A new modified deep learning technique based on physics-informed neural networks (PINNs) for the shock-induced coupled thermoelasticity analysis in a porous material

References

• S. Rashidi, J. A. Esfahani and A. Rashidi, “A review on the applications of porous materials in solar energy systems,” Renew. Sustain. Energy Rev., vol. 73, pp. 1198–1210, Jun. 2017. DOI: 10.1016/j.rser.2017.02.028.
   Web of Science ®Google Scholar
• S. Rashidi, J. A. Esfahani and N. Karimi, “Porous materials in building energy technologies—A review of the applications, modelling and experiments,” Renew. Sustain. Energy Rev., vol. 91, pp. 229–247, Aug. 2018. DOI: 10.1016/j.rser.2018.03.092.
   Web of Science ®Google Scholar
• J. W. Nunziato and S. C. Cowin, “A nonlinear theory of elastic materials with voids,” Arch. Rational Mech. Anal., vol. 72, no. 2, pp. 175–201, Jun. 1979. DOI: 10.1007/BF00249363.
   Web of Science ®Google Scholar
• S. C. Cowin and J. W. Nunziato, “Linear elastic materials with voids,” J. Elasticity, vol. 13, no. 2, pp. 125–147, Jul. 1983. DOI: 10.1007/BF00041230.
   Web of Science ®Google Scholar
• D. Ieşan, “A theory of thermoelastic materials with voids,” Acta Mech., vol. 60, no. 1–2, pp. 67–89, Jun. 1986. DOI: 10.1007/BF01302942.
   Web of Science ®Google Scholar
• M. A. Biot, “Thermoelasticity and irreversible thermodynamics,” J. Appl. Phys., vol. 27, no. 3, pp. 240–253, 1956. DOI: 10.1063/1.1722351/title/thermoelasticity_and_irreversible_thermodynamics.
   Web of Science ®Google Scholar
• H. W. Lord and Y. Shulman, “A generalized dynamical theory of thermoelasticity,” J. Mech. Phys. Solids, vol. 15, no. 5, pp. 299–309, Sep. 1967. DOI: 10.1016/0022-5096(67)90024-5.
   Web of Science ®Google Scholar
• A. E. Green and K. A. Lindsay, “Thermoelasticity,” J. Elasticity, vol. 2, no. 1, pp. 1–7, Mar. 1972. DOI: 10.1007/BF00045689.
   Google Scholar
• . E. Green and P. M. Naghdi, “ON UNDAMPED HEAT WAVES IN AN ELASTIC SOLID,” J. Thermal Stresses, vol. 15, no. 2, pp. 253–264, Apr. 1992. DOI: 10.1080/01495739208946136.
   Web of Science ®Google Scholar
• A. E. Green and P. M. Naghdi, “Thermoelasticity without energy dissipation,” J. Elasticity, vol. 31, no. 3, pp. 189–208, Jun. 1993. DOI: 10.1007/BF00044969.
   Web of Science ®Google Scholar
• C. Li, T. He and X. Tian, “Transient responses of nanosandwich structure based on size-dependent generalized thermoelastic diffusion theory,” J. Thermal Stresses, vol. 42, no. 9, pp. 1171–1191, Sep. 2019. DOI: 10.1080/01495739.2019.1623140.
   Web of Science ®Google Scholar
• C. Li, H. Guo and X. Tian, “Time-domain finite element analysis to nonlinear transient responses of generalized diffusion-thermoelasticity with variable thermal conductivity and diffusivity,” Int. J. Mech. Sci., vol. 131–132, pp. 234–244, Oct. 2017. DOI: 10.1016/j.ijmecsci.2017.07.008.
   Web of Science ®Google Scholar
• B. Singh, “On theory of generalized thermoelastic solids with voids and diffusion,” Eur. J. Mech. - A/Solids, vol. 30, no. 6, pp. 976–982, Nov. 2011. DOI: 10.1016/j.euromechsol.2011.06.007.
   Google Scholar
• M. A. Ezzat, “Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer,” Phys. B Condens Matter., vol. 406, no. 1, pp. 30–35, Jan. 2011. DOI: 10.1016/j.physb.2010.10.005.
