Umbala we-Left Abstract and 1. Introduction Abstract futhi 1. Ukuhambisana 1.1 Izinzuzo ezivela 1.2 Izifundo ze-neural networks 1.3. mayelana ne-entropy ye-Direct PINN methods 1.4. Ukwakhiwa kwePapers 
 
 
 
 
 
 
 
 
 
 
 
 
 Non-diffusive neural network solver for one dimensional scalar HCLs 2.1. One shock wave 2.2. Arbitrary number of shock waves 2.3. Shock wave generation 2.4. Shock wave interaction 2.5. Non-diffusive neural network solver for one dimensional systems of CLs 2.6. Efficient initial wave decomposition 
 
 
 
 Gradient descent algorithm and efficient implementation 3.1. Classical gradient descent algorithm for HCLs 3.2. Gradient descent and domain decomposition methods 
 
 
 
 
 
 
 
 
 Numerics 4.1. Practical implementations 4.2. Basic tests and convergence for 1 and 2 shock wave problems 4.3. Shock wave generation 4.4. Shock-Shock interaction 4.5. Entropy solution 4.6. Domain decomposition 4.7. Nonlinear systems 
 
 Conclusion and References I-non-diffusive neural network solver ye-one-dimensional scalar HCLs Ukuhlaziywa & References 5. Ukungena ngemvume Kule isihloko, sinikeza indlela original yokuxhumana nezinsizakalo ze-hyperbolic using a non-diffusive neural network method. The DLs futhi ukuguqulwa izinsizakalo zokuhlobisa ezisetshenziselwa i-DLs, futhi lapho isixazululo ifanelekayo. Siye zisebenzisa networks ne-neural ukuguqulwa kwe-DLs kanye nesisezinsizakalo zokuhlobisa kwelinye subdomains. I-networks ibhekwa ngokunciphisa isizukulwane esebenzayo esilinganiselwe (norm of the) izinsizakalo zokuhlobisa izinsizakalo, izimo zokuhlobisa kanye nezimo zokuqala kanye nezimo ze-Rankine-Hugoniot. Lolu hlobo lwezinhlobonhlobo lwezocingo ngaphandle kokusetshenziswa isisombululo esincane se-HCL. Izinsizakalo ezinye zingasetshenziselwa ukulungiselela isixazululo kanye ne-DL Uma ingcindezi global yama-loss functional iyatholakala ngokushesha, i-convergent ngokushesha futhi enhle (Schwarz) method of decomposition ye-domain ingasetshenziselwa. Lezi zilandelayo ivumela ukuxuba inqubo ye-optimization ngokusebenzisa ukuphuculwa kwezinkampani zezinkampani zezinkampani zezinkampani zezinkampani ezivela ku-global approximate isixazululwa, njenge-showed ku [23]. Ngezinye umsebenzi esizayo, sinikezela isicelo le-methodology ku-high-dimensional problems. CRediT authorship contribution statement Umbhali wahlanganyela ngokulinganayo. Declaration of competing interest I-Authors ibhalisele ukuthi akuyona izinzuzo ezithakazelisayo ze-financial noma izinhlobonhlobo zebhizinisi ezinokuthi ziye ziye ziye ziye ziye ziye ziye ziye ziye zihlanganisa umsebenzi esifundeni. Data availability Akukho idatha asetshenziselwa isifundo esifundisiwe kulesi sihloko. Ukuhlobisa [1] J.-M. Ghidaglia no-F. Pascal. 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Zhang, futhi G. E. Karniadakis. I-physics-informed generative adversarial networks for stochastic differential equations. SIAM J. Sci. Comput., 42(1):A292–A317, 2020. [13] J. Han, A. Jentzen, no-Weinan E. Ukuphendula izinhlayiya ze-differential e-high-dimensional nge-deep learning. Proc. Natl. Acad. Sci. USA, 115(34):8505–8510, 2018. [14] H. Gao, M. J. Zahr, no-J.-X. Wang. I-Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems. Comput. Methods Appl. Mech. Engrg., 390:Paper No. 114502, 18, 2022. [15] J. Sirignano noK. Spiliopoulos. DGM: algorithm yokufunda okuphambili ukuguqulwa izinhlayiya ezingenalutho. J. Comput. Phys., 375:1339-1364, 2018. [16] A.D. Jagtap, E. Kharazmi, futhi G.E. Karniadakis. I-Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and reverse problems. I-Computer Methods in Applied Mechanics and Engineering, 365, 2020. [17] R. Rodriguez-Torrado, P. Ruiz, no. L. Cueto-Felgueroso. I-physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley-Leverett problem. Sci Rep., 12:7557, 2022. [18] R. G. Patel, I. Manickam, N. A. Trask, M. A. Wood, M. Lee, I. Tomas, futhi E. C. Cyr. I-physics-informed neural networks ye-thermodynamically consistent for hyperbolic systems. J. of Comput. Phys., 449:110754, 2022. [19] E. Lorin no-X. Yang Schwarz waveform relaxation-learning for advection-diffusion reaction equations. J. Comput. Phys., 473: Paper No. 111657, 2023. [20] I-E. Lorin no-X. Yang. Izindlela ze-quasi-optimum ye-domain decomposition ezisekelwe ku-network ye-neural ekubunjweni ye-Schr ̈odinger. I-Comput. Phys. Commun., I-Accepted. 