Học máy khoa học thông qua mạng nơ-ron có thông tin vật lý: Chúng ta đang ở đâu và điều gì đang chờ đón?

Salvatore Cuomo1, Vincenzo Schiano di Cola2, Fabio Giampaolo1, Gianluigi Rozza3, Maziar Raissi4, Francesco Piccialli1
1Department of Mathematics and Applications “Renato Caccioppoli”, University of Naples Federico II, Napoli, 80126, Italy
2Department of Electrical Engineering and Information Technology, University of Naples Federico II, Via Claudio, Napoli, 80125, Italy
3SISSA, International School for Advanced Studies, Mathematics Area, mathLab, via Bonomea 265, Trieste 34136, Italy
4Department of Applied Mathematics, University of Colorado – Boulder, Boulder, CO 80309, USA

Tóm tắt

Tóm tắt

Các Mạng Nơ-ron Có Thông Tin Vật Lý (PINN) là các mạng nơ-ron (NN) mà trong đó nội dung các phương trình mô hình, như Phương Trình Vi Phân Bộ (PDE), được mã hóa như một thành phần của chính mạng nơ-ron. Hiện nay, PINNs được sử dụng để giải các phương trình PDE, phương trình phân thức, phương trình tích phân-vi phân, và các phương trình PDE ngẫu nhiên. Phương pháp mới này đã xuất hiện như một khuôn khổ học tập đa nhiệm trong đó một NN phải khớp với dữ liệu quan sát trong khi giảm thiểu phần dư của PDE. Bài viết này cung cấp một đánh giá toàn diện về tài liệu liên quan đến PINNs: trong khi mục tiêu chính của nghiên cứu là xác định đặc điểm của các mạng này cùng những ưu điểm và nhược điểm liên quan của chúng. Đánh giá này cũng cố gắng đưa vào các công bố về một phạm vi rộng hơn của các mạng nơ-ron có thông tin vật lý dựa trên phương pháp phân bố điểm, mà xuất phát từ PINN cơ bản, cũng như nhiều biến thể khác, chẳng hạn như mạng nơ-ron bị hạn chế bởi vật lý (PCNN), hp-VPINN biến thiên, và PINN bảo tồn (CPINN). Nghiên cứu chỉ ra rằng hầu hết các nghiên cứu đã tập trung vào việc tùy chỉnh PINN qua các hàm kích hoạt khác nhau, kỹ thuật tối ưu hóa gradient, cấu trúc mạng nơ-ron, và cấu trúc hàm mất mát. Mặc dù có nhiều ứng dụng mà PINN đã được sử dụng, thông qua việc chứng minh khả năng của chúng dễ thực hiện hơn trong một số bối cảnh so với các kỹ thuật số truyền thống như Phương Pháp Phần Tử Hữu Hạn (FEM), vẫn còn những tiến bộ có thể xảy ra, đặc biệt là các vấn đề lý thuyết vẫn chưa được giải quyết.

Từ khóa


Tài liệu tham khảo

Abreu, E., Florindo, J.B.: A Study on a Feedforward Neural Network to Solve Partial Differential Equations in Hyperbolic-Transport Problems. In: Paszynski M, Kranzlmüller D, Krzhizhanovskaya VV, et al (eds) Computational Science – ICCS 2021. Springer International Publishing, Cham, Lecture Notes in Comput. Sci. pp. 398–411, (2021) https://doi.org/10.1007/978-3-030-77964-1_31

Aldweesh, A., Derhab, A., Emam, A.Z.: Deep learning approaches for anomaly-based intrusion detection systems: A survey, taxonomy, and open issues. Knowledge-Based Systems 189, 105,124 (2020). https://doi.org/10.1016/j.knosys.2019.105124, https://www.sciencedirect.com/science/article/pii/S0950705119304897

Alkhadhr, S., Liu, X., Almekkawy, M: Modeling of the Forward Wave Propagation Using Physics-Informed Neural Networks. In: 2021 IEEE International Ultrasonics Symposium (IUS), pp. 1–4, (2021) https://doi.org/10.1109/IUS52206.2021.9593574, iSSN: 1948-5727

Almajid, M.M., Abu-Al-Saud, M.O.: Prediction of porous media fluid flow using physics informed neural networks. J. Pet. Sci, Eng. 208, 109,205 (2022). https://doi.org/10.1016/j.petrol.2021.109205, https://www.sciencedirect.com/science/article/pii/S0920410521008597

Alom, M.Z., Taha, T.M., Yakopcic, C., et al: A state-of-the-art survey on deep learning theory and architectures. Electron. 8(3) (2019). https://doi.org/10.3390/electronics8030292, https://www.mdpi.com/2079-9292/8/3/292

Amini Niaki, S., Haghighat, E., Campbell, T., et al.: Physics-informed neural network for modelling the thermochemical curing process of composite-tool systems during manufacture. Computer Methods in Applied Mechanics and Engineering 384, 113,959 (2021). https://doi.org/10.1016/j.cma.2021.113959, https://www.sciencedirect.com/science/article/pii/S0045782521002966

Araz, J.Y., Criado, J.C., Spannowsky, M: Elvet – a neural network-based differential equation and variational problem solver (2021). arXiv:2103.14575 [hep-lat, physics:hep-ph, physics:hep-th, stat] , arXiv: 2103.14575

Arnold, D.N.: Stability, Consistency, and Convergence of Numerical Discretizations, pp. 1358–1364. Springer, Berlin, Heidelberg (2015). https://doi.org/10.1007/978-3-540-70529-1_407

Arnold, F., King, R: State–space modeling for control based on physics-informed neural networks. Eng. Appl. Artif. Intell. 101, 104,195 . https://doi.org/10.1016/j.engappai.2021.104195, https://www.sciencedirect.com/science/article/pii/S0952197621000427

Arthurs, C.J., King, A.P.: Active training of physics-informed neural networks to aggregate and interpolate parametric solutions to the Navier-Stokes equations. J. Comput. Phys. 438:110,364 (2021). https://doi.org/10.1016/j.jcp.2021.110364, https://www.sciencedirect.com/science/article/pii/S002199912100259X

Arulkumaran, K., Deisenroth, M.P., Brundage, M., et al.: Deep Reinforcement Learning: A Brief Survey. IEEE Signal Process. Mag. 34(6), 26–38 (2017). https://doi.org/10.1109/MSP.2017.2743240

Balu, A., Botelho, S., Khara, B., et al.: Distributed multigrid neural solvers on megavoxel domains. In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis. Association for Computing Machinery, New York, NY, USA, SC ’21, (2021) https://doi.org/10.1145/3458817.3476218

Bauer, B.: Kohler, M: On deep learning as a remedy for the curse of dimensionality in nonparametric regression. Ann. Statist. 47(4), 2261–2285 (2019). https://doi.org/10.1214/18-AOS1747, http://projecteuclid.org/journals/annals-of-statistics/volume-47/issue-4/On-deep-learning-as-a-remedy-for-the-curse-of/10.1214/18-AOS1747.full

