to the Burgers equation.

*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.

M. Vais. Deep learning for the solution of differential equations.

L. Novák, H. Sharma, & M. D. Shields. Physics-informed polynomial chaos expansions.

*arXiv preprint arXiv:2309.01697*, 2023.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.

*In AIP Conference Proceedings (Vol. 2849, No. 1). AIP Publishing*, 2023.S. Song, & H. Jin. Identifying constitutive parameters for complex hyperelastic solids using physics-informed neural networks.

*arXiv preprint arXiv:2308.15640*, 2023.A. Moreira, M. Philipps, & N. Van Riel. Parameter estimation of a physiological diabetes model using neural networks.

*In 2023 IEEE Conference on Computational Intelligence in Bioinformatics and Computational Biology (CIBCB) (pp. 1-8). IEEE*, 2023.T. Sahin, M. von Danwitz, & M. Popp. Solving forward and inverse problems of contact mechanics using physics-informed neural networks.

*arXiv preprint arXiv:2308.12716*, 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.

*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.

*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.

*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.

*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.

*arXiv preprint arXiv:2308.08010*, 2023.D. Gazoulis, I. Gkanis, & C. G. Makridakis. On the stability and convergence of physics informed neural networks.

*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.

*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.

*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.

M. Z. Asadzadeh, K. Roppert, & P. Raninger. Material data identification in an induction hardening test rig with physics-informed neural networks.

*Materials*, 16(14), 5013, 2023.A. Ogueda, E. Martinez, V. Arunachalam, & P. Seshaiyer. Machine learning for predicting the dynamics of infectious diseases during travel through physics informed neural networks.

*Journal of Machine Learning for Modeling and Computing*, 2023.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*, 119636, 2023.W. Xuan, H. Lou, S. Fu, Z. Zhang, & N. Ding. Physics-informed deep learning method for the refrigerant filling mass flow metering.

*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.

*Sensors*, 23(5), 2023.M. Bazmara, M. Mianroodi, and M. Silani. Application of physics-informed neural networks for nonlinear buckling analysis of beams.

*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.

*Structures*, 49:152-162, 2023.J. Duan and H. Zhao. PINNs for sound propagation and sound speed field estimation simultaneously. 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.

*Engineering with Computers*, 2023.F. Fonseca. A solution of a 3D cartesian poisson-boltzmann equation.

*Contemporary Engineering Sciences*, 16(1):1-10, 2023.L. Fritschi and K. Lenk. Parameter inference for an astrocyte model using machine learning approaches.

*bioRxiv*, p. 2023-05, 2023.Z. Gong, Y. Chu, and S. Yang. Physics-informed neural networks for solving 2-D magnetostatic fields.

*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). 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.

*arXiv preprint arXiv:2306.08827*, 2023.J. H. Harmening, F. Pioch, L. Fuhrig, F.-J. Peitzmann, D. Schramm, and el Moctar. Data-assisted training of a physics-informed neural network to predict the Reynolds-averaged turbulent flow field around a stalled airfoil under variable angles of attack.

*Preprints*, 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.

*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).

*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. 2023.

N. V. Jagtap, M. Mudunuru, and K. Nakshatrala. CoolPINNs: A physics-informed neural network modeling of active cooling in vascular systems.

*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.

*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.

*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.

*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.

*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.

*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.

*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? 2023.

C. McDevitt. A physics-informed deep learning model of the hot tail runaway electron seed. 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.

*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. 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.

*Fluids*, 8(2), 2023.P. Sharma, L. Evans, M. Tindall, and P. Nithiarasu. Stiff-PDEs and physics-informed neural networks.

*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.

*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.

*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.

*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. 2023.

X. Zeng, S. Zhang, C. Ren, and T. Shao. Physics informed neural networks for electric field distribution characteristics analysis.

*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. 2023.

Z. Zhang and Z. Li. Haemoglobin response modelling under erythropoietin treatment: Physiological model-informed machine learning method.