   Web of Science ®Google Scholar
• Y. J. Yu, X. G. Tian and T. J. Lu, “Fractional order generalized electro-magneto-thermo-elasticity,” Eur. J. Mech. - A/Solids, vol. 42, pp. 188–202, Nov. 2013. DOI: 10.1016/j.euromechsol.2013.05.006.
   Google Scholar
• H. M. Youssef, “Theory of fractional order generalized thermoelasticity,” J. Heat Transf., vol. 132, no. 6, pp. 061301, Jun. 2010. DOI: 10.1115/1.4000705.
   Web of Science ®Google Scholar
• Y. Li, P. Zhang, C. Li and T. He, “Fractional order and memory-dependent analysis to the dynamic response of a bi-layered structure due to laser pulse heating,” Int. J. Heat Mass Transf., vol. 144, pp. 118664, Dec. 2019. DOI: 10.1016/j.ijheatmasstransfer.2019.118664.
   Web of Science ®Google Scholar
• H. H. Sherief, A. M. A. El-Sayed and A. M. Abd El-Latief, “Fractional order theory of thermoelasticity,” Int. J. Solids Struct., vol. 47, no. 2, pp. 269–275, Jan. 2010. DOI: 10.1016/j.ijsolstr.2009.09.034.
   Web of Science ®Google Scholar
• Y.-J. Yu, W. Hu and X.-G. Tian, “A novel generalized thermoelasticity model based on memory-dependent derivative,” Int. J. Eng. Sci., vol. 81, pp. 123–134, Aug. 2014. DOI: 10.1016/j.ijengsci.2014.04.014.
   Web of Science ®Google Scholar
• Y. Li and T. He, “A generalized thermoelastic diffusion problem with memory-dependent derivative,” Math. Mech. Solids, vol. 24, no. 5, pp. 1438–1462, May 2019. DOI: 10.1177/1081286518797988.
   Web of Science ®Google Scholar
• T. He and Y. Li, “Transient responses of sandwich structure based on the generalized thermoelastic diffusion theory with memory-dependent derivative,” J. Sandw. Struct. Mater., vol. 22, no. 8, pp. 2505–2543, Nov. 2020. DOI: 10.1177/1099636218802574.
   Web of Science ®Google Scholar
• A. Behravan Rad and M. Shariyat, “Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations,” Compos. Struct., vol. 125, pp. 558–574, Jul. 2015. DOI: 10.1016/j.compstruct.2015.02.049.
   Web of Science ®Google Scholar
• M. M. Alipour and M. Shariyat, “Nonlocal zigzag analytical solution for Laplacian hygrothermal stress analysis of annular sandwich macro/nanoplates with poor adhesions and 2D-FGM porous cores,” Arch. Civil Mech. Eng., vol. 19, no. 4, pp. 1211–1234, Aug. 2019. DOI: 10.1016/j.acme.2019.06.008.
   Web of Science ®Google Scholar
• Y. J. Yu, Z.-N. Xue and X.-G. Tian, “A modified Green–Lindsay thermoelasticity with strain rate to eliminate the discontinuity,” Meccanica, vol. 53, no. 10, pp. 2543–2554, Aug. 2018. DOI: 10.1007/s11012-018-0843-1.
   Web of Science ®Google Scholar
• S. He, W. Peng, Y. Ma and T. He, “Investigation on the transient response of a porous half-space with strain and thermal relaxations,” Eur. J. Mech. - A/Solids, vol. 84, pp. 104064, Nov. 2020. DOI: 10.1016/j.euromechsol.2020.104064.
   Google Scholar
• S. He and T. He, “Investigation on the transient response of a porous half-space with strain and thermal relaxations due to laser pulse heating,” Waves Random Complex Media, vol. 34, no. 2, pp. 601–625, Apr. 2024. DOI: 10.1080/17455030.2021.1916126.
   Google Scholar
• Z. Qi, W. Peng and T. He, “Investigation on the thermoelastic response of a nanobeam in modified couple stress theory considering size-dependent and memory-dependent effects,” J. Thermal Stresses, vol. 45, no. 10, pp. 773–792, Oct. 2022. DOI: 10.1080/01495739.2022.2109543.