Belkin, M., Hsu, D., Ma, S., et al.: Reconciling modern machine-learning practice and the classical bias-variance trade-off. Proc. Nat. Acad. Sci. India Sect. 116(32), 15849–15854 (2019). https://doi.org/10.1073/pnas.1903070116, www.pnas.org/doi/10.1073/pnas.1903070116

Bellman, R.: Dynamic programming. Sci. 153(3731), 34–37 (1966)

Berg, J., Nyström, K.: A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317, 28–41 (2018). https://doi.org/10.1016/j.neucom.2018.06.056, www.sciencedirect.com/science/article/pii/S092523121830794X

Berman, D.S., Buczak, A.L., Chavis, J.S., et al.: A survey of deep learning methods for cyber security. Inform. 10(4) (2019). https://doi.org/10.3390/info10040122, https://www.mdpi.com/2078-2489/10/4/122

Biswas, A., Tian, J., Ulusoy, S.: Error estimates for deep learning methods in fluid dynamics. Numer. Math. 151(3), 753–777 (2022). https://doi.org/10.1007/s00211-022-01294-z

Blechschmidt, J., Ernst, O.G.: Three ways to solve partial differential equations with neural networks – A review. GAMM-Mitteilungen 44(2), e202100,006 (2021). https://doi.org/10.1002/gamm.202100006, https://onlinelibrary.wiley.com/doi/abs/10.1002/gamm.202100006

Cai, S., Mao, Z., Wang, Z., et al.: Physics-informed neural networks (PINNs) for fluid mechanics: a review. Acta. Mech. Sin. 37(12), 1727–1738 (2021). https://doi.org/10.1007/s10409-021-01148-1

Cai, S., Wang, Z., Lu, L., et al.: DeepM &Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks. J. Comput. Phys. 436, 110,296 (2021b). https://doi.org/10.1016/j.jcp.2021.110296, https://www.sciencedirect.com/science/article/pii/S0021999121001911

Cai, S., Wang, Z., Wang, S., et al.: Physics-Informed Neural Networks for Heat Transfer Problems. J. Heat Transf. 143(6) (2021c). https://doi.org/10.1115/1.4050542

Calin, O.: Convolutional Networks, pp. 517–542. Springer International Publishing, Cham (2020). https://doi.org/10.1007/978-3-030-36721-3_16

Calin, O.: Universal Approximators, pp. 251–284. Springer Series in the Data Sciences, Springer International Publishing, Cham (2020b). https://doi.org/10.1007/978-3-030-36721-3_9

Caterini, A.L., Chang, D.E.: In: Caterini, A.L., Chang, D.E. (eds.) Deep Neural Networks in a Mathematical Framework. Generic Representation of Neural Networks, pp. 23–34. SpringerBriefs in Computer Science, Springer International Publishing, Cham (2018). https://doi.org/10.1007/978-3-319-75304-1_3

Caterini, A.L., Chang, D.E.: Specific Network Descriptions. In: Caterini, A.L., Chang, D.E. (eds.) Deep Neural Networks in a Mathematical Framework, pp. 35–58. Springer International Publishing, Cham, SpringerBriefs in Computer Science (2018). https://doi.org/10.1007/978-3-319-75304-1_4

Cavanagh, H., Mosbach, A., Scalliet, G., et al.: Physics-informed deep learning characterizes morphodynamics of asian soybean rust disease. Nat. Commun. 12(1), 6424 (2021). https://doi.org/10.1038/s41467-021-26577-1

Chen, F., Sondak, D., Protopapas, P., et al.: Neurodiffeq: A python package for solving differential equations with neural networks. J. Open Source Softw. 5(46), 1931 (2020)

Chen, H., Engkvist, O., Wang, Y., et al.: The rise of deep learning in drug discovery. Drug Discov. Today 23(6), 1241–1250 (2018). https://doi.org/10.1016/j.drudis.2018.01.039, www.sciencedirect.com/science/article/pii/S1359644617303598

Chen, Y., Lu, L., Karniadakis, G.E., et al.: Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Opt. Express 28(8), 11618–11633 (2020). https://doi.org/10.1364/OE.384875, www.osapublishing.org/oe/abstract.cfm?uri=oe-28-8-11618

Cheng, C., Zhang, G.T.: Deep Learning Method Based on Physics Informed Neural Network with Resnet Block for Solving Fluid Flow Problems. Water 13(4), 423 (2021). https://doi.org/10.3390/w13040423, www.mdpi.com/2073-4441/13/4/423

Cheung, K.C., See, S.: Recent advance in machine learning for partial differential equation. CCF Trans. High Performance Comput. 3(3), 298–310 (2021). https://doi.org/10.1007/s42514-021-00076-7

Chiu, P.H., Wong, J.C., Ooi, C., et al.: CAN-PINN: A fast physics-informed neural network based on coupled-automatic–numerical differentiation method. Comput. Methods Appl. Mech. Engrg. 395, 114,909 (2022). https://doi.org/10.1016/j.cma.2022.114909, https://www.sciencedirect.com/science/article/pii/S0045782522001906

Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Systems 2(4), 303–314 (1989). https://doi.org/10.1007/BF02551274

Dargan, S., Kumar, M., Ayyagari, M.R., et al.: A Survey of Deep Learning and Its Applications: A New Paradigm to Machine Learning. Arch. Comput. Methods Engrg. 27(4), 1071–1092 (2020). https://doi.org/10.1007/s11831-019-09344-w

De Ryck, T., Mishra, S.: Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs. (2021) arXiv:2106.14473 [cs, math]

De Ryck, T., Lanthaler, S., Mishra, S.: On the approximation of functions by tanh neural networks. Neural Netw. 143, 732–750 (2021). https://doi.org/10.1016/j.neunet.2021.08.015, www.sciencedirect.com/science/article/pii/S0893608021003208

De Ryck, T., Jagtap, A.D., Mishra, S.: Error estimates for physics informed neural networks approximating the Navier-Stokes equations. (2022) arXiv:2203.09346 [cs, math]

Dissanayake, M.W.M.G., Phan-Thien, N.: Neural-network-based approximations for solving partial differential equations. Commun. Numer. Methods Eng. 10(3), 195–201 (1994). https://doi.org/10.1002/cnm.1640100303, https://onlinelibrary.wiley.com/doi/abs/10.1002/cnm.1640100303

Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide. Pafnuty Publications, http://www.chebfun.org/docs/guide/ (2014)

Dwivedi, V., Srinivasan, B.: Physics Informed Extreme Learning Machine (PIELM)-A rapid method for the numerical solution of partial differential equations. Neurocomputing 391, 96–118 (2020). https://doi.org/10.1016/j.neucom.2019.12.099, www.sciencedirect.com/science/article/pii/S0925231219318144

EW, Yu. B.: The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems. Commun. Math. Stat. 6(1), 1–12 (2018). https://doi.org/10.1007/s40304-018-0127-z

Elbrächter, D., Perekrestenko, D., Grohs, P., et al.: Deep Neural Network Approximation Theory. IEEE Trans. Inf. Theory 67(5), 2581–2623 (2021). https://doi.org/10.1109/TIT.2021.3062161

