References
[1] Lagaris IE, Likas A, Fotiadis DI. Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw. 1998;9(5):987-1000. doi:10.1109/72.712178.
[2] Yadav N, Yadav A, Kumar M. An introduction to neural network methods for differential equations. Dordrecht: Springer; 2015. doi:10.1007/978-94-017-9816-7.
[3] Raissi M, Perdikaris P, Karniadakis GE. Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. arXiv. 2017. Preprint. doi:10.48550/arXiv.1711.10561.
[4] Raissi M, Perdikaris P, Karniadakis GE. Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations. arXiv. 2017. Preprint. doi:10.48550/arXiv.1711.10566.
[5] Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys. 2019;378:686-707. doi:10.1016/j.jcp.2018.10.045.
[6] Lakshmanan V, Robinson S, Munn M. Machine Learning Design Patterns: Solutions to Common Challenges in Data Preparation, Model Building, and MLOps. 1st ed. Sebastopol: O'Reilly Media; 2020.
[7] Jin X, Cai S, Li H, Karniadakis GE. NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. J Comput Phys. 2021;426:109951. doi:10.1016/j.jcp.2020.109951.
[8] Averweg S, Schwarz A, Schwarz C, Schröder J. 3D modeling of generalized Newtonian fluid flow with data assimilation using the least-squares finite element method. Comput Methods Appl Mech Eng. 2022;392:114668. doi:10.1016/j.cma.2022.114668.
[9] Zhang L-C, Lee D. Design-based individual prediction. arXiv. 2023. Preprint. doi:10.48550/arXiv.2301.09117.
[10] Karniadakis GE, Kevrekidis IG, Lu L, Perdikaris P, Wang S, Yang L. Physics-informed machine learning. Nat Rev Phys. 2021;3:422-440. doi:10.1038/s42254-021-00314-5.
[11] Zubov K, McCarthy Z, Ma Y, et al. NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations. arXiv. 2021. Preprint. doi:10.48550/arXiv.2107.09443.
[12] Taha WM, Taha AE, Thunberg J. Cyber-Physical Systems: A Model-Based Approach. Cham: Springer; 2021. doi:10.1007/978-3-030-36071-9.
[13] Lu Y, Liu C, Wang K, Huang H, Xu X. Digital Twin-driven smart manufacturing: Connotation, reference model, applications and research issues. Robot Comput Integr Manuf. 2020;61:101837. doi:10.1016/j.rcim.2019.101837.
[14] Uhlemann THJ, Steinhilper CLR, Steinhilper R. The Digital Twin: Realizing the Cyber-Physical Production System for Industry 4.0. Procedia CIRP. 2017;61:335-340. doi:10.1016/j.procir.2016.11.152.
[15] Tarkhov DA, Malykhina GF. Neural network modelling methods for creating digital twins of real objects. J Phys Conf Ser. 2019;1236:012068. doi:10.1088/1742-6596/1236/1/012068.
[16] Tarkhov DA, Vasilyev AN. Semi-Empirical Neural Network Modeling and Digital Twins Development. Cambridge: Academic Press; 2019.
[17] Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys. 2019;378:686-707. doi:10.1016/j.jcp.2018.10.045.
[18] Lu L, Meng X, Mao Z, Karniadakis GE. DeepXDE: A deep learning library for solving differential equations. SIAM Rev. 2021;63(1):208-228. doi:10.1137/19M1274067.
[19] Cai S, Wang Z, Wang S, Perdikaris P, Karniadakis GE. Physics-Informed Neural Networks (PINNs) for Heat Transfer. J Heat Transfer. 2021;143(6):060801. doi:10.1115/1.4050542.
[20] Geron A. Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intel-ligent Systems. 3rd ed. Sebastopol: O'Reilly Media; 2022.
[21] Bavdin AG, Pearlmutter BF, Radul AA, Siskind JM. Automatic Differentiation in Machine Learning: a Survey. J Mach Learn Res. 2018;18(153):1-43. doi:10.48550/arXiv.1502.05767.
[22] Strizhak SV, Koshelev KB. The use of a physically based neural network on the example of modeling hydrodynamic processes that allow an analytical solution. Proc Inst Syst Program RAS. 2023;35(5):245-258. doi:10.15514/ISPRAS-2023-35(5)-16.
[23] Sophiya AA, Nair AK, Maleki S, Krishnababu SK. A comprehensive analysis of PINNs: Variants, Applications, and Challenges. arXiv. 2025. Preprint. doi:10.48550/arXiv.2505.22761.
[24] Gorbachenko VI, Stenkin DA. Solving Equations Describing Processes in a Piecewise Homogeneous Medium on Radial Basis Functions Networks. In: Studies in Computational Intelligence. Vol 925. Cham: Springer; 2021:412-419. doi:10.1007/978-3-030-60577-3_49.
[25] Gorbachenko VI, Stenkin DA. Solving of Inverse Coefficient Problems on Networks of Radial Basis Functions. In: Studies in Computational Intelligence. Vol 1008. Cham: Springer; 2022:230-237. doi:10.1007/978-3-030-91581-0_31.
[26] Cybenko G. Approximation by Superposition of a Sigmoidal Function. Math Control Signals Syst. 1989;2:303-314. doi:10.1007/BF02134016.