   Web of Science ®Google Scholar
• M. Raissi, P. Perdikaris and G. E. Karniadakis, “Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” J. Comput. Phys., vol. 378, pp. 686–707, Feb. 2019. DOI: 10.1016/j.jcp.2018.10.045.
   Web of Science ®Google Scholar
• Q. Zhu, Z. Liu and J. Yan, “Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks,” Comput. Mech., vol. 67, no. 2, pp. 619–635, Feb. 2021. DOI: 10.1007/s00466-020-01952-9.
   Web of Science ®Google Scholar
• X. Jin, S. Cai, H. Li and G. E. Karniadakis, “NSFnets (Navier-Stokes flow nets): physics-informed neural networks for the incompressible Navier-Stokes equations,” J. Comput. Phys., vol. 426, pp. 109951, Feb. 2021. DOI: 10.1016/j.jcp.2020.109951.
   Web of Science ®Google Scholar
• T. Kadeethum, T. M. Jørgensen and H. M. Nick, “Physics-informed neural networks for solving nonlinear diffusivity and Biot’s equations,” PLoS One vol. 15, no. 5, pp. e0232683, May 2020. DOI: 10.1371/journal.pone.0232683.
   PubMed Web of Science ®Google Scholar
• L. Sun, H. Gao, S. Pan and J. X. Wang, “Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data,” Comput. Methods Appl. Mech. Eng., vol. 361, pp. 112732, Apr. 2020. DOI: 10.1016/j.cma.2019.112732.
   Web of Science ®Google Scholar
• A. Mathews, M. Francisquez, J. W. Hughes, D. R. Hatch, B. Zhu and B. N. Rogers, “Uncovering turbulent plasma dynamics via deep learning from partial observations,” Phys. Rev. E, vol. 104, no. 2–2, pp. 025205, Aug. 2021. DOI: 10.1103/PhysRevE.104.025205.
   PubMed Web of Science ®Google Scholar
• C. Cheng and G. T. Zhang, “Deep learning method based on physics informed neural network with resnet block for solving fluid flow problems,” Water (Basel, vol. 13, no. 4, pp. 423, Feb. 2021. DOI: 10.3390/w13040423.
   Web of Science ®Google Scholar
• Z. Mao, A. D. Jagtap and G. E. Karniadakis, “Physics-informed neural networks for high-speed flows,” Comput. Methods Appl. Mech. Eng., vol. 360, pp. 112789, Mar. 2020. DOI: 10.1016/j.cma.2019.112789.
   Web of Science ®Google Scholar
• T. Zhou, E. L. Droguett and A. Mosleh, “Physics-informed deep learning: a promising technique for system reliability assessment,” Appl. Soft Comput., vol. 126, pp. 109217, Sep. 2022. DOI: 10.1016/j.asoc.2022.109217.
   Web of Science ®Google Scholar
• T. Zhou, X. Zhang, E. L. Droguett and A. Mosleh, “A generic physics-informed neural network-based framework for reliability assessment of multi-state systems,” Reliab. Eng. Syst. Saf., vol. 229, pp. 108835, Jan. 2023. DOI: 10.1016/j.ress.2022.108835.
   Web of Science ®Google Scholar
• C. Zhang and A. Shafieezadeh, “Simulation-free reliability analysis with active learning and physics-informed neural network,” Reliab. Eng. Syst. Saf., vol. 226, pp. 108716, Oct. 2022. DOI: 10.1016/j.ress.2022.108716.
   Web of Science ®Google Scholar
• M. Sepe et al., “A physics-informed machine learning framework for predictive maintenance applied to turbomachinery assets,” J. Glob. Power Propuls. Soc., pp. 1–15, May 2021. DOI: 10.33737/jgpps/134845.
   Google Scholar
• S. Shen et al., “A physics-informed deep learning approach for bearing fault detection,” Eng. Appl. Artif. Intell., vol. 103, pp. 104295, Aug. 2021. DOI: 10.1016/j.engappai.2021.104295.
   Web of Science ®Google Scholar
• A. D. J and G. E. Karniadakis, “Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations,” CiCP, vol. 28, no. 5, pp. 2002–2041, Jun. 2020. DOI: 10.4208/cicp.OA-2020-0164.