Fang, Z.: A High-Efficient Hybrid Physics-Informed Neural Networks Based on Convolutional Neural Network. IEEE Transactions on Neural Networks and Learning Systems pp. 1–13. (2021) https://doi.org/10.1109/TNNLS.2021.3070878

Fang, Z., Zhan, J.: Deep Physical Informed Neural Networks for Metamaterial Design. IEEE Access 8, 24506–24513 (2020). https://doi.org/10.1109/ACCESS.2019.2963375

Fang, Z., Zhan, J.: A Physics-Informed Neural Network Framework for PDEs on 3D Surfaces: Time Independent Problems. IEEE Access 8, 26328–26335 (2020). https://doi.org/10.1109/ACCESS.2019.2963390

Fuks, O., Tchelepi, H.A.: LIMITATIONS OF PHYSICS INFORMED MACHINE LEARNING FOR NONLINEAR TWO-PHASE TRANSPORT IN POROUS MEDIA. Journal of Machine Learning for Modeling and Computing 1(1) (2020). https://doi.org/10.1615/.2020033905, https://www.dl.begellhouse.com/journals/558048804a15188a,583c4e56625ba94e,415f83b5707fde65.html

Gao, H., Sun, L., Wang, J.X.: PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain. J. Comput. Phys. 428, 110,079 (2021). https://doi.org/10.1016/j.jcp.2020.110079, https://www.sciencedirect.com/science/article/pii/S0021999120308536

Gardner, J.R., Pleiss, G., Bindel, D., et al.: Gpytorch: Blackbox matrix-matrix gaussian process inference with gpu acceleration. In: Advances in Neural Information Processing Systems (2018)

Garnelo, M., Shanahan, M.: Reconciling deep learning with symbolic artificial intelligence: representing objects and relations. Curr. Opinion in Behav. Sci. 29, 17–23 (2019). https://doi.org/10.1016/j.cobeha.2018.12.010, www.sciencedirect.com/science/article/pii/S2352154618301943

Geneva, N., Zabaras, N.: Modeling the dynamics of pde systems with physics-constrained deep auto-regressive networks. J. Comput. Phys. 403, 109,056 (2020). https://doi.org/10.1016/j.jcp.2019.109056, https://www.sciencedirect.com/science/article/pii/S0021999119307612

Goswami, S., Anitescu, C., Chakraborty, S., et al.: Transfer learning enhanced physics informed neural network for phase-field modeling of fracture. Theoret. Appl. Fracture Mech. 106, 102,447 (2020). https://doi.org/10.1016/j.tafmec.2019.102447, https://www.sciencedirect.com/science/article/pii/S016784421930357X

Grandits, T., Pezzuto, S., Costabal, F.S., et al.: Learning Atrial Fiber Orientations and Conductivity Tensors from Intracardiac Maps Using Physics-Informed Neural Networks. In: Ennis, D.B., Perotti, L.E., Wang, V.Y. (eds) Functional Imaging and Modeling of the Heart. Springer International Publishing, Cham, Lecture Notes in Comput. Sci., pp. 650–658 (2021), https://doi.org/10.1007/978-3-030-78710-3_62

Grubišić, L., Hajba, M., Lacmanović, D.: Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential. Entropy 23(1), 95 (2021). https://doi.org/10.3390/e23010095, www.mdpi.com/1099-4300/23/1/95

Haghighat, E., Juanes, R.: SciANN: A Keras/Tensorflow wrapper for scientific computations and physics-informed deep learning using artificial neural networks. Comput. Methods Appl. Mech. Engrg. 373, 113,552 (2021). https://doi.org/10.1016/j.cma.2020.113552, arXiv: 2005.08803

Haghighat, E., Bekar, A.C., Madenci, E., et al.: A nonlocal physics-informed deep learning framework using the peridynamic differential operator. Comput. Methods Appl. Mech. Engrg. 385, 114,012 (2021a). https://doi.org/10.1016/j.cma.2021.114012, https://www.sciencedirect.com/science/article/pii/S0045782521003431

Haghighat, E., Raissi, M., Moure, A., et al.: A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Comput. Methods Appl. Mech. Engrg. 379, 113,741 (2021b). https://doi.org/10.1016/j.cma.2021.113741, https://www.sciencedirect.com/science/article/pii/S0045782521000773

Haitsiukevich, K., Ilin, A.: Improved Training of Physics-Informed Neural Networks with Model Ensembles. (2022) arXiv:2204.05108 [cs, stat]

He, Q., Tartakovsky, A.M.: Physics-informed neural network method for forward and backward advection-dispersion equations. Water Resources Research 57(7), e2020WR029,479 (2021). https://doi.org/10.1029/2020WR029479, https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020WR029479, e2020WR029479 2020WR029479

He, Q., Barajas-Solano, D., Tartakovsky, G., et al.: Physics-informed neural networks for multiphysics data assimilation with application to subsurface transport. Adv. Water Resources 141, 103,610 (2020). https://doi.org/10.1016/j.advwatres.2020.103610, https://www.sciencedirect.com/science/article/pii/S0309170819311649

Hennigh, O., Narasimhan, S., Nabian, M.A., et al.: NVIDIA SimNet: An AI-Accelerated Multi-Physics Simulation Framework. In: Paszynski M, Kranzlmüller D, Krzhizhanovskaya VV, et al (eds) Computational Science – ICCS 2021. Springer International Publishing, Cham, Lecture Notes in Comput. Sci., pp. 447–461 (2021), https://doi.org/10.1007/978-3-030-77977-1_36

Hillebrecht, B., Unger, B.: Certified machine learning: A posteriori error estimation for physics-informed neural networks. Tech. rep., (2022) https://doi.org/10.48550/arXiv.2203.17055, arXiv:2203.17055 [cs, math] type: article

Hinze, M., Pinnau, R., Ulbrich, M., et al.: Optimization with PDE constraints, vol. 23. Springer Science & Business Media, Berlin (2008)

Hoffer, J.G., Geiger, B.C., Ofner, P., et al.: Mesh-Free Surrogate Models for Structural Mechanic FEM Simulation: A Comparative Study of Approaches. Appl. Sci. 11(20), 9411 (2021). https://doi.org/10.3390/app11209411, www.mdpi.com/2076-3417/11/20/9411

Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Networks 2(5), 359–366 (1989). https://doi.org/10.1016/0893-6080(89)90020-8, www.sciencedirect.com/science/article/pii/0893608089900208

Huang, G.B., Wang, D.H., Lan, Y.: Extreme learning machines: a survey. Int. J. Mach. Learn. Cybern. 2(2), 107–122 (2011). https://doi.org/10.1007/s13042-011-0019-y

Huang, X., Liu, H., Shi, B., et al.: Solving Partial Differential Equations with Point Source Based on Physics-Informed Neural Networks. (2021) arXiv:2111.01394 [physics]