[27] Maiorov V, Pinkus A. Lower bounds for approximation by MLP neural networks. Neurocomputing. 1999;25(1-3):81-91. doi:10.1016/S0925-2312(98)00111-8.
[28] Hornik K. Approximation capabilities of multilayer feedforward networks. Neural Netw. 1991;4(2):251-257. doi:10.1016/0893-6080(91)90009-T.
[29] Hornik K, Stinchcombe M, White H. Multilayer feedforward networks are universal approximators. Neural Netw. 1989;2(5):359-366. doi:10.1016/0893-6080(89)90020-8.
[30] Leshno M, Lin VY, Pinkus A, Schocken S. Multilayer Feedforward Networks with a Non-Polynomial Activation Function Can Approximate Any Function. Neural Netw. 1993;6(6):861-867. doi:10.1016/S0893-6080(05)80131-5.
[31] Osowski S. Sieci neuronowe do przetwarzania informacji. Warszawa: Oficyna Wydawnicza Politechniki Warszawskiej; 2000.
[32] Haykin S. Neural Networks and Learning Machines. 3rd ed. London: Pearson; 2009.
[33] Park J, Sandberg IW. Approximation and Radial-Basis-Function Networks. Neural Comput. 1993;5(2):305-316. doi:10.1162/neco.1993.5.2.305.
[34] Park J, Sandberg IW. Universal Approximation Using Radial-Basis-Function Networks. Neural Comput. 1991;3(2):246-257. doi:10.1162/neco.1991.3.2.246.
[35] Buhmann MD. Radial Basis Functions: Theory and Implementations. Cambridge: Cambridge University Press; 2003.
[36] Micchelli CA. Interpolation of scattered data: Distance matrices and conditionally positive definite functions. Constr Approx. 1986;2(1):11-22. doi:10.1007/BF01893414.
[37] Eivazi H, Tahani M, Schlatter P, Vinuesa R. Physics-informed neural networks for solving Reynolds-averaged Navier–Stokes equations. Phys Fluids. 2022;34(7):075117. doi:10.1063/5.0095270.
[38] De Ryck T, Jagtap AD, Mishra S. Error estimates for physics-informed neural networks approximating the Navier–Stokes equations. IMA J Numer Anal. 2024;44(1):83-119. doi:10.1093/imanum/drac085.
[39] Demirkaya G, Wafo SC, Ilegbusi OJ. Direct solution of Navier–Stokes equations by radial basis functions. Appl Math Model. 2008;32(9):1848-1858. doi:10.1016/j.apm.2007.06.019.
[40] Chinchapatnam PP, Djidjeli K, Nair PB, Tan M. A compact RBF-FD based meshless method for the incompressible Navier–Stokes equations. Proc Inst Mech Eng M: J Eng Marit Environ. 2009;223(3):275-290. doi:10.1243/14750902JEME151.
[41] Abdukhakimov F, Xiang C, Kamzolov D, Gower R, Takáč M. SANIA: Polyak-type Optimization Framework Leads to Scale In-variant Stochastic Algorithms. arXiv. 2023. Preprint. doi:10.48550/arXiv.2312.17369.
[42] Nagwekar A. Towards Guided Descent: Optimization Algorithms for Training Neural Networks At Scale. arXiv. 2024. Preprint. doi:10.48550/arXiv.2512.18373.
[43] Liu H, Li Z, Hall D, Liang P, Ma T. Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training. arXiv. 2023. Preprint. doi:10.48550/arXiv.2305.14342.
[44] Hutchinson MF. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Commun Stat Simul Comput. 1989;18(3):1059-1076. doi:10.1080/03610919008812866.
[45] Schraudolph NN. Fast curvature matrix-vector products for second-order gradient descent. Neural Comput. 2002;14(7):1723-1738. doi:10.1162/08997660260028683.
[46] Pearlmutter BA. Fast Exact Multiplication by the Hessian. Neural Comput. 1994;6(1):147-160. doi:10.1162/neco.1994.6.1.147.
[47] Kovasznay LSG. Laminar flow behind a two-dimensional grid. Math Proc Camb Philos Soc. 1948;44(1):58-62. doi:10.1017/S0305004100023999.
[48] Slezkin NA. Dynamics of viscous incompressible fluid. Moscow: Gostekhizdat; 1955. Russian.
[49] Taylor GI, Green AE. Mechanism of the Production of Small Eddies from Large Ones. Proc R Soc Lond A. 1937;158(895):499-521. doi:10.1098/rspa.1937.0036.
[50] Botella O, Peyret R. Benchmark spectral results on the lid-driven cavity flow. Comput Fluids. 1998;27(4):421-433. doi:10.1016/S0045-7930(98)00002-4.
[51] Ghia U, Ghia KN, Shin CT. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J Comput Phys. 1982;48(3):387-411. doi:10.1016/0021-9991(82)90058-4.
[52] Xie T, Yu H, Wilamowski BM. Comparison between Traditional Neural Networks and Radial Basis Function Networks. In: 2011 IEEE International Symposium on Industrial Electronics; 2011 Jun 27-30; Gdansk, Poland. IEEE; 2011. p. 1194-1199. doi:10.1109/ISIE.2011.5984334.
[53] Wang S, Teng Y, Perdikaris P. Understanding and mitigating gradient pathologies in physics-informed neural networks. arXiv. 2020. Preprint. doi:10.48550/arXiv.2001.04536.