Stiff-PDEs and Physics-Informed Neural Networks
In recent years, physics-informed neural networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problems with conflicting boundary conditions at adjacent edges and corners. These problems have disconti...
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| Published in | Archives of computational methods in engineering Vol. 30; no. 5; pp. 2929 - 2958 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.06.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1134-3060 1886-1784 1886-1784 |
| DOI | 10.1007/s11831-023-09890-4 |
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| Abstract | In recent years, physics-informed neural networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problems with conflicting boundary conditions at adjacent edges and corners. These problems have discontinuous solutions at edges and corners that are difficult to learn for neural networks with a continuous activation function. In this review paper, we have investigated various PINN frameworks that are designed to solve stiff-PDEs. We took two heat conduction problems (2D and 3D) with a discontinuous solution at corners as test cases. We investigated these problems with a number of PINN frameworks, discussed and analysed the results against the FEM solution. It appears that PINNs provide a more general platform for parameterisation compared to conventional solvers. Thus, we have investigated the 2D heat conduction problem with parametric conductivity and geometry separately. We also discuss the challenges associated with PINNs and identify areas for further investigation. |
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| AbstractList | In recent years, physics-informed neural networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problems with conflicting boundary conditions at adjacent edges and corners. These problems have discontinuous solutions at edges and corners that are difficult to learn for neural networks with a continuous activation function. In this review paper, we have investigated various PINN frameworks that are designed to solve stiff-PDEs. We took two heat conduction problems (2D and 3D) with a discontinuous solution at corners as test cases. We investigated these problems with a number of PINN frameworks, discussed and analysed the results against the FEM solution. It appears that PINNs provide a more general platform for parameterisation compared to conventional solvers. Thus, we have investigated the 2D heat conduction problem with parametric conductivity and geometry separately. We also discuss the challenges associated with PINNs and identify areas for further investigation. |
| Author | Sharma, Prakhar Evans, Llion Nithiarasu, Perumal Tindall, Michelle |
| Author_xml | – sequence: 1 givenname: Prakhar orcidid: 0000-0002-7635-1857 surname: Sharma fullname: Sharma, Prakhar email: prakhars962@gmail.com organization: Faculty of Science and Engineering, Swansea University, Zienkiewicz Institute for Modelling, Data and AI, Swansea University, Bay Campus – sequence: 2 givenname: Llion surname: Evans fullname: Evans, Llion organization: Faculty of Science and Engineering, Swansea University, Culham Science Centre, United Kingdom Atomic Energy Authority, Zienkiewicz Institute for Modelling, Data and AI, Swansea University, Bay Campus – sequence: 3 givenname: Michelle surname: Tindall fullname: Tindall, Michelle organization: Fusion Technology Facilities, United Kingdom Atomic Energy Authority – sequence: 4 givenname: Perumal surname: Nithiarasu fullname: Nithiarasu, Perumal organization: Faculty of Science and Engineering, Swansea University, Zienkiewicz Institute for Modelling, Data and AI, Swansea University, Bay Campus |
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| CitedBy_id | crossref_primary_10_1080_10407790_2023_2264489 crossref_primary_10_1016_j_advwatres_2023_104564 crossref_primary_10_1063_5_0218611 crossref_primary_10_1016_j_jcp_2023_112349 crossref_primary_10_1063_5_0226562 crossref_primary_10_1088_1873_7005_adb32e crossref_primary_10_1063_5_0233607 crossref_primary_10_1109_ACCESS_2024_3482711 |
| Cites_doi | 10.1137/19M1260141 10.1016/j.cma.2021.114502 10.1016/j.jcp.2019.109020 10.23919/CCC52363.2021.9550487 10.1016/j.jcp.2019.109136 10.1109/MSP.2017.2743240 10.1007/s40305-020-00309-6 10.1038/s42256-021-00302-5 10.1002/qre.1924 10.1063/5.0086649 10.2139/ssrn.4086448 10.1007/s41965-019-00023-0 10.1016/j.jcp.2021.110296 10.1137/070709359 10.1137/S0895479895283409 10.1007/BF01386213 10.1016/j.jcp.2021.110768 10.1016/j.sigpro.2016.08.025 10.1016/j.cma.2022.115852 10.1016/j.cma.2020.113552 10.1007/978-3-540-68942-3_4 10.1137/17M1120762 10.1109/CVPR.2017.632 10.1007/978-3-030-77977-1_36 10.1137/0905013 10.1017/jfm.2018.872 10.1002/widm.1305 10.1109/CVPR.2017.19 10.1002/nme.7176 10.1007/978-94-009-0369-2_5 10.1016/j.neunet.2017.12.012 10.1088/1757-899X/495/1/012003 10.1063/5.0055600 10.1016/j.neucom.2014.11.058 10.1111/j.1749-6632.1960.tb42846.x 