   Google Scholar
• L. Yang, X. Meng and G. E. Karniadakis, “B-PINNs: bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data,” J. Comput. Phys., vol. 425, pp. 109913, Jan. 2021. DOI: 10.1016/j.jcp.2020.109913.
   Web of Science ®Google Scholar
• A. D. Jagtap, E. Kharazmi and G. E. Karniadakis, “Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems,” Comput. Methods Appl. Mech. Eng., vol. 365, pp. 113028, Jun. 2020. DOI: 10.1016/j.cma.2020.113028.
   Web of Science ®Google Scholar
• E. Haghighat and R. Juanes, “SciANN: a Keras/TensorFlow wrapper for scientific computations and physics-informed deep learning using artificial neural networks,” Comput. Methods Appl. Mech. Eng., vol. 373, pp. 113552, Jan. 2021. DOI: 10.1016/j.cma.2020.113552.
   Web of Science ®Google Scholar
• M. Liu, L. Liang and W. Sun, “A generic physics-informed neural network-based constitutive model for soft biological tissues,” Comput. Methods Appl. Mech. Eng., vol. 372, pp. 113402, Dec. 2020. DOI: 10.1016/j.cma.2020.113402.
   PubMed Web of Science ®Google Scholar
• A. Yazdani, L. Lu, M. Raissi and G. E. Karniadakis, “Systems biology informed deep learning for inferring parameters and hidden dynamics,” PLoS Comput. Biol., vol. 16, no. 11, pp. e1007575, Nov. 2020. DOI: 10.1371/journal.pcbi.1007575.
   PubMed Web of Science ®Google Scholar
• M. Raissi, A. Yazdani and G. E. Karniadakis, “Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations,” Science, vol. 367, no. 6481, pp. 1026–1030, Feb. 2020. DOI: 10.1126/science.aaw4741.
   PubMed Web of Science ®Google Scholar
• A. D. Jagtap, D. Mitsotakis and G. E. Karniadakis, “Deep learning of inverse water waves problems using multi-fidelity data: application to Serre–Green–Naghdi equations,” Ocean Eng., vol. 248, pp. 110775, Mar. 2022. DOI: 10.1016/j.oceaneng.2022.110775.
   Web of Science ®Google Scholar
• K. Shukla, A. D. Jagtap, J. L. Blackshire, D. Sparkman and G. Em Karniadakis, “A physics-informed neural network for quantifying the microstructural properties of polycrystalline nickel using ultrasound data: a promising approach for solving inverse problems,” IEEE Signal Proc. Mag., vol. 39, no. 1, pp. 68–77, Jan. 2022. DOI: 10.1109/MSP.2021.3118904.
   Web of Science ®Google Scholar
• A. D. Jagtap, Z. Mao, N. Adams and G. E. Karniadakis, “Physics-informed neural networks for inverse problems in supersonic flows,” J. Comput. Phys., vol. 466, pp. 111402, Oct. 2022. DOI: 10.1016/j.jcp.2022.111402.
   Web of Science ®Google Scholar
• E. Haghighat, M. Raissi, A. Moure, H. Gomez and R. Juanes, “A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics,” Comput. Methods Appl. Mech. Eng., vol. 379, pp. 113741, Jun. 2021. DOI: 10.1016/j.cma.2021.113741.
   Web of Science ®Google Scholar
• M. Rasht‐Behesht, C. Huber, K. Shukla and G. E. Karniadakis, “Physics‐informed neural networks (PINNs) for wave propagation and full waveform inversions,” JGR Solid Earth, vol. 127, no. 5, pp. e2021JB023120, May 2022. DOI: 10.1029/2021JB023120.
   Web of Science ®Google Scholar
• W. Li, M. Z. Bazant and J. Zhu, “A physics-guided neural network framework for elastic plates: comparison of governing equations-based and energy-based approaches,” Comput. Methods Appl. Mech. Eng., vol. 383, pp. 113933, Sep. 2021. DOI: 10.1016/j.cma.2021.113933.
   Web of Science ®Google Scholar
• A. Habib and U. Yildirim, “Developing a physics-informed and physics-penalized neural network model for preliminary design of multi-stage friction pendulum bearings,” Eng. Appl. Artif. Intell., vol. 113, pp. 104953, Aug. 2022. DOI: 10.1016/j.engappai.2022.104953.