*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.

*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.

*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.

*arXiv preprint arXiv:2305.09783*, 2023.T. Grossmann, U. Komorowska, J. Latz, & C. Schönlieb. Can physics-informed neural networks beat the finite element method?

*arXiv preprint arXiv:2302.04107*, 2023.L. Sliwinski, & G. Rigas. Mean flow reconstruction of unsteady flows using physics-informed neural networks.

*Data-Centric Engineering*, 4, p.e4, 2023.E. Lorin, & X. Yang. Schwarz waveform relaxation-learning for advection-diffusion-reaction equations.

*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.

*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.

*arXiv preprint arXiv:2212.13531*, 2022.R. Usman, & D. Amato. ML-Ops pipeline for improved physics-informed ODE modeling. 2022.

S. Saqlain, W. Zhu, E. Charalampidis, & P. Kevrekidis. Discovering governing equations in discrete systems using PINNs.

*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.

*Applied Mathematics and Mechanics*, 44(7), 1039-1068, 2023.C. McDevitt, E. Fowler, & S. Roy. Physics-constrained deep learning of incompressible cavity flows.

*arXiv preprint arXiv:2211.06375*, 2022.E. Lorin, & X. Yang. Time-dependent Dirac equation with physics-informed neural networks: Computation and properties.

*Computer Physics Communications*, 280, p.108474, 2022.Y. Ji. Solving singular Liouville equations using deep learning.

*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.

*Transport in Porous Media*, 1-24, 2022.A. Cornell, A. Ncube, & G. Harmsen. Determining QNMs using PINNs.

*arXiv preprint arXiv:2205.08284*, 2022.M. Mukhametzhanov. High precision differentiation techniques for data-driven solution of nonlinear PDEs by physics-informed neural networks.

*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.

*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).

*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.

*Energies*, 15(20), 7697, 2022.A. Zhu. Accelerating parameter inference in diffusion-reaction models of glioblastoma using physics-informed neural networks. 2022.

Y. Wang, J. Xing, K. Luo, H. Wang, & J. Fan. Solving combustion chemical differential equations via physics-informed neural network.

*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.

*Nano Energy*, 101, p.107586, 2022.M. Ferrante, A. Duggento, & N. Toschi. Physically constrained neural networks to solve the inverse problem for neuron models.

*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.

*arXiv preprint arXiv:2209.11929*, 2022.D. Sana. Approximating the wave equation via physics informed neural networks: Various forward and inverse problems. 2022.

C. Garcia-Cervera, M. Kessler, & F. Periago. Control of partial differential equations via physics-informed neural networks.

*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.

*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.

*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.

*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.

*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.

*arXiv preprint arXiv:2208.04319*, 2022.J. Yu. Indifference computer experiment for mathematical identification of two variables.

*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.

*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.

*arXiv preprint arXiv:2206.01545*, 2022.R. Anelli. Physics-informed neural networks for shallow water equations. 2022.

A. Konradsson. Physics-informed neural networks for charge dynamics in air.

*Master’s thesis in Complex Adaptive Systems*, 2022.X. Wang, J. Li, & J. Li. A deep learning based numerical PDE method for option pricing.

*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.

*arXiv preprint arXiv:2205.13748*, 2022.B. Dalen. Characterization of Cardiac cellular dynamics using physics-informed neural networks. 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.

*Computer Aided Geometric Design*, 95, p.102087, 2022.A. Ncube. Investigating new computational approaches for solving black hole perturbation equations.

*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. 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.

*arXiv preprint arXiv:2203.08501*, 2022.P. Escapil-Inchauspé, & G. A. Ruz. Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems.

*Neurocomputing*, 126826, 2023.P. Escapil-Inchauspé, & G. Ruz. Physics-informed neural networks for operator equations with stochastic data.

*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.

*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).

*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.

*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.

*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.

*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.

*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.

*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.

*Frontiers in cardiovascular medicine*, 2179, 2022.V. Schäfer. Generalization of physics-informed neural networks for various boundary and initial conditions.

*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.

*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.

*The University of Bergen*, 2021.S. Markidis. The old and the new: Can physics-informed deep-learning replace traditional linear solvers?.

*Frontiers in Big Data*, 4:669097, 2021.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.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.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.

*arXiv preprint arXiv:2111.03794*, 2021.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.C. Hennigan. The primal Hamiltonian: A new global approach to monetary policy. 2021.

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

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

*arXiv preprint arXiv:2108.05867*, 2021.M. Almajid, & M. Abu-Alsaud. Prediction of porous media fluid flow using physics informed neural networks.

*Journal of Petroleum Science and Engineering*, 109205, 2021.J. Kuhlmann. Development of a physics-informed machine learning method for aerodynamic and fluids simulation. 2021.

E. Whalen. Enhancing surrogate models of engineering structures with graph-based and physics-informed learning.

*PhD dissertation, Massachusetts Institute of Technology*, 2021.M. Merkle. Boosting the training of physics-informed neural networks with transfer learning. 2021.

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.L. Lu, X. Meng, Z. Mao, & G. Karniadakis. DeepXDE: A deep learning library for solving differential equations.

*SIAM Review*, 63(1), 208–228, 2021.V. Liu, & H. Yoon. Prediction of advection and diffusion transport using physics informed neural networks.

*2020 AGU Fall Meeting*, 2020.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.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.

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

*Preprints*, 2020060258, 2020.W. Peng, W. Zhou, J. Zhang, & W. Yao. Accelerating physics-informed neural network training with prior dictionaries.

*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.

*Optics Express*, 28(8), 11618–11633, 2020.G. Pang, L. Lu, & G. Karniadakis. fPINNs: Fractional physics-informed neural networks.

*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.

*Journal of Computational Physics*, 397, 108850, 2019.

## DeepONet

K. Leng, M. Shankar, & J. Thiyagalingam. Zero coordinate shift: Whetted automatic differentiation for physics-informed operator learning.

*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.

*arXiv preprint arXiv:2401.02511*, 2024.A. Xavier. Solving Heat Conduction Problems with DeepONets. 2023.

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.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.J. He, S. Kushwaha, J. Park, S. Koric, D. Abueidda, & I. Jasiuk. Multi-component predictions of transient solution fields with sequential deep operator network.

*arXiv preprint arXiv:2311.11500*, 2023.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.K. Kobayashi, & S. B. Alam. Potential of deep operator networks in digital twin-enabling technology for nuclear system.

*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.

*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. 2023.

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

*Computer Methods in Applied Mechanics and Engineering*, 415:116277, 2023.Z. Jiang, M. Zhu, D. Li, Q. Li, Y. Yuan, & L. Lu. Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration.

*arXiv preprint arXiv:2303.04778*, 2023.K. Kobayashi, J. Daniell, & S. B. Alam. Operator learning framework for digital twin and complex engineering systems. 2023.

O. Ovadia, A. Kahana, P. Stinis, E. Turkel, & G. E. Karniadakis. ViTO: Vision transformer-operator. 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.

*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.

*The Astrophysical Journal Letters*, 950(2), L12, 2023.S. Wang, & P. Perdikaris. Long-time integration of parametric evolution equations with physics-informed deeponets.

*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.

*Nature Computational Science*, 2(12), pp.823-833, 2022.S. Dhulipala, & R. Hruska. Efficient interdependent systems recovery modeling with DeepONets.

*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.

*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.

*Journal of Computational Physics*, 474, 111793, 2023.P. Jin, S. Meng, & L. Lu. MIONet: Learning multiple-input operators via tensor product.

*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.

*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.

*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.

*arXiv preprint arXiv:2202.08942*, 2022.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.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.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.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.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

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.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.

## Others

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*, 2021.