2024. [21] I-M. Gander no-L. Halpern. Izindlela ze-Schwarz ye-waveform eyenziwe ngempumelelo ze-advection reaction diffusion problems. SIAM J. on Numer. Anal., 45(2):666-697-2007. [22] M.J. Gander, F. Kwok, futhi B.C. Mandal. Dirichlet-Neumann izindlela waveform ukunciphisa imiphumela parabolic kanye hyperbolic e subdomains eziningi. BIT Numerical Mathematics, 61(1):173-207, 2021. [23] M.J. Gander noC. Rohde. Ukuphefumula ukuguqulwa kwe-Schwarz waveform ngenxa ye-convection-dominated nonlinear conservation laws. SIAM Journal on Scientific Computing, 27(2):415-439, 2006. [24] X. Antoine futhi E. Lorin. Ukuhlolwa kwe-Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrodinger and Gross-Pitaevskii equations. Numerische Mathematik, 137(4):923–958, 2017. [25] L. Bottou, F. E. Curtis, no. J. Nocedal. Izindlela ze-optimization ye-machine learning. SIAM Rev., 60(2):223-311, 2018. [26] Bradbury J, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, futhi Q. Zhang. JAX: Ukuhlobisa okuhlobisa ku-Python+NumPy izinhlelo, 2018. [ 27] J.L. Montagne, H.C. Yee, no. M. Vinokur. Ukuhlolwa okuhlobene kwe-high-resolution shockcapturing schemes for a real gas. in: I-Proc. seventh gamm conf. on numerical methods in fluid mechanics, (Lauvain-la-Neuve, Belgium: Sep. 9-11, 1987), m. Devill, 20 (ISBN 3-528-08094-9), 1987. [28] P. L. Roe. I-approximate Riemann solvers, i-parameter vectors, ne-difference schemes. J. Comput. Phys., 43(2):357-372, 1981. [29] L. Halpern no-J. Szeftel. Ukuphumelela kwe-Schwarz ye-waveform e-optimized kanye ne-quasi-optimal ye-Schwarz ye-Schrödinger ye-dimensional equation. Amamathematical Models and Methods in Applied Sciences, 20(12):2167-2199, 2010. [30] X. Antoine, F. Hou, futhi E. Lorin. Ukubuyekezwa kwe-asymptotic ye-convergence ye-classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves. ESAIM: Numerical Analysis and Mathematical Modeling (M2AN), 52(4):1569-1596, 2018. [31] X. Antoine futhi E. Lorin. On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schr ̈odinger equation. J. Comput. Appl. Math., 354:15-30, 2019. [32] X. Antoine, C. Besse, futhi P. Klein. Ukupholisa izimo ezimbini ye-onedimensional Schr ́odinger equation nge-potency repulsive ephakathi. J. Comput. Phys., 228(2):312-335, 2009. [33] A. Modave, A. Royer, X. Antoine, no. C. Geuzaine. Uhlelo olungaphakathi yokuhlukaniswa okungenani, ngezimo zokudluliselwa okuphezulu kanye nokwelashwa kwe-cross-point ye-Helmholtz. Comput. Methods Appl. Mech. Engrg., 368:113162, 23, 2020. [34] I. Badia, B. Caudron, X. Antoine, noC. Geuzaine. Ukubambisana okungenani okungenani we-limit element ne-high-order finite element methods for time-harmonic electromagnetic scattering by inhomogeneous objects. SIAM J. Sci. Comput., 44(3):B640-B667, 2022. [35] X. Antoine noC. Besse. Izinhlelo ezinzima ezinzima ezivamile ze-discretization ze-non-reflecting conditions for the one-dimensional Schrodinger equation. 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 Umbhali: (1) Emmanuel LORIN, School of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6 futhi Centre de Recherches Mathematiques, Universit ́e de Montr ́eal, Montreal, Canada, H3T 1J4 (elorin@math.carleton.ca); (2) Arian NOVRUZI, umbhali wokubhalisa we-Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada (novruzi@uottawa.ca). Authors: (1) Emmanuel LORIN, School of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6 futhi Centre de Recherches Mathematiques, Universit ́e de Montr ́eal, Montreal, Canada, H3T 1J4 (elorin@math.carleton.ca); (2) Arian NOVRUZI, umbhali wokubhalisa we-Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada (novruzi@uottawa.ca). 
 
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