Irrgang, C., Boers, N., Sonnewald, M., et al.: Towards neural Earth system modelling by integrating artificial intelligence in Earth system science. Nat. Mach. Intelligence 3(8), 667–674 (2021). https://doi.org/10.1038/s42256-021-00374-3, www.nature.com/articles/s42256-021-00374-3

Islam, M., Thakur, M.S.H., Mojumder, S., et al.: Extraction of material properties through multi-fidelity deep learning from molecular dynamics simulation. Comput. Mater. Sci. 188, 110,187 (2021). https://doi.org/10.1016/j.commatsci.2020.110187, https://www.sciencedirect.com/science/article/pii/S0927025620306789

Jagtap, A.D., Kawaguchi, K., Karniadakis, G.E.: Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J. Comput. Phys. 404, 109,136 (2020a). https://doi.org/10.1016/j.jcp.2019.109136, https://www.sciencedirect.com/science/article/pii/S0021999119308411

Jagtap, A.D., Kharazmi, E., Karniadakis, G.E.: Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems. Comput. Methods Appl. Mech. Engrg. 365, 113,028 (2020b). https://doi.org/10.1016/j.cma.2020.113028, https://www.sciencedirect.com/science/article/pii/S0045782520302127

Jamali, B., Haghighat, E., Ignjatovic, A., et al.: Machine learning for accelerating 2D flood models: Potential and challenges. Hydrological Processes 35(4), e14,064 (2021). https://doi.org/10.1002/hyp.14064, https://onlinelibrary.wiley.com/doi/abs/10.1002/hyp.14064

Jin, X., Cai, S., Li, H., et al.: NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. J. Comput. Phys. 426, 109,951 (2021). https://doi.org/10.1016/j.jcp.2020.109951, https://www.sciencedirect.com/science/article/pii/S0021999120307257

Karniadakis, G.E., Kevrekidis, I.G., Lu, L., et al.: Physics-informed machine learning. Nature Reviews Phys. 3(6), 422–440 (2021). https://doi.org/10.1038/s42254-021-00314-5, www.nature.com/articles/s42254-021-00314-5

Kashinath, K., Mustafa, M., Albert, A., et al.: Physics-informed machine learning: case studies for weather and climate modelling. Philosophical Transactions of the Royal Society A: Mathematical, Phys. Eng. Sci. 379(2194), 20200,093 (2021). https://doi.org/10.1098/rsta.2020.0093, https://royalsocietypublishing.org/doi/full/10.1098/rsta.2020.0093

Kharazmi, E., Zhang, Z., Karniadakis, G.E.: Variational Physics-Informed Neural Networks For Solving Partial Differential Equations. (2019) arXiv:1912.00873 [physics, stat]

Kharazmi, E., Cai, M., Zheng, X., et al.: Identifiability and predictability of integer- and fractional-order epidemiological models using physics-informed neural networks. Nature Comput. Sci. 1(11), 744–753 (2021). https://doi.org/10.1038/s43588-021-00158-0

Kharazmi, E., Zhang, Z., Karniadakis, G.E.M.: hp-VPINNs: Variational physics-informed neural networks with domain decomposition. Comput. Methods Appl. Mech. Engrg. 374, 113,547 (2021b). https://doi.org/10.1016/j.cma.2020.113547, https://www.sciencedirect.com/science/article/pii/S0045782520307325

Kim, J., Lee, K., Lee, D., et al.: DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation. Proc. AAAI Conf. Artif. Intell. 35(9), 8146–8154 (2021a). https://ojs.aaai.org/index.php/AAAI/article/view/16992

Kim, S.W., Kim, I., Lee, J., et al.: Knowledge Integration into deep learning in dynamical systems: an overview and taxonomy. J. Mech. Sci. Technol. 35(4), 1331–1342 (2021). https://doi.org/10.1007/s12206-021-0342-5

Kissas, G., Yang, Y., Hwuang, E., et al.: Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks. Comput. Methods Appl. Mech. Engrg. 358, 112,623 (2020). https://doi.org/10.1016/j.cma.2019.112623, https://www.sciencedirect.com/science/article/pii/S0045782519305055

Kollmannsberger, S., D’Angella, D., Jokeit, M., et al.: Physics-Informed Neural Networks. In: Kollmannsberger, S., D’Angella, D., Jokeit, M., et al. (eds.) Deep Learning in Computational Mechanics, pp. 55–84. Studies in Computational Intelligence, Springer International Publishing, Cham (2021). https://doi.org/10.1007/978-3-030-76587-3_5

Kondor, R., Trivedi, S.: On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups. In: Dy J, Krause A (eds) Proceedings of the 35th International Conference on Machine Learning, Proc. Mach. Learn. Res., vol 80. PMLR, pp. 2747–2755 (2018), https://proceedings.mlr.press/v80/kondor18a.html

Koryagin, A., Khudorozkov, R., Tsimfer, S.: PyDEns: a Python Framework for Solving Differential Equations with Neural Networks. (2019) arXiv:1909.11544 [cs, stat]

Kovacs, A., Exl, L., Kornell, A., et al.: Conditional physics informed neural networks. Commun. Nonlinear Sci. Numer. Simulation 104, 106,041 (2022). https://doi.org/10.1016/j.cnsns.2021.106041, https://www.sciencedirect.com/science/article/pii/S1007570421003531

Krishnapriyan, A., Gholami, A., Zhe, S., et al.: Characterizing possible failure modes in physics-informed neural networks. In: Ranzato, M., Beygelzimer, A., Dauphin, Y., et al (eds) Advances in Neural Information Processing Systems, vol 34. Curran Associates, Inc., pp. 26,548–26,560 (2021), https://proceedings.neurips.cc/paper/2021/file/df438e5206f31600e6ae4af72f2725f1-Paper.pdf

Kumar, M., Yadav, N.: Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey. Computers & Mathematics with Applications 62(10), 3796–3811 (2011). https://doi.org/10.1016/j.camwa.2011.09.028, www.sciencedirect.com/science/article/pii/S0898122111007966

Kutyniok, G.: The Mathematics of Artificial Intelligence (2022). arXiv:2203.08890 [cs, math, stat]

Lagaris, I., Likas, A., Fotiadis, D.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Networks 9(5), 987–1000 (1998). https://doi.org/10.1109/72.712178

Lagaris, I., Likas, A., Papageorgiou, D.: Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans. Neural Networks 11(5), 1041–1049 (2000). https://doi.org/10.1109/72.870037

Lai, Z., Mylonas, C., Nagarajaiah, S., et al.: Structural identification with physics-informed neural ordinary differential equations. J. Sound and Vibration 508, 116,196 (2021). https://doi.org/10.1016/j.jsv.2021.116196, https://www.sciencedirect.com/science/article/pii/S0022460X21002686

LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436–444 (2015). https://doi.org/10.1038/nature14539

Lee, H., Kang, I.S.: Neural algorithm for solving differential equations. J. Comput. Phys. 91(1), 110–131 (1990). https://doi.org/10.1016/0021-9991(90)90007-N, www.sciencedirect.com/science/article/pii/002199919090007N