10.1016/j.cma.2021.113938 10.1137/21M1397908 10.3390/info11040193 10.1111/mice.12685 10.7551/mitpress/11301.001.0001 10.1109/CVPR.2005.212 10.2139/ssrn.4280307 10.1016/j.cma.2021.114333 10.1016/j.neucom.2021.06.015 10.1016/0038-0121(96)00010-9 10.1016/j.ijheatfluidflow.2022.109002 10.1007/s00033-022-01767-z 10.1126/sciadv.abi8605 10.1016/S0927-0507(06)13012-1 10.1137/19M1274067 10.1007/s10586-021-03240-4 10.1006/jcph.1995.1209 10.1137/22M1477751 10.1109/ICICCS48265.2020.9120874 10.1109/CVPR.2018.00917 10.4208/cicp.OA-2020-0086 10.1137/18M1229845 10.1214/aos/1176350842 10.1137/20M1318043 10.1007/978-1-4842-2766-4_8 10.1109/72.392253 10.1162/neco_a_{0}1199 10.1016/0021-9991(90)90007-N 10.1016/j.cma.2022.114823 10.1016/j.jcp.2021.110698 10.1109/72.712178 10.1090/S0025-5718-01-01307-2 10.1038/s41598-022-19157-w 10.1109/72.870037 10.1145/3483595 10.1103/PhysRevResearch.4.023210 10.1016/j.amc.2006.05.068 10.1016/j.jcp.2018.08.029 10.1007/s40745-020-00253-5 10.1016/j.jcp.2018.10.045 10.1145/1596519.1596520 10.1007/978-1-4757-3071-5_3 10.1073/pnas.1922210117 10.1007/s11277-017-5224-x 10.1029/JC094iC05p06177 10.1016/0041-5553(67)90144-9 |
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| References | Lu L, Pestourie R, Johnson SG, Romano G (2022) Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport. Technical Report arXiv:2204.06684 Guibas J, Mardani M, Li Z, Tao A, Anandkumar A, Catanzaro B (2022) Adaptive Fourier neural operators: efficient token mixers for transformers. Technical Report. arXiv:2111.13587 Wang T-C, Liu M-Y, Zhu J-Y, Tao A, Kautz J, Catanzaro B (2018) High-resolution image synthesis and semantic manipulation with conditional GANs. In: Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp 8798–8807 Kingma DP, Ba J (2017) Adam: a method for stochastic optimization. Technical Report. arXiv arXiv:1412.6980 Lu L, Pestourie R, Yao W, Wang Z, Verdugo F, Johnson SG (2021a) Physics-informed neural networks with hard constraints for inverse design. SIAM J Sci Comput 43(6):B1105–B1132. ISSN 1064-8275. https://doi.org/10.1137/21M1397908 Arzani A, Wang J-X, D’Souza RM (2021) Uncovering near-wall blood flow from sparse data with physics-informed neural networks. Phys Fluids 33(7):071905. ISSN 1070-6631. https://doi.org/10.1063/5.0055600 Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez AN, Kaiser Ł, Polosukhin I (2017b) Attention is all you need. In: Advances in neural information processing systems, vol 30. Curran Associates, Inc. https://proceedings.neurips.cc/paper/2017/hash/3f5ee243547dee91fbd053c1c4a845aa-Abstract.html Shi S, Liu D, Zhao Z (2021) Non-Fourier heat conduction based on self-adaptive weight physics-informed neural networks. In: 2021 40th Chinese control conference (CCC), pp 8451–8456. ISSN 1934-1768. https://doi.org/10.23919/CCC52363.2021.9550487 WangSWangHPerdikarisPLearning the solution operator of parametric partial differential equations with physics-informed DeepONetsSci Adv2021740eabi860510.1126/sciadv.abi8605 Srivastava RK, Greff K, Schmidhuber J (2015) Training very deep networks. In: Advances in neural information processing systems, vol 8. Curran Associates, Inc. https://proceedings.neurips.cc/paper/2015/hash/215a71a12769b056c3c32e7299f1c5ed-Abstract.html Isola P, Zhu J-Y, Zhou T, Efros AA (2017) Image-to-image translation with conditional adversarial networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp 1125–1134 Elfwing S, Uchibe E, Doya K (2017) Sigmoid-weighted linear units for neural network function approximation in reinforcement learning. Technical Report. arXiv arXiv:1702.03118 HammersleyJMonte Carlo methods2013SingaporeSpringer Ketkar N (2017) Stochastic gradient descent. In: Ketkar N (ed) Deep learning with Python: a hands-on introduction. Apress, Berkeley, pp 113–132. ISBN 978-1-4842-2766-4. https://doi.org/10.1007/978-1-4842-2766-4_8 Chen T, Chen H (1995) Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Trans Neural Netw 6(4):911–917. ISSN 1941-0093. https://doi.org/10.1109/72.392253 Partial differential equation toolbox (R2022a). https://uk.mathworks.com/products/pde.html Cao S (2021) Choose a transformer: Fourier or Galerkin. In: Advances in neural information processing systems, 2021, vol 34. Curran Associates, Inc., pp 24924–24940. https://proceedings.neurips.cc/paper/2021/hash/d0921d442ee91b896ad95059d13df618-Abstract.html Zhang D, Guo L, Karniadakis GE (2020) Learning in modal space: solving time-dependent stochastic PDEs using physics-informed neural networks. SIAM J Sci Comput 42(2):A639–A665. ISSN 1064-8275. https://doi.org/10.1137/19M1260141 Mathias R (1996) A chain rule for matrix functions and applications. SIAM J Matrix Anal Appl 17(3):610–620. ISBN: 0895-4798 Andonie R (2019) Hyperparameter optimization in learning systems. J Membr Comput 1(4):279–291. ISSN 2523-8914. https://doi.org/10.1007/s41965-019-00023-0 Gopakumar V, Pamela S, Samaddar D (2022) Loss landscape engineering via data regulation on PINNs. Technical Report. arXiv arXiv:2205.07843 Hammersley JM (1960) Monte Carlo methods for solving multivariable problems. Ann NY Acad Sci 86(3):844–874. ISSN 1749-6632. https://doi.org/10.1111/j.1749-6632.1960.tb42846.x Raissi M, Perdikaris P, Karniadakis GE (2019b) 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. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2018.10.045 Lee H, Kang IS (1990) Neural algorithm for solving differential equations. J Comput Phys 91(1):110–131. ISSN 0021-9991. https://doi.org/10.1016/0021-9991(90)90007-N Yu J, Lu L, Meng X, Karniadakis GE (2022) Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Comput Methods Appl Mech Eng 393:114823. ISSN 0045-7825. https://doi.org/10.1016/j.cma.2022.114823 Li Z, Zheng H, Kovachki N, Jin D, Chen H, Liu B, Azizzadenesheli K, Anandkumar A (2021) Physics-informed neural operator for learning partial differential equations. arXiv preprint arXiv:2111.03794 Lagaris IE, Likas AC, Papageorgiou DG (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Netw 11(5):1041–1049. ISSN 1941-0093. https://doi.org/10.1109/72.870037 Freedman D (2009) Statistical models: theory and practice. Cambridge University Press. ISBN 978-0-521-11243-7. Google-Books-ID 4N3KOEitRe8C Raissi M, Perdikaris P, Karniadakis GE (2018) Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. SIAM J Sci Comput. https://doi.org/10.1137/17M1120762 Gao H, Zahr MJ, Wang J-X (2022) Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems. Comput Methods Appl Mech Eng 390:114502. ISSN 0045-7825. https://doi.org/10.1016/j.cma.2021.114502 Lemieux C (2006) Chapter 12: quasi-random number techniques. In: Henderson SG, Nelson BL (eds) Handbooks in operations research and management science. Simulation, vol 13. Elsevier, pp 351–379. https://doi.org/10.1016/S0927-0507(06)13012-1 Barros CDT, Mendonça MRF, Vieira AB, Ziviani A (2021) A survey on embedding dynamic graphs. ACM Comput Surv 55(1):1–37. ISSN: 0360-0300 Xiang Z, Peng W, Zhou W, Yao W (2022) Hybrid finite difference with the physics-informed neural network for solving PDE in complex geometries. arXiv:2202.07926 [physics] Xu C, Cao TB, Yuan Y, Meschke G (2022) Transfer learning based physics-informed neural networks for solving inverse problems in tunneling. Technical Report. arXiv arXiv:2205.07731 Malek A, Shekari Beidokhti R (2006) Numerical solution for high order differential equations using a hybrid neural network-optimization method. Appl Math Comput 183(1):260–271. ISSN 0096-3003. https://doi.org/10.1016/j.amc.2006.05.068 Arulkumaran K, Deisenroth MP, Brundage M, Bharath AA (2017) Deep reinforcement learning: a brief survey. IEEE Signal Process Mag 34(6):26–38. ISSN 1558-0792. https://doi.org/10.1109/MSP.2017.2743240 Faure H, Lemieux C (2009) Generalized Halton sequences in 2008: a comparative study. ACM Trans Model Comput Simul 19(4):15:1–15:31. ISSN 1049-3301. https://doi.org/10.1145/1596519.1596520 Schiassi E, Leake C, De Florio M, Johnston H, Furfaro R, Mortari D (2020) Extreme theory of functional connections: a physics-informed neural network method for solving parametric differential equations. Technical Report. arXiv arXiv:2005.10632 Raschka S, Patterson J, Nolet C (2020) Machine learning in Python: main developments and technology trends in data science, machine learning, and artificial intelligence. Information 11(4):193. ISSN 2078-2489. https://doi.org/10.3390/info11040193 Sukumar N, Srivastava A (2022) Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Comput Methods Appl Mech Eng 389:114333. ISSN 0045-7825. https://doi.org/10.1016/j.cma.2021.114333 Ledig C, Theis L, Huszár F, Caballero J, Cunningham A, Acosta A, Aitken A, Tejani A, Totz J, Wang Z, Shi W (2017) Photo-realistic single image super-resolution using a generative adversarial network. In: Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp 4681–4690 Morokoff WJ, Caflisch RE (1995) Quasi-Monte Carlo integration. J Comput Phys 122(2):218–230. ISSN 0021-9991. https://doi.org/10.1006/jcph.1995.1209 Raissi M, Wang Z, Triantafyllou MS, Karniadakis GE (2019a) Deep learning of vortex-induced vibrations. J Fluid Mech 861:119–137. ISSN 0022-1120, 1469-7645. https://doi.org/10.1017/jfm.2018.872 Jagtap AD, Kawaguchi K, Karniadakis GE (2020) Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J Comput Phys 404:109136. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2019.109136 Tan HH, Lim KH (2019) Review of second-order optimization techniques in artificial neural networks backpropagation. IOP Conf Ser Mater Sci Eng 495:012003. ISSN 1757-899X. https://doi.org/10.1088/1757-899X/495/1/012003 Mao Z, Lu L, Marxen O, Zaki TA, Karniadakis GE (2021) 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 447:110698. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2021.110698 Chan T, Zhu W (2005) Level set based shape prior segmentation. In: 2005 IEEE Computer Society conference on computer vision and pattern recognition (CVPR’05), vol 2, pp 1164–1170. ISSN 1063-6919. https://doi.org/10.1109/CVPR.2005.212 Sun R-Y (2020) Optimization for deep learning: an overview. J Oper Res Soc China 8(2):249–294. ISSN 2194-6698. https://doi.org/10.1007/s40305-020-00309-6 Fang B, Yang E, Xie F (2020) Symbolic techniques for deep learning: challenges and opportunities. arXiv preprint arXiv:2010.02727 Zubov K, McCarthy Z, Ma Y, Calisto F, Pagliarino V, Azeglio S, Bottero L, Luján E, Sulzer V, Bharambe A, Vinchhi N, Balakrishnan K, Upadhyay D 9890_CR19 9890_CR20 9890_CR21 9890_CR26 9890_CR27 9890_CR28 9890_CR29 9890_CR22 9890_CR23 9890_CR24 9890_CR25 9890_CR30 9890_CR31 9890_CR32 9890_CR37 9890_CR38 9890_CR39 9890_CR33 9890_CR34 9890_CR35 9890_CR36 9890_CR40 9890_CR41 9890_CR42 9890_CR43 9890_CR48 9890_CR49 9890_CR44 9890_CR45 9890_CR46 9890_CR47 9890_CR51 9890_CR52 9890_CR53 9890_CR54 9890_CR50 9890_CR59 9890_CR55 9890_CR56 9890_CR57 9890_CR58 9890_CR102 9890_CR101 9890_CR100 9890_CR106 9890_CR105 9890_CR104 9890_CR103 9890_CR62 9890_CR63 9890_CR64 9890_CR65 9890_CR60 9890_CR61 9890_CR66 9890_CR67 9890_CR68 9890_CR69 9890_CR112 9890_CR111 9890_CR110 9890_CR73 9890_CR74 9890_CR109 9890_CR75 9890_CR108 9890_CR76 9890_CR107 9890_CR70 9890_CR71 9890_CR72 9890_CR77 9890_CR78 9890_CR79 9890_CR4 9890_CR5 9890_CR6 9890_CR7 9890_CR1 9890_CR2 9890_CR3 9890_CR84 9890_CR85 9890_CR86 9890_CR87 9890_CR80 9890_CR81 9890_CR82 9890_CR83 S Wang (9890_CR94) 2021; 7 9890_CR88 9890_CR89 9890_CR90 9890_CR95 9890_CR96 9890_CR97 9890_CR10 9890_CR98 9890_CR8 9890_CR9 9890_CR92 9890_CR93 9890_CR15 9890_CR16 9890_CR17 9890_CR18 9890_CR11 9890_CR99 9890_CR12 9890_CR13 9890_CR14 J Hammersley (9890_CR91) 2013 |
| References_xml | – reference: Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez AN, Kaiser Ł, Polosukhin I (2017b) Attention is all you need. In: Advances in neural information processing systems, vol 30. Curran Associates, Inc. https://proceedings.neurips.cc/paper/2017/hash/3f5ee243547dee91fbd053c1c4a845aa-Abstract.html – reference: Xu C, Cao TB, Yuan Y, Meschke G (2022) Transfer learning based physics-informed neural networks for solving inverse problems in tunneling. Technical Report. arXiv arXiv:2205.07731 – reference: Raissi M, Wang Z, Triantafyllou MS, Karniadakis GE (2019a) Deep learning of vortex-induced vibrations. J Fluid Mech 861:119–137. ISSN 0022-1120, 1469-7645. https://doi.org/10.1017/jfm.2018.872 – reference: Wang S, Yu X, Perdikaris P (2022a) When and why PINNs fail to train: a neural tangent kernel perspective. J Comput Phys 449:110768. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2021.110768 – reference: Davenport JH, Siret Y, Tournier É (1993) Computer algebra systems and algorithms for algebraic computation. Academic Press Professional, Inc. ISBN 0-12-204232-8 – reference: Srivastava RK, Greff K, Schmidhuber J (2015) Training very deep networks. In: Advances in neural information processing systems, vol 8. Curran Associates, Inc. https://proceedings.neurips.cc/paper/2015/hash/215a71a12769b056c3c32e7299f1c5ed-Abstract.html – reference: Lu L, Jin P, Pang G, Zhang Z, Karniadakis GE (2021c) Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat Mach Intell 3(3):218–229. ISSN 2522-5839. https://doi.org/10.1038/s42256-021-00302-5 – reference: Lee H, Kang IS (1990) Neural algorithm for solving differential equations. J Comput Phys 91(1):110–131. ISSN 0021-9991. https://doi.org/10.1016/0021-9991(90)90007-N – reference: Margossian CC (2019) A review of automatic differentiation and its efficient implementation. WIREs Data Min Knowl Discov 9(4):e1305. ISSN 1942-4795. https://doi.org/10.1002/widm.1305 – reference: Diaconis P, Shahshahani M (1984) On nonlinear functions of linear combinations. SIAM J Sci Stat Comput 5(1):175–191. ISSN 0196-5204. https://doi.org/10.1137/0905013 – reference: Abueidda DW, Koric S, Guleryuz E, Sobh NA (2022) Enhanced physics-informed neural networks for hyperelasticity. Technical Report. arXiv:2205.14148 – reference: Raissi M, Perdikaris P, Karniadakis GE (2019b) 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. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2018.10.045 – reference: Morokoff WJ, Caflisch RE (1995) Quasi-Monte Carlo integration. J Comput Phys 122(2):218–230. ISSN 0021-9991. https://doi.org/10.1006/jcph.1995.1209 – reference: Faure H, Lemieux C (2009) Generalized Halton sequences in 2008: a comparative study. ACM Trans Model Comput Simul 19(4):15:1–15:31. ISSN 1049-3301. https://doi.org/10.1145/1596519.1596520 – reference: Thacker WC (1989) The role of the Hessian matrix in fitting models to measurements. J Geophys Res Oceans 94(C5):6177–6196. ISSN 2156-2202. https://doi.org/10.1029/JC094iC05p06177 – reference: Barros CDT, Mendonça MRF, Vieira AB, Ziviani A (2021) A survey on embedding dynamic graphs. ACM Comput Surv 55(1):1–37. ISSN: 0360-0300 – reference: Lu L, Dao M, Kumar P, Ramamurty U, Karniadakis GE, Suresh S (2020) Extraction of mechanical properties of materials through deep learning from instrumented indentation. Proc Natl Acad Sci USA 117(13):7052–7062. https://doi.org/10.1073/pnas.1922210117 – reference: Jin P, Meng S, Lu L (2022) MIONet: learning multiple-input operators via tensor product. Technical Report. arXiv:2202.06137 – reference: Lu L, Meng X, Mao Z, Karniadakis GE (2021b) DeepXDE: a deep learning library for solving differential equations. SIAM Rev 63(1):208–228. ISSN 0036-1445. https://doi.org/10.1137/19M1274067 – reference: Yu Y, Si X, Hu C, Zhang J (2019) A review of recurrent neural networks: LSTM cells and network architectures. Neural Comput 31(7):1235–1270. ISSN 0899-7667. https://doi.org/10.1162/neco_a_{0}1199 – reference: Meng X, Karniadakis GE (2020) A composite neural network that learns from multi-fidelity data: application to function approximation and inverse PDE problems. J Comput Phys 401:109020. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2019.109020 – reference: Robert CP, Casella G (1999) Monte Carlo integration. In: Robert CP, Casella G (eds) Monte Carlo statistical methods, Springer texts in statistics. Springer, New York, pp 71–138. ISBN 978-1-4757-3071-5. https://doi.org/10.1007/978-1-4757-3071-5_3 – reference: Raissi M, Perdikaris P, Karniadakis GE (2018) Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. SIAM J Sci Comput. https://doi.org/10.1137/17M1120762 – reference: Jagtap AD, Kawaguchi K, Karniadakis GE (2020) Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J Comput Phys 404:109136. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2019.109136 – reference: Hennigh O, Narasimhan S, Nabian MA, Subramaniam A, Tangsali K, Fang Z, Rietmann M, Byeon W, Choudhry S (2021) NVIDIA SimNetTM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{TM}$$\end{document}: an AI-accelerated multi-physics simulation framework. In: Paszynski M, Kranzlmüller D, Krzhizhanovskaya VV, Dongarra JJ, Sloot PMA (eds) Computational science—ICCS 2021, Lecture notes in computer science, 2021. Springer, Cham, pp 447–461. ISBN 978-3-030-77977-1. https://doi.org/10.1007/978-3-030-77977-1_36 – reference: Isola P, Zhu J-Y, Zhou T, Efros AA (2017) Image-to-image translation with conditional adversarial networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp 1125–1134 – reference: Lagaris IE, Likas AC, Papageorgiou DG (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Netw 11(5):1041–1049. ISSN 1941-0093. https://doi.org/10.1109/72.870037 – reference: Tancik M, Srinivasan P, Mildenhall B, Fridovich-Keil S, Raghavan N, Singhal U, Ramamoorthi R, Barron J, Ng R (2020) Fourier features let networks learn high frequency functions in low dimensional domains. In: Advances in neural information processing systems, 2020, vol 33. Curran Associates, Inc., pp 7537–7547. https://proceedings.neurips.cc/paper/2020/hash/55053683268957697aa39fba6f231c68-Abstract.html – reference: Shaw JEH (1988) A quasirandom approach to integration in Bayesian statistics. Ann Stat 16(2):895–914. ISSN 0090-5364. https://www.jstor.org/stable/2241763 – reference: Zapf B, Haubner J, Kuchta M, Ringstad G, Eide PK, Mardal K-A (2022) Investigating molecular transport in the human brain from MRI with physics-informed neural networks. Technical Report. arXiv arXiv:2205.02592 – reference: Stiff differential equations. https://uk.mathworks.com/company/newsletters/articles/stiff-differential-equations.html – reference: Trehan D (2020) Non-convex optimization: a review. In: 2020 4th International conference on intelligent computing and control systems (ICICCS), pp 418–423. https://doi.org/10.1109/ICICCS48265.2020.9120874 – reference: Yu J, Lu L, Meng X, Karniadakis GE (2022) Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Comput Methods Appl Mech Eng 393:114823. ISSN 0045-7825. https://doi.org/10.1016/j.cma.2022.114823 – reference: Kingma DP, Ba J (2017) Adam: a method for stochastic optimization. Technical Report. arXiv arXiv:1412.6980 – reference: Molnar C (2020) Interpretable machine learning. Lulu.com. ISBN 0-244-76852-8 – reference: Joe S, Kuo FY (2008) Constructing Sobol sequences with better two-dimensional projections. SIAM J Sci Comput 30(5):2635–2654. ISSN 1064-8275. https://doi.org/10.1137/070709359 – reference: Sukumar N, Srivastava A (2022) Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Comput Methods Appl Mech Eng 389:114333. ISSN 0045-7825. https://doi.org/10.1016/j.cma.2021.114333 – reference: Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9(5): 987–1000. ISSN 1941-0093. https://doi.org/10.1109/72.712178 – reference: Nabian MA, Gladstone RJ, Meidani H (2021) Efficient training of physics-informed neural networks via importance sampling. Comput Aided Civ Infrastruct Eng 36(8):962–977. ISSN 1467-8667. https://doi.org/10.1111/mice.12685 – reference: WangSWangHPerdikarisPLearning the solution operator of parametric partial differential equations with physics-informed DeepONetsSci Adv2021740eabi860510.1126/sciadv.abi8605 – reference: Mathias R (1996) A chain rule for matrix functions and applications. SIAM J Matrix Anal Appl 17(3):610–620. ISBN: 0895-4798 – reference: Ramm A, Smirnova A (2001) On stable numerical differentiation. Math Comput 70(235):1131–1153. ISSN 