   Web of Science ®Google Scholar
• H. Guo, X. Zhuang, X. Fu, Y. Zhu and T. Rabczuk, “Physics-informed deep learning for three-dimensional transient heat transfer analysis of functionally graded materials,” Comput. Mech., vol. 72, no. 3, pp. 513–524, Apr. 2023. DOI: 10.1007/s00466-023-02287-x.
   Web of Science ®Google Scholar
• K. Shukla, P. C. Di Leoni, J. Blackshire, D. Sparkman and G. E. Karniadakis, “Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks,” J. Nondestruct. Eval., vol. 39, no. 3, pp. 61, Sep. 2020. DOI: 10.1007/s10921-020-00705-1.
   Web of Science ®Google Scholar
• C. Rao, H. Sun and Y. Liu, “Physics-informed deep learning for computational elastodynamics without labeled data,” J. Eng. Mech., vol. 147, no. 8, pp. 04021043, Aug. 2021. DOI: 10.1061/(ASCE)EM.1943-7889.0001947.
   Web of Science ®Google Scholar
• A. Fallah and M. M. Aghdam, “Physics-informed neural network for bending and free vibration analysis of three-dimensional functionally graded porous beam resting on elastic foundation,” Eng. Comput., vol. 40, no. 1, pp. 437–454, Mar. 2023. DOI: 10.1007/s00366-023-01799-7.
   Web of Science ®Google Scholar
• K. Shukla, A. D. Jagtap and G. E. Karniadakis, “Parallel physics-informed neural networks via domain decomposition,” J. Comput. Phys., vol. 447, pp. 110683, Dec. 2021. DOI: 10.1016/j.jcp.2021.110683.
   Web of Science ®Google Scholar
• M. Penwarden, A. D. Jagtap, S. Zhe, G. E. Karniadakis and R. M. Kirby, “A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions,” J. Comput. Phys., vol. 493, pp. 112464, Nov. 2023. DOI: 10.1016/j.jcp.2023.112464.
   Web of Science ®Google Scholar
• Z. Hu, A. D. Jagtap, G. E. Karniadakis and K. Kawaguchi, “Augmented Physics-Informed Neural Networks (APINNs): a gating network-based soft domain decomposition methodology,” Eng. Appl. Artif. Intell., vol. 126, pp. 107183, Nov. 2023. DOI: 10.1016/j.engappai.2023.107183.
   Web of Science ®Google Scholar
• B. Zhang, G. Wu, Y. Gu, X. Wang and F. Wang, “Multi-domain physics-informed neural network for solving forward and inverse problems of steady-state heat conduction in multilayer media,” Phys. Fluids, vol. 34, no. 11, pp. 116116, Nov. 2022. DOI: 10.1063/5.0116038.
   Web of Science ®Google Scholar
• B. Zhang, F. Wang and L. Qiu, “Multi-domain physics-informed neural networks for solving transient heat conduction problems in multilayer materials,” J. Appl. Phys., vol. 133, no. 24, pp. 245103, Jun. 2023. DOI: 10.1063/5.0153705.
   Web of Science ®Google Scholar
• G. Wu, F. Wang and L. Qiu, “Physics-informed neural network for solving Hausdorff derivative poisson equations,” Fractals, vol. 31, no. 06, pp. 2340103, Jan. 2023. DOI: 10.1142/S0218348X23401035.
   Web of Science ®Google Scholar
• S. Mishra and R. Molinaro, “Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs,” IMA J. Numer. Anal., vol. 42, no. 2, pp. 981–1022, Apr. 2022. DOI: 10.1093/imanum/drab032.
   Web of Science ®Google Scholar
• T. De Ryck, A. D. Jagtap and S. Mishra, “Error estimates for physics-informed neural networks approximating the Navier–Stokes equations,” IMA J. Numer. Anal., vol. 44, no. 1, pp. 83–119, Jan. 2023. DOI: 10.1093/imanum/drac085.