Li, W., Bazant, M.Z., Zhu, J.: A physics-guided neural network framework for elastic plates: Comparison of governing equations-based and energy-based approaches. Comput. Methods Appl. Mech. Engrg. 383, 113,933 (2021). https://doi.org/10.1016/j.cma.2021.113933, https://www.sciencedirect.com/science/article/pii/S004578252100270X

Lin, C., Li, Z., Lu, L., et al.: Operator learning for predicting multiscale bubble growth dynamics. J. Chem. Phys. 154(10), 104,118 (2021a). https://doi.org/10.1063/5.0041203, https://aip.scitation.org/doi/10.1063/5.0041203

Lin, C., Maxey, M., Li, Z., et al.: A seamless multiscale operator neural network for inferring bubble dynamics. J. Fluid Mech. 929 (2021b). https://doi.org/10.1017/jfm.2021.866, https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/seamless-multiscale-operator-neural-network-for-inferring-bubble-dynamics/D516AB0EF954D0FF56AD864DB2618E94

Liu, D., Wang, Y.: A Dual-Dimer method for training physics-constrained neural networks with minimax architecture. Neural Netw. 136, 112–125 (2021). https://doi.org/10.1016/j.neunet.2020.12.028, www.sciencedirect.com/science/article/pii/S0893608020304536

Lu, L., Dao, M., Kumar, P., et al.: Extraction of mechanical properties of materials through deep learning from instrumented indentation. Proc. Nat. Acad. Sci. India Sect. 117(13), 7052–7062 (2020). https://doi.org/10.1073/pnas.1922210117, www.pnas.org/content/117/13/7052

Lu, L., Jin, P., Pang, G., et al.: Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat. Mac. Intell. 3(3), 218–229 (2021). https://doi.org/10.1038/s42256-021-00302-5, www.nature.com/articles/s42256-021-00302-5

Lu, L., Meng, X., Mao, Z., et al.: DeepXDE: A deep learning library for solving differential equations. SIAM Rev. 63(1), 208–228 (2021). https://doi.org/10.1137/19M1274067

Lu, L., Pestourie, R., Yao, W., et al.: Physics-informed neural networks with hard constraints for inverse design. SIAM J. Sci. Comput. 43(6), B1105–B1132 (2021). https://doi.org/10.1137/21M1397908

Mallat, S.: Understanding deep convolutional networks. Philosophical Transactions of the Royal Society A: Mathematical, Phys. Eng. Sci. 374(2065), 20150,203 (2016). https://doi.org/10.1098/rsta.2015.0203, https://royalsocietypublishing.org/doi/10.1098/rsta.2015.0203

Mao, Z., Jagtap, A.D., Karniadakis, G.E.: Physics-informed neural networks for high-speed flows. Comput. Methods Appl. Mech. Engrg. 360, 112,789 (2020). https://doi.org/10.1016/j.cma.2019.112789, https://www.sciencedirect.com/science/article/pii/S0045782519306814

Mao, Z., Lu, L., Marxen, O., et al.: DeepM &Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators. J. Comput. Phys. 447, 110,698 (2021). https://doi.org/10.1016/j.jcp.2021.110698, https://www.sciencedirect.com/science/article/pii/S0021999121005933

Markidis, S.: The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers? Frontiers in Big Data 4 (2021). https://www.frontiersin.org/article/10.3389/fdata.2021.669097

Mathews, A., Francisquez, M., Hughes, J.W., et al.: Uncovering turbulent plasma dynamics via deep learning from partial observations. Phys. Review E 104(2) (2021). https://doi.org/10.1103/physreve.104.025205, https://www.osti.gov/pages/biblio/1813020

McClenny, L.D., Haile, M.A., Braga-Neto, U.M.: Tensordiffeq: Scalable multi-gpu forward and inverse solvers for physics informed neural networks. (2021) arXiv preprint arXiv:2103.16034

Mehta, P.P., Pang, G., Song, F., et al.: Discovering a universal variable-order fractional model for turbulent couette flow using a physics-informed neural network. Fract. Calc. Appl. Anal. 22(6), 1675–1688 (2019). https://doi.org/10.1515/fca-2019-0086

Meng, X., Li, Z., Zhang, D., et al.: Ppinn: Parareal physics-informed neural network for time-dependent pdes. Comput. Methods Appl. Mech. Engrg. 370, 113,250 (2020). https://doi.org/10.1016/j.cma.2020.113250, https://www.sciencedirect.com/science/article/pii/S0045782520304357

Minh Nguyen-Thanh, V., Trong Khiem Nguyen, L., Rabczuk, T., et al.: A surrogate model for computational homogenization of elastostatics at finite strain using high-dimensional model representation-based neural network. Int. J. Numer. Methods Eng. 121(21), 4811–4842 (2020). https://doi.org/10.1002/nme.6493, https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.6493

Mishra, S., Molinaro, R.: Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs. IMA J. Numer. Anal. (2021). https://doi.org/10.1093/imanum/drab032

Mishra, S., Molinaro, R.: Physics informed neural networks for simulating radiative transfer. J. Quant. Spectroscopy and Radiative Transf. 270, 107,705 (2021b). https://doi.org/10.1016/j.jqsrt.2021.107705, https://www.sciencedirect.com/science/article/pii/S0022407321001989

Mishra, S., Molinaro, R.: Estimates on the generalization error of physics-informed neural networks for approximating PDEs. IMA J. Numer. Anal. p drab093 (2022). https://doi.org/10.1093/imanum/drab093

Misyris, G.S., Venzke, A., Chatzivasileiadis, S.: Physics-informed neural networks for power systems. 2020 IEEE Power & Energy Society General Meeting (PESGM) pp. 1–5 (2020)

Mo, Y., Ling, L., Zeng, D.: Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithm. Phys. Lett. A 421, 127,739 (2022). https://doi.org/10.1016/j.physleta.2021.127739, https://www.sciencedirect.com/science/article/pii/S0375960121006034

Moseley, B., Markham, A., Nissen-Meyer, T.: Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations. (2021) arXiv:2107.07871 [physics]

Muhammad, A.N., Aseere, A.M., Chiroma, H., et al.: Deep learning application in smart cities: recent development, taxonomy, challenges and research prospects. Neural Comput. Appl. 33(7), 2973–3009 (2021). https://doi.org/10.1007/s00521-020-05151-8

Nabian, M.A., Gladstone, R.J., Meidani, H.: Efficient training of physics-informed neural networks via importance sampling. Comput. Aided Civil Infrastruct. Eng. 36(8), 962–977 (2021). https://doi.org/10.1111/mice.12685, https://onlinelibrary.wiley.com/doi/abs/10.1111/mice.12685

Nandi, T., Hennigh, O., Nabian, M., et al.: Progress Towards Solving High Reynolds Number Reacting Flows in SimNet. Tech. rep., (2021) https://www.osti.gov/biblio/1846970-progress-towards-solving-high-reynolds-number-reacting-flows-simnet

Nandi, T., Hennigh, O., Nabian, M., et al.: Developing Digital Twins for Energy Applications Using Modulus. Tech. rep., (2022) https://www.osti.gov/biblio/1866819