0025-5718, 1088-6842. https://doi.org/10.1090/S0025-5718-01-01307-2 – reference: Demo N, Strazzullo M, Rozza G (2021) An extended physics informed neural network for preliminary analysis of parametric optimal control problems. Technical Report. arXiv arXiv:2110.13530 – reference: Partial differential equation toolbox (R2022a). https://uk.mathworks.com/products/pde.html – reference: De Florio M, Schiassi E, Furfaro R (2022b) Physics-informed neural networks and functional interpolation for stiff chemical kinetics. Chaos Interdiscip J Nonlinear Sci 32(6):063107. ISSN 1054-1500. https://doi.org/10.1063/5.0086649 – reference: Pang G, Lu L, Karniadakis GE (2019) fPINNs: fractional physics-informed neural networks. SIAM J Sci Comput 41(4):A2603–A2626. ISSN 1064-8275. https://doi.org/10.1137/18M1229845 – reference: Bajaj C, McLennan L, Andeen T, Roy A (2021) Robust learning of physics informed neural networks. Technical Report. arXiv arXiv:2110.13330 – reference: Aliakbari M, Mahmoudi M, Vadasz P, Arzani A (2022) Predicting high-fidelity multiphysics data from low-fidelity fluid flow and transport solvers using physics-informed neural networks. Int J Heat Fluid Flow 96:109002. ISSN 0142-727X. https://doi.org/10.1016/j.ijheatfluidflow.2022.109002 – reference: McClenny L, Braga-Neto U (2019) Self-adaptive physics-informed neural networks using a soft attention mechanism. Technical Report 68. http://ceur-ws.org/Vol-2964/article_68.pdf – reference: Modulus user guide, release v21.06 (2021). https://developer.nvidia.com/modulus-user-guide-v2106 – reference: Raj M, Kumbhar P, Annabattula RK (2022) Physics-informed neural networks for solving thermo-mechanics problems of functionally graded material. Technical Report. arXiv arXiv:2111.10751 – reference: Ketkar N (2017) Stochastic gradient descent. In: Ketkar N (ed) Deep learning with Python: a hands-on introduction. Apress, Berkeley, pp 113–132. ISBN 978-1-4842-2766-4. https://doi.org/10.1007/978-1-4842-2766-4_8 – reference: Minsky M, Papert SA (2017) Perceptrons: an introduction to computational geometry. The MIT Press. ISBN 978-0-262-34393-0. https://doi.org/10.7551/mitpress/11301.001.0001 – reference: Haghighat E, Juanes R (2021) SciANN: a Keras/TensorFlow wrapper for scientific computations and physics-informed deep learning using artificial neural networks. Comput Methods Appl Mech Eng 373:113552. ISSN 0045-7825. https://doi.org/10.1016/j.cma.2020.113552 – reference: Cao S (2021) Choose a transformer: Fourier or Galerkin. In: Advances in neural information processing systems, 2021, vol 34. Curran Associates, Inc., pp 24924–24940. https://proceedings.neurips.cc/paper/2021/hash/d0921d442ee91b896ad95059d13df618-Abstract.html – reference: Sun R-Y (2020) Optimization for deep learning: an overview. J Oper Res Soc China 8(2):249–294. ISSN 2194-6698. https://doi.org/10.1007/s40305-020-00309-6 – reference: Andonie R (2019) Hyperparameter optimization in learning systems. J Membr Comput 1(4):279–291. ISSN 2523-8914. https://doi.org/10.1007/s41965-019-00023-0 – reference: Pascanu R, Mikolov T, Bengio Y (2013) On the difficulty of training recurrent neural networks. In: Proceedings of the 30th international conference on machine learning. PMLR, pp 1310–1318. ISSN 1938-7228. https://proceedings.mlr.press/v28/pascanu13.html – reference: Schiassi E, Leake C, De Florio M, Johnston H, Furfaro R, Mortari D (2020) Extreme theory of functional connections: a physics-informed neural network method for solving parametric differential equations. Technical Report. arXiv arXiv:2005.10632 – reference: Sitzmann V, Martel J, Bergman A, Lindell D, Wetzstein G (2020) Implicit neural representations with periodic activation functions. In: Advances in neural information processing systems, 2020, vol 33. Curran Associates, Inc., pp 7462–7473. https://proceedings.neurips.cc/paper/2020/hash/53c04118df112c13a8c34b38343b9c10-Abstract.html – reference: Lu L, Pestourie R, Johnson SG, Romano G (2022) Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport. Technical Report arXiv:2204.06684 – reference: Arulkumaran K, Deisenroth MP, Brundage M, Bharath AA (2017) Deep reinforcement learning: a brief survey. IEEE Signal Process Mag 34(6):26–38. ISSN 1558-0792. https://doi.org/10.1109/MSP.2017.2743240 – reference: Zubov K, McCarthy Z, Ma Y, Calisto F, Pagliarino V, Azeglio S, Bottero L, Luján E, Sulzer V, Bharambe A, Vinchhi N, Balakrishnan K, Upadhyay D, Rackauckas C (2021) NeuralPDE: automating physics-informed neural networks (PINNs) with error approximations. Technical Report. arXiv arXiv:2107.09443 – reference: Lewkowycz A (2021) How to decay your learning rate. arXiv:2103.12682 [cs] – reference: Wang S, Teng Y, Perdikaris P (2021a) Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM J Sci Comput 43(5):A3055–A3081. ISSN 1064-8275. https://doi.org/10.1137/20M1318043 – reference: Cai S, Wang Z, Lu L, Zaki TA, Karniadakis GE (2021) DeepM&Mnet: inferring the electroconvection multiphysics fields based on operator approximation by neural networks. J Comput Phys 436:110296. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2021.110296 – reference: Gao H, Zahr MJ, Wang J-X (2022) Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems. Comput Methods Appl Mech Eng 390:114502. ISSN 0045-7825. https://doi.org/10.1016/j.cma.2021.114502 – reference: Xiang Z, Peng W, Zhou W, Yao W (2022) Hybrid finite difference with the physics-informed neural network for solving PDE in complex geometries. arXiv:2202.07926 [physics] – reference: Ghojogh B, Ghodsi A, Karray F, Crowley M (2021) KKT conditions, first-order and second-order optimization, and distributed optimization: tutorial and survey. arXiv:2110.01858 [cs, math] – reference: Heger P, Full M, Hilger D, Hosters N (2022) Investigation of physics-informed deep learning for the prediction of parametric, three-dimensional flow based on boundary data. Technical Report. arXiv arXiv:2203.09204 – reference: Freedman D (2009) Statistical models: theory and practice. Cambridge University Press. ISBN 978-0-521-11243-7. Google-Books-ID 4N3KOEitRe8C – reference: Sobol’ IM (1967) On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput Math Math Phys 7(4):86–112. ISSN 00415553. https://doi.org/10.1016/0041-5553(67)90144-9 – reference: Giles M (2008) An extended collection of matrix derivative results for forward and reverse mode automatic differentiation. Report – reference: Mao Z, Lu L, Marxen O, Zaki TA, Karniadakis GE (2021) 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 447:110698. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2021.110698 – reference: Wang Q, Ma Y, Zhao K, Tian Y (2022b) A comprehensive survey of loss functions in machine learning. Ann Data Sci 9(2):187–212. ISSN 2198-5812. https://doi.org/10.1007/s40745-020-00253-5 – reference: Lu L, Pestourie R, Yao W, Wang Z, Verdugo F, Johnson SG (2021a) Physics-informed neural networks with hard constraints for inverse design. SIAM J Sci Comput 43(6):B1105–B1132. ISSN 1064-8275. https://doi.org/10.1137/21M1397908 – reference: Fletcher R (1994) An overview of unconstrained optimization. In: Spedicato E (ed) Algorithms for continuous optimization: the state of the art, NATO ASI series. Springer, Dordrecht, pp 109–143. ISBN 978-94-009-0369-2. https://doi.org/10.1007/978-94-009-0369-2_5 – reference: Werbos P (1974) Beyond regression: new tools for prediction and analysis in the behavior science. Doctoral Dissertation, Harvard University. https://ci.nii.ac.jp/naid/10012540025/ – reference: Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez AN, Kaiser Ł, Polosukhin I (2017a) Attention is all you need. In: Advances in neural information processing systems, 2017, vol 30 – reference: Sirignano J, Spiliopoulos K (2018) DGM: a deep learning algorithm for solving partial differential equations. J Comput Phys 375:1339–1364. ISSN 0021-9991. https://doi.org/10.1016/j.jcp.2018.08.029 – reference: Chan T, Zhu W (2005) Level set based shape prior segmentation. In: 2005 IEEE Computer Society conference on computer vision and pattern recognition (CVPR’05), vol 2, pp 1164–1170. ISSN 1063-6919. https://doi.org/10.1109/CVPR.2005.212 – reference: Raschka S, Patterson J, Nolet C (2020) Machine learning in Python: main developments and technology trends in data science, machine learning, and artificial intelligence. Information 11(4):193. ISSN 2078-2489. https://doi.org/10.3390/info11040193 – reference: Martino L, Elvira V, Louzada F (2017) Effective sample size for importance sampling based on discrepancy measures. Signal Process 131:386–401. ISSN 0165-1684. https://doi.org/10.1016/j.sigpro.2016.08.025 – reference: HammersleyJMonte Carlo methods2013SingaporeSpringer – reference: Lemieux C (2006) Chapter 12: quasi-random number techniques. In: Henderson SG, Nelson BL (eds) Handbooks in operations research and management science. Simulation, vol 13. Elsevier, pp 351–379. https://doi.org/10.1016/S0927-0507(06)13012-1 – reference: Wang T-C, Liu M-Y, Zhu J-Y, Tao A, Kautz J, Catanzaro B (2018) High-resolution image synthesis and semantic manipulation with conditional GANs. In: Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp 8798–8807 – reference: Rahaman N, Baratin A, Arpit D, Draxler F, Lin M, Hamprecht F, Bengio Y, Courville A (2019) On the spectral bias of neural networks. In: Proceedings of the 36th international conference on machine learning, 2019. PMLR, pp 5301–5310. ISSN 2640-3498. https://proceedings.mlr.press/v97/rahaman19a.html – reference: Son H, Jang JW, Han WJ, Hwang HJ (2021) Sobolev training for physics informed neural networks. Technical Report. arXiv arXiv:2101.08932 – reference: De Florio M, Schiassi E, Ganapol BD, Furfaro R (2022a) Physics-Informed Neural Networks for rarefied-gas dynamics: Poiseuille flow in the BGK approximation. Z angew Math Phys 73(3):126. ISSN 1420-9039. https://doi.org/10.1007/s00033-022-01767-z – reference: Tan HH, Lim KH (2019) Review of second-order optimization techniques in artificial neural networks backpropagation. IOP Conf Ser Mater Sci Eng 495:012003. ISSN 1757-899X. https://doi.org/10.1088/1757-899X/495/1/012003 – reference: Malek A, Shekari Beidokhti R (2006) Numerical solution for high order differential equations using a hybrid neural network-optimization method. Appl Math Comput 183(1):260–271. ISSN 0096-3003. https://doi.org/10.1016/j.amc.2006.05.068 – reference: Berrada I, Ferland JA, Michelon P (1996) A multi-objective approach to nurse scheduling with both hard and soft constraints. Socio-Econ Plan Sci 30(3):183–193. ISSN 0038-0121. https://doi.org/10.1016/0038-0121(96)00010-9 – reference: Wu Y, Feng J (2018) Development and application of artificial neural network. Wirel Pers Commun 102(2):1645–1656. ISSN 1572-834X. https://doi.org/10.1007/s11277-017-5224-x – reference: Wang S, Wang H, Perdikaris P (2021c) On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks. Comput Methods Appl Mech Eng 384:113938. ISSN 0045-7825. https://doi.org/10.1016/j.cma.2021.113938 – reference: McClenny LD, Haile MA, Braga-Neto UM (2021) TensorDiffEq: scalable multi-GPU forward and inverse solvers for physics informed neural networks. Technical Report. arXiv arXiv:2103.16034 – reference: Halton JH (1960) On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer Math 2(1):84–90. ISSN 0945-3245. https://doi.org/10.1007/BF01386213 – reference: Guibas J, Mardani M, Li Z, Tao A, Anandkumar A, Catanzaro B (2022) Adaptive Fourier neural operators: efficient token mixers for transformers. Technical Report. arXiv:2111.13587 – reference: Zhang D, Guo L, Karniadakis GE (2020) Learning in modal space: solving time-dependent stochastic PDEs using physics-informed neural networks. SIAM J Sci Comput 42(2):A639–A665. ISSN 1064-8275. https://doi.org/10.1137/19M1260141 – reference: Rudd K, Ferrari S (2015) A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks. Neurocomputing 155:277–285. ISSN 0925-2312. https://doi.org/10.1016/j.neucom.2014.11.058 – reference: Li Z, Zheng H, Kovachki N, Jin D, Chen H, Liu B, Azizzadenesheli K, Anandkumar A (2021) Physics-informed neural operator for learning partial differential equations. arXiv preprint arXiv:2111.03794 – reference: Samsami MR, Alimadad H (2020) Distributed deep reinforcement learning: an overview. Technical Report. arXiv arXiv:2011.11012 – reference: Zhao CL (2020) Solving Allen–Cahn and Cahn–Hilliard equations using the adaptive physics informed neural networks. Commun Comput Phys 29(3). https://doi.org/10.4208/cicp.OA-2020-0086 – reference: Fang B, Yang E, Xie F (2020) Symbolic techniques for deep learning: challenges and opportunities. arXiv preprint arXiv:2010.02727 – reference: Gopakumar V, Pamela S, Samaddar D (2022) Loss landscape engineering via data regulation on PINNs. Technical Report. arXiv arXiv:2205.07843 – reference: Hammersley JM (1960) Monte Carlo methods for solving multivariable problems. Ann NY Acad Sci 86(3):844–874. ISSN 1749-6632. https://doi.org/10.1111/j.1749-6632.1960.tb42846.x – reference: Elfwing S, Uchibe E, Doya K (2017) Sigmoid-weighted linear units for neural network function approximation in reinforcement learning. Technical Report. arXiv arXiv:1702.03118 – reference: Chen T, Chen H (1995) Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Trans Neural Netw 6(4):911–917. ISSN 1941-0093. https://doi.org/10.1109/72.392253 – reference: Elshawi R, Wahab A, Barnawi A, Sakr S (2021) DLBench: a comprehensive experimental evaluation of deep learning frameworks. Clust Comput 24(3):2017–2038. ISSN 1573-7543. https://doi.org/10.1007/s10586-021-03240-4 – reference: Shi S, Liu D, Zhao Z (2021) Non-Fourier heat conduction based on self-adaptive weight physics-informed neural networks. In: 2021 40th Chinese control conference (CCC), pp 8451–8456. ISSN 1934-1768. https://doi.org/10.23919/CCC52363.2021.9550487 – reference: Viana FAC (2016) A tutorial on Latin hypercube design of experiments. Qual Reliab Eng Int 32(5):1975–1985. ISSN 1099-1638. https://doi.org/10.1002/qre.1924 – reference: Ledig C, Theis L, Huszár F, Caballero J, Cunningham A, Acosta A, Aitken A, Tejani A, Totz J, Wang Z, Shi W (2017) Photo-realistic single image super-resolution using a generative adversarial network. In: Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp 4681–4690 – reference: Arzani A, Wang J-X, D’Souza RM (2021) Uncovering near-wall blood flow from sparse data with physics-informed neural networks. Phys Fluids 33(7):071905. 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| SubjectTerms | Algorithms Boundary conditions Conduction heating Conductive heat transfer Corners Engineering Field programmable gate arrays Hypotheses Inverse problems Libraries Machine learning Mathematical and Computational Engineering Neural networks Ordinary differential equations Parameterization Partial differential equations Physics Regression analysis Review Article |
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| Title | Stiff-PDEs and Physics-Informed Neural Networks |
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