   Web of Science ®Google Scholar
• Z. Hu, A. D. Jagtap, G. E. Karniadakis and K. Kawaguchi, “When do extended physics-informed neural networks (XPINNs) improve generalization,” SIAM J. Sci. Comput., vol. 44, no. 5, pp. A3158–A3182, Oct. 2022. DOI: 10.1137/21M1447039.
   Web of Science ®Google Scholar
• A. D. Jagtap, K. Kawaguchi and G. E. Karniadakis, “Adaptive activation functions accelerate convergence in deep and physics-informed neural networks,” J. Comput. Phys., vol. 404, pp. 109136, Mar. 2020. DOI: 10.1016/j.jcp.2019.109136.
   Web of Science ®Google Scholar
• A. D. Jagtap, K. Kawaguchi and G. Em Karniadakis, “Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks,” Proc. Math. Phys. Eng. Sci., vol. 476, no. 2239, pp. 20200334, Jul. 2020. DOI: 10.1098/rspa.2020.0334.
   PubMed Web of Science ®Google Scholar
• A. D. Jagtap, Y. Shin, K. Kawaguchi and G. E. Karniadakis, “Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions,” Neurocomputing, vol. 468, pp. 165–180, Jan. 2022. DOI: 10.1016/j.neucom.2021.10.036.
   Web of Science ®Google Scholar
• A. D. Jagtap and G. E. Karniadakis, “How important are activation functions in regression and classification? a survey, performance comparison, and future directions,” J. Mach. Learn. Model. Comput., vol. 4, no. 1, pp. 21–75, 2023. DOI: 10.1615/JMachLearnModelComput.2023047367.
   Google Scholar
• K. Eshkofti and S. M. Hosseini, “A deep learning approach based on the physics-informed neural networks for Gaussian thermal shock-induced thermoelastic wave propagation analysis in a thick hollow cylinder with energy dissipation,” Waves Random Complex Media, pp. 1–40, Jun. 2022. DOI: 10.1080/17455030.2022.2083264.
   Web of Science ®Google Scholar
• K. Eshkofti and S. M. Hosseini, “The novel PINN/gPINN-based deep learning schemes for non-Fickian coupled diffusion-elastic wave propagation analysis,” Waves Random Complex Media, pp. 1–24, Feb. 2023. DOI: 10.1080/17455030.2023.2177499.
   Google Scholar
• K. Eshkofti and S. M. Hosseini, “A gradient-enhanced physics-informed neural network (gPINN) scheme for the coupled non-fickian/non-fourierian diffusion-thermoelasticity analysis: A novel gPINN structure,” Eng. Appl. Artif. Intell., vol. 126, pp. 106908, Nov. 2023. DOI: 10.1016/j.engappai.2023.106908.
   Web of Science ®Google Scholar
• S. Shi, D. Liu and Z. Huo, “Simulation of thermoelastic coupling in silicon single crystal growth based on alternate two-stage physics-informed neural network,” Eng. Appl. Artif. Intell., vol. 123, pp. 106468, Aug. 2023. DOI: 10.1016/j.engappai.2023.106468.
   Web of Science ®Google Scholar
• C. Wu, M. Zhu, Q. Tan, Y. Kartha and L. Lu, “A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks,” Comput. Methods Appl. Mech. Eng., vol. 403, pp. 115671, Jan. 2023. DOI: 10.1016/j.cma.2022.115671.
   Web of Science ®Google Scholar
• S. Wang, Y. Teng and P. Perdikaris, “Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks,” SIAM J. Sci. Comput., vol. 43, no. 5, pp. A3055–A3081, Jan. 2021. DOI: 10.1137/20M1318043.
   Web of Science ®Google Scholar
• M. Bachher, N. Sarkar and A. Lahiri, “Fractional order thermoelastic interactions in an infinite porous material due to distributed time-dependent heat sources,” Meccanica, vol. 50, no. 8, pp. 2167–2178, Aug. 2015. DOI: 10.1007/s11012-015-0152-x.
   Web of Science ®Google Scholar
• M. A. Nabian, R. J. Gladstone and H. Meidani, “Efficient training of physics‐informed neural networks via importance sampling,” Comput. Aided Civil Eng., vol. 36, no. 8, pp. 962–977, Aug. 2021. DOI: 10.1111/mice.12685.
   Web of Science ®Google Scholar