Nascimento, R.G., Fricke, K., Viana, F.A.: A tutorial on solving ordinary differential equations using python and hybrid physics-informed neural network. Eng. Appl. Artif. Intell. 96, 103,996. (2020) https://doi.org/10.1016/j.engappai.2020.103996, https://www.sciencedirect.com/science/article/pii/S095219762030292X

Novak, R., Xiao, L., Hron, J., et al.: Neural tangents: Fast and easy infinite neural networks in python. In: International Conference on Learning Representations, (2020) https://github.com/google/neural-tangents

NVIDIA Corporation (2021) Modulus User Guide. https://developer.nvidia.com/modulus-user-guide-v2106, release v21.06 – November 9, (2021)

Oreshkin, B.N., Carpov, D., Chapados, N., et al.: N-BEATS: Neural basis expansion analysis for interpretable time series forecasting. (2020) arXiv:1905.10437 [cs, stat]

Owhadi, H.: Bayesian numerical homogenization. Multiscale Model. Simul 13(3), 812–828 (2015). https://doi.org/10.1137/140974596

Owhadi, H., Yoo, G.R.: Kernel Flows: From learning kernels from data into the abyss. J. Comput. Phys. 389, 22–47 (2019). https://doi.org/10.1016/j.jcp.2019.03.040, www.sciencedirect.com/science/article/pii/S0021999119302232

Özbay, A.G., Hamzehloo, A., Laizet, S., et al.: Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh. Data-Centric Engineering 2. (2021) https://doi.org/10.1017/dce.2021.7, https://www.cambridge.org/core/journals/data-centric-engineering/article/poisson-cnn-convolutional-neural-networks-for-the-solution-of-the-poisson-equation-on-a-cartesian-mesh/8CDFD5C9D5172E51B924E9AA1BA253A1

Pang, G., Lu, L., Karniadakis, G.E.: fPINNs: Fractional Physics-Informed Neural Networks. SIAM J. Sci. Comput. 41(4), A2603–A2626 (2019). https://doi.org/10.1137/18M1229845, https://epubs.siam.org/doi/abs/10.1137/18M1229845

Paszke, A., Gross, S., Chintala, S., et al.: Automatic differentiation in PyTorch. Tech. rep., (2017) https://openreview.net/forum?id=BJJsrmfCZ

Patel, R.G., Manickam, I., Trask, N.A., et al.: Thermodynamically consistent physics-informed neural networks for hyperbolic systems. J. Comput. Phys. 449, 110,754 (2022). https://doi.org/10.1016/j.jcp.2021.110754, https://www.sciencedirect.com/science/article/pii/S0021999121006495

Pedro, J.B., Maroñas, J., Paredes, R.: Solving Partial Differential Equations with Neural Networks. (2019) arXiv:1912.04737 [physics]

Peng, W., Zhang, J., Zhou, W., et al.: IDRLnet: A Physics-Informed Neural Network Library. (2021) arXiv:2107.04320 [cs, math]

Pinkus, A.: Approximation theory of the MLP model in neural networks. Acta Numer. 8, 143–195 (1999). https://doi.org/10.1017/S0962492900002919, www.cambridge.org/core/journals/acta-numerica/article/abs/approximation-theory-of-the-mlp-model-in-neural-networks/18072C558C8410C4F92A82BCC8FC8CF9

Pratama, D.A., Bakar, M.A., Man, M., et al.: ANNs-Based Method for Solving Partial Differential Equations : A Survey. (2021) Preprint https://doi.org/10.20944/preprints202102.0160.v1, https://www.preprints.org/manuscript/202102.0160/v1

Psichogios, D.C., Ungar, L.H.: A hybrid neural network-first principles approach to process modeling. AIChE J. 38(10), 1499–1511 (1992). https://doi.org/10.1002/aic.690381003, https://onlinelibrary.wiley.com/doi/abs/10.1002/aic.690381003

Quarteroni, A.: Numerical Models for Differential Problems, 2nd edn. Springer Publishing Company, Incorporated (2013)

Rackauckas, C., Ma, Y., Martensen, J., et al.: Universal Differential Equations for Scientific Machine Learning. (2021) arXiv:2001.04385 [cs, math, q-bio, stat]

Rafiq, M., Rafiq, G., Choi, G.S.: DSFA-PINN: Deep Spectral Feature Aggregation Physics Informed Neural Network. IEEE Access 10, 1 (2022). https://doi.org/10.1109/ACCESS.2022.3153056

Raissi, M.: Deep hidden physics models: Deep learning of nonlinear partial differential equations. J. Mach. Learn. Res. 19(25), 1–24 (2018). http://jmlr.org/papers/v19/18-046.html

Raissi, M., Karniadakis, G.E.: Hidden physics models: Machine learning of nonlinear partial differential equations. J. Comput. Phys. 357, 125–141 (2018). https://doi.org/10.1016/j.jcp.2017.11.039, www.sciencedirect.com/science/article/pii/S0021999117309014

Raissi, M., Perdikaris, P., Karniadakis, G.E.: Inferring solutions of differential equations using noisy multi-fidelity data. J. Comput. Phys. 335, 736–746 (2017). https://doi.org/10.1016/j.jcp.2017.01.060, www.sciencedirect.com/science/article/pii/S0021999117300761

Raissi, M., Perdikaris, P., Karniadakis, G.E.: Machine learning of linear differential equations using Gaussian processes. J. Comput. Phys. 348, 683–693 (2017). https://doi.org/10.1016/j.jcp.2017.07.050, www.sciencedirect.com/science/article/pii/S0021999117305582

Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. (2017c) arXiv:1711.10561 [cs, math, stat]

Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations. (2017d) arXiv:1711.10566 [cs, math, stat]

Raissi, M., Perdikaris, P., Karniadakis, G.E.: Numerical gaussian processes for time-dependent and nonlinear partial differential equations. SIAM J. Sci. Comput. 40(1), A172–A198 (2018). https://doi.org/10.1137/17M1120762

Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019). https://doi.org/10.1016/j.jcp.2018.10.045, www.sciencedirect.com/science/article/pii/S0021999118307125

Raissi, M., Yazdani, A., Karniadakis, G.E.: Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Sci. 367(6481), 1026–1030 (2020). https://doi.org/10.1126/science.aaw4741, www.science.org/doi/10.1126/science.aaw4741

Ramabathiran, A.A., Ramachandran, P.: SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs. J. Comput. Phys. 445, 110,600 (2021). https://doi.org/10.1016/j.jcp.2021.110600, https://www.sciencedirect.com/science/article/pii/S0021999121004952

Rudin, C., Chen, C., Chen, Z., et al.: Interpretable machine learning: Fundamental principles and 10 grand challenges. Stat. Surveys 16(none), 1–85 (2022). https://doi.org/10.1214/21-SS133, https://projecteuclid.org/journals/statistics-surveys/volume-16/issue-none/Interpretable-machine-learning-Fundamental-principles-and-10-grand-challenges/10.1214/21-SS133.full

Ryaben’kii, V.S., Tsynkov, S.V.: A Theoretical Introduction to Numerical Analysis. CRC Press, Boca Raton, FL (2006)

