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SIKIŞTIRILAMAZ ISIL TAŞINIM PROBLEMLERİNİN FİZİKLE ÖĞRENEN YAPAY SİNİR AĞLARI İLE ÇÖZÜMÜ

Yıl 2022, Cilt: 42 Sayı: 2, 221 - 232, 31.10.2022
https://doi.org/10.47480/isibted.1194992

Öz

Fizikle öğrenen yapay sinir ağları (PINN'ler), etkinlikleri ve karmaşık ağlar oluşturmadan problemlerin üstesinden gelme yetenekleri nedeniyle son yıllarda mühendislik problemlerinde dikkat çekmiştir. PINN'ler, koruma yasalarında diferansiyel operatörleri değerlendirmek için otomatik türevlenmeyi kullanır ve bu nedenle bir ayrıklaştırma şemasına ihtiyaç duymaz. Bu yeteneği kullanarak, PINN'ler herhangi bir eğitim verisi olmadan kayıp fonksiyonunda geçerli fizik yasalarını karşılar. Bu çalışmada, gerçek uygulamalar ve karşılaştırılabilir sayısal veya analitik sonuçlara sahip problemler de dahil olmak üzere çeşitli sıkıştırılamaz ısıl taşınım problemlerini çözüyoruz. Modelin performansını değerlendirmek için analitik çözümü olan bir kanal problemini çözüyoruz. Model, bireysel kayıp terimlerinin ağırlıklarına büyük ölçüde bağımlıdır. Alan içindeki akış çok karmaşık değilse, sınır koşulu kaybının ağırlığının arttırılması doğruluğu artırır. Farklı tipteki ağların performansını ve Neumann sınır koşullarını yakalama yeteneğini değerlendirmek için, sınırlardaki sıcaklık gradyanlarından dolayı akışın meydana geldiği kapalı bir muhafazada bir termal konveksiyon problemini çözüyoruz. Basit tam bağlantılı ağ, termal konveksiyon problemlerinde iyi performans gösterir ve çok ölçekli davranış olmadığından ağda Fourier dönüşümüne ihtiyacımız yoktur. Son olarak, endüstriyel uygulamaları güç elektroniğine benzeyen sabit ve kararsız kısmen bloke kanal problemlerini ele alıyoruz ve yöