# Research

DeepXDE has been used in

> 80 universities, e.g., 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, University of British Columbia, Tsinghua University, Peking University, University of Texas at Austin, Georgia Institute of Technology, University of Manchester, University of California, Santa Barbara, Boston University, University of Colorado Boulder, University of Southern California, Pennsylvania State University, University of California Irvine, University of Oslo, Zhejiang University, University of Florida, University of California, Santa Cruz, Brown University, Purdue University, Kyoto University, Texas A&M University, University of Basel, Arizona State University, University of Massachusetts Amherst, Delft University of Technology, Tongji University, Rice University, University of Bergen, University of Naples Federico II, University of Electronic Science and Technology of China, KTH Royal Institute of Technology, Florida State University, Xiamen University, Beihang University, University of Strasbourg, China University of Geosciences, University Duisburg-Essen, University of Rome Tor Vergata, Chalmers University of Technology, University of Victoria, Wuhan University of Technology, University of Delaware, University of Kentucky, University of Surrey, Federal University of Rio de Janeiro, University of Johannesburg, University of Houston, Carleton University, University of Stuttgart, National University of Colombia, University of Calabria, Clemson University, Isfahan University of Technology, Nanchang University, Graz University of Technology, Missouri University of Science and Technology, California Polytechnic State University, University of Nevada, Las Vegas, Universiti Teknologi PETRONAS, Hangzhou Dianzi University, University of Kaiserslautern, Shanghai Normal University, Worcester Polytechnic Institute, Technical University of Cartagena, 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

> 15 national labs and research institutes, e.g., Pacific Northwest National Laboratory, Sandia National Laboratories, Argonne 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

industry, e.g., Anailytica, Ansys, BirenTech Research, ExxonMobil, General Motors, RocketML, Saudi Aramco

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

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, 16840, 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.S. Song, & H. Jin. Identifying constitutive parameters for complex hyperelastic solids using physics-informed neural networks.

*arXiv preprint arXiv:2308.15640*, 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.S. Alkhadhr, & M. Almekkawy. Wave equation modeling via physics-informed neural networks: models of soft and hard constraints for initial and boundary conditions.

*Sensors*, 23(5), 2792, 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.S. Li, G. Wang, Y. Di, L. Wang, H. Wang, & 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, p.105908, 2023.M. Bazmara, M. Silani, & M. Mianroodi. Physics-informed neural networks for nonlinear bending of 3D functionally graded beam.

*Structures*, Vol. 49, Elsevier, 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.P. Sharma, L. Evans, M. Tindall, & P. Nithiarasu. Stiff-PDEs and physics-informed neural networks.

*Archives of Computational Methods in Engineering*, 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.F. Pioch, J. Harmening, A. Müller, F. Peitzmann, D. Schramm, & O. Moctar. Turbulence modeling for physics-informed neural networks: Comparison of different RANS models for the backward-facing step flow.

*Fluids*, 8(2), p.43, 2023.Q. Jiang, M. Yu, M. Tang, B. Zhan, & S. Dong. Study on pile driving and sound propagation in shallow water using Physics-informed neural network. 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.F. Fonseca. A solution of a 3d Cartesian Poisson-Boltzmann equation.

*Contemporary Engineering Sciences*, Vol. 16, no. 1, 1-10, 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.N. Ma, B. Sun, K. Mao, B. Chen, & Y. Zhai. Physics-informed neural networks for solving nonlinear bloch equations in atomic magnetometer. 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.N. Ma, B. Sun, K. Mao, B. Chen, & Y. Zhai. Physics-informed neural networks for solving nonlinear bloch equations in atomic magnetometer. 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

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.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.S. Wang, & P. Perdikaris. Long-time integration of parametric evolution equations with physics-informed deeponets.

*Journal of Computational Physics*, 475, p.111855, 2023.K. Kobayashi, J. Daniell, & S. Alam. Operator learning framework for digital twin and complex engineering systems.

*arXiv e-prints*, pp.arXiv-2301, 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.