Sahli Costabal, F., Yang, Y., Perdikaris, P., et al.: Physics-Informed Neural Networks for Cardiac Activation Mapping. Front. Phys. 8, 42 (2020). https://doi.org/10.3389/fphy.2020.00042, www.frontiersin.org/article/10.3389/fphy.2020.00042

Scharzenberger, C., Hays, J.: Learning To Estimate Regions Of Attraction Of Autonomous Dynamical Systems Using Physics-Informed Neural Networks. Tech. rep., (2021) https://doi.org/10.48550/arXiv.2111.09930, arXiv:2111.09930 [cs] type: article

Schiassi, E., Furfaro, R., Leake, C., et al.: Extreme theory of functional connections: A fast physics-informed neural network method for solving ordinary and partial differential equations. Neurocomputing 457, 334–356 (2021). https://doi.org/10.1016/j.neucom.2021.06.015, www.sciencedirect.com/science/article/pii/S0925231221009140

Schölkopf, B., Locatello, F., Bauer, S., et al.: Toward Causal Representation Learning. Proc. IEEE 109(5), 612–634 (2021). https://doi.org/10.1109/JPROC.2021.3058954

Sengupta, S., Basak, S., Saikia, P., et al.: A review of deep learning with special emphasis on architectures, applications and recent trends. Knowledge-Based Systems 194, 105,596 (2020). https://doi.org/10.1016/j.knosys.2020.105596, https://www.sciencedirect.com/science/article/pii/S095070512030071X

Sergeev, A., Del Balso, M.: Horovod: fast and easy distributed deep learning in TensorFlow. Tech. rep., (2018) https://doi.org/10.48550/arXiv.1802.05799, arXiv:1802.05799 [cs, stat] type: article

Shin, Y., Darbon, J., Karniadakis, G.E.: On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Commun. Comput. Phys. 28(5), 2042–2074 (2020a). https://doi.org/10.4208/cicp.OA-2020-0193, arXiv: 2004.01806

Shin, Y., Zhang, Z., Karniadakis, G.E.: Error estimates of residual minimization using neural networks for linear PDEs. (2020b) arXiv:2010.08019 [cs, math]

Shrestha, A., Mahmood, A.: Review of Deep Learning Algorithms and Architectures. IEEE Access 7, 53040–53065 (2019). https://doi.org/10.1109/ACCESS.2019.2912200

Sirignano, J., Spiliopoulos, K.: DGM: A deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 1339–1364 (2018). https://doi.org/10.1016/j.jcp.2018.08.029, www.sciencedirect.com/science/article/pii/S0021999118305527

Sitzmann, V., Martel, J.N.P., Bergman, A.W., et al.: Implicit Neural Representations with Periodic Activation Functions (2020). arXiv:2006.09661 [cs, eess]

Smith, J.D., Azizzadenesheli, K., Ross, Z.E.: EikoNet: Solving the Eikonal Equation With Deep Neural Networks. IEEE Trans. Geosci. Remote Sens. 59(12), 10685–10696 (2021). https://doi.org/10.1109/TGRS.2020.3039165

Smith, J.D., Ross, Z.E., Azizzadenesheli, K., et al.: HypoSVI: Hypocentre inversion with Stein variational inference and physics informed neural networks. Geophys. J. Int. 228(1), 698–710 (2021). https://doi.org/10.1093/gji/ggab309

Stein, M.L.: Large sample properties of simulations using latin hypercube sampling. Technometrics 29, 143–151 (1987)

Stiasny J, Misyris GS, Chatzivasileiadis S (2021) Physics-Informed Neural Networks for Non-linear System Identification for Power System Dynamics. In: 2021 IEEE Madrid PowerTech, pp 1–6, https://doi.org/10.1109/PowerTech46648.2021.9495063

Stielow, T., Scheel, S.: Reconstruction of nanoscale particles from single-shot wide-angle free-electron-laser diffraction patterns with physics-informed neural networks. Phys. Review E 103(5), 053,312 (2021). https://doi.org/10.1103/PhysRevE.103.053312, https://link.aps.org/doi/10.1103/PhysRevE.103.053312

Stiller, P., Bethke, F., Böhme, M., et al.: Large-Scale Neural Solvers for Partial Differential Equations. In: Nichols J, Verastegui B, Maccabe AB, et al (eds) Driving Scientific and Engineering Discoveries Through the Convergence of HPC, Big Data and AI. Springer International Publishing, Cham, Communications in Computer and Information Science, pp. 20–34, (2020) https://doi.org/10.1007/978-3-030-63393-6_2

Sun, L., Gao, H., Pan, S., et al.: Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data. Comput. Methods Appl. Mech. Engrg. 361, 112,732 (2020a). https://doi.org/10.1016/j.cma.2019.112732, https://www.sciencedirect.com/science/article/pii/S004578251930622X

Sun, S., Cao, Z., Zhu, H., et al.: A Survey of Optimization Methods From a Machine Learning Perspective. IEEE Trans. Cybernet. 50(8), 3668–3681 (2020). https://doi.org/10.1109/TCYB.2019.2950779

Tartakovsky, A.M., Marrero, C.O., Perdikaris, P., et al.: Physics-Informed Deep Neural Networks for Learning Parameters and Constitutive Relationships in Subsurface Flow Problems. Water Resources Research 56(5), e2019WR026,731 (2020). https://doi.org/10.1029/2019WR026731, https://onlinelibrary.wiley.com/doi/abs/10.1029/2019WR026731

Thompson, D.B.: Numerical Methods 101 – Convergence of Numerical Models. ASCE, pp. 398–403 (1992), https://cedb.asce.org/CEDBsearch/record.jsp?dockey=0078142

Tong, Y., Xiong, S., He, X., et al.: Symplectic neural networks in taylor series form for hamiltonian systems. J. Comput. Phys. 437, 110,325 (2021). https://doi.org/10.1016/j.jcp.2021.110325, https://www.sciencedirect.com/science/article/pii/S0021999121002205

Torabi Rad, M., Viardin, A., Schmitz, G.J., et al.: Theory-training deep neural networks for an alloy solidification benchmark problem. Comput. Mater. Sci. 180, 109,687 (2020). https://doi.org/10.1016/j.commatsci.2020.109687, https://www.sciencedirect.com/science/article/pii/S0927025620301786

Viana, F.A.C., Nascimento, R.G., Dourado, A., et al.: Estimating model inadequacy in ordinary differential equations with physics-informed neural networks. Comput. Structures 245, 106,458 (2021). https://doi.org/10.1016/j.compstruc.2020.106458, https://www.sciencedirect.com/science/article/pii/S0045794920302613

Waheed, U.b., Haghighat, E., Alkhalifah, T., et al.: PINNeik: Eikonal solution using physics-informed neural networks. Comput. Geosci. 155, 104,833 (2021). https://doi.org/10.1016/j.cageo.2021.104833, https://www.sciencedirect.com/science/article/pii/S009830042100131X

Wang, L., Yan, Z.: Data-driven rogue waves and parameter discovery in the defocusing nonlinear Schrödinger equation with a potential using the PINN deep learning. Phys. Lett. A 404, 127,408 (2021). https://doi.org/10.1016/j.physleta.2021.127408, https://www.sciencedirect.com/science/article/pii/S0375960121002723

Wang, N., Chang, H., Zhang, D.: Theory-guided auto-encoder for surrogate construction and inverse modeling. Comput. Methods Appl. Mech. Engrg. 385, 114,037 (2021a). https://doi.org/10.1016/j.cma.2021.114037, https://www.sciencedirect.com/science/article/pii/S0045782521003686

Wang, S., Perdikaris, P.: Deep learning of free boundary and Stefan problems. J. Comput. Phys. 428, 109,914 (2021). https://doi.org/10.1016/j.jcp.2020.109914, https://www.sciencedirect.com/science/article/pii/S0021999120306884

Wang, S., Teng, Y., Perdikaris, P.: Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks. SIAM J. Sci. Comput. 43(5), A3055–A3081 (2021). https://doi.org/10.1137/20M1318043, https://epubs.siam.org/doi/abs/10.1137/20M1318043

Wang, S., Sankaran, S., Perdikaris, P.: Respecting causality is all you need for training physics-informed neural networks. (2022a) arXiv:2203.07404 [nlin, physics:physics, stat]

Wang, S., Yu, X., Perdikaris, P.: When and why PINNs fail to train: A neural tangent kernel perspective. J. Comput. Phys. 449, 110,768 (2022b). https://doi.org/10.1016/j.jcp.2021.110768, https://www.sciencedirect.com/science/article/pii/S002199912100663X

Wen, G., Li, Z., Azizzadenesheli, K., et al.: U-FNO–An enhanced Fourier neural operator-based deep-learning model for multiphase flow. Adv. Water Resources 163, 104,180 (2022). https://doi.org/10.1016/j.advwatres.2022.104180, https://www.sciencedirect.com/science/article/pii/S0309170822000562

Wiecha, P.R., Arbouet, A., Arbouet, A., et al.: Deep learning in nano-photonics: inverse design and beyond. Photonics Research 9(5), B182–B200 (2021). https://doi.org/10.1364/PRJ.415960, www.osapublishing.org/prj/abstract.cfm?uri=prj-9-5-B182

Wong, J.C., Gupta, A., Ong, Y.S.: Can Transfer Neuroevolution Tractably Solve Your Differential Equations? IEEE Comput. Intell. Mag. 16(2), 14–30 (2021). https://doi.org/10.1109/MCI.2021.3061854

Wong, J.C., Ooi, C., Gupta, A., et al.: Learning in Sinusoidal Spaces with Physics-Informed Neural Networks. (2022) arXiv:2109.09338 [physics]

Xiao, H., Wu, J.L., Laizet, S., et al.: Flows over periodic hills of parameterized geometries: A dataset for data-driven turbulence modeling from direct simulations. Comput. Fluids 200, 104,431 (2020). https://doi.org/10.1016/j.compfluid.2020.104431, https://www.sciencedirect.com/science/article/pii/S0045793020300074

Xu, K., Darve, E.: ADCME: Learning Spatially-varying Physical Fields using Deep Neural Networks. (2020) arXiv:2011.11955 [cs, math]

Xu, K., Darve, E.: Solving inverse problems in stochastic models using deep neural networks and adversarial training. Comput. Methods Appl. Mech. Engrg. 384, 113,976 (2021). https://doi.org/10.1016/j.cma.2021.113976, https://www.sciencedirect.com/science/article/pii/S0045782521003078

Yang, L., Zhang, D., Karniadakis, G.E.: Physics-informed generative adversarial networks for stochastic differential equations. SIAM J. Sci. Comput. 42(1), A292–A317 (2020). https://doi.org/10.1137/18M1225409

Yang, L., Meng, X., Karniadakis, G.E.: B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. J. Comput. Phys. 425, 109,913 (2021). https://doi.org/10.1016/j.jcp.2020.109913, https://www.sciencedirect.com/science/article/pii/S0021999120306872

Yang, Y., Perdikaris, P.: Adversarial uncertainty quantification in physics-informed neural networks. J. Comput. Phys. 394, 136–152 (2019). https://doi.org/10.1016/j.jcp.2019.05.027, www.sciencedirect.com/science/article/pii/S0021999119303584

Yarotsky, D.: Error bounds for approximations with deep relu networks. Neural Netw. 94, 103–114 (2017). https://doi.org/10.1016/j.neunet.2017.07.002, www.sciencedirect.com/science/article/pii/S0893608017301545

Yuan, L., Ni, Y.Q., Deng, X.Y., et al.: A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations. J. Comput. Phys. 462, 111,260 (2022). https://doi.org/10.1016/j.jcp.2022.111260,https://www.sciencedirect.com/science/article/pii/S0021999122003229

Yucesan, Y.A., Viana, F.A.C.: Hybrid physics-informed neural networks for main bearing fatigue prognosis with visual grease inspection. Comput. Ind. 125:103,386 (2021). https://doi.org/10.1016/j.compind.2020.103386, https://www.sciencedirect.com/science/article/pii/S0166361520306205

Zhang, D., Lu, L., Guo, L., et al.: Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. J. Comput. Phys. 397, 108,850 (2019). https://doi.org/10.1016/j.jcp.2019.07.048, https://www.sciencedirect.com/science/article/pii/S0021999119305340

Zhang, D., Guo, L., Karniadakis, G.E.: Learning in modal space: Solving time-dependent stochastic pdes using physics-informed neural networks. SIAM J. Sci. Comput. 42(2), A639–A665 (2020). https://doi.org/10.1137/19M1260141

Zhang, R., Liu, Y., Sun, H.: Physics-informed multi-LSTM networks for metamodeling of nonlinear structures. Comput. Methods Appl. Mech. Engrg. 369, 113,226 (2020b). https://doi.org/10.1016/j.cma.2020.113226, https://www.sciencedirect.com/science/article/pii/S0045782520304114

Zhi-Qin, Xu, J., et al.: Frequency principle: Fourier analysis sheds light on deep neural networks. Commun. Comput. Phys. 28(5), 1746–1767 (2020). https://doi.org/10.4208/cicp.OA-2020-0085, http://global-sci.org/intro/article_detail/cicp/18395.html

Zhu, Q., Liu, Z., Yan, J.: Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Comput. Mech. 67(2), 619–635 (2021). https://doi.org/10.1007/s00466-020-01952-9

Zhu, Y., Zabaras, N., Koutsourelakis, P.S., et al.: Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. J. Comput. Phys. 394, 56–81 (2019). https://doi.org/10.1016/j.jcp.2019.05.024, www.sciencedirect.com/science/article/pii/S0021999119303559

Zubov, K., McCarthy, Z., Ma, Y., et al.: NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations. (2021a) arXiv:2107.09443 [cs]

Zubov, K., McCarthy, Z., Ma, Y., et al.: NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations. (2021b) arXiv:2107.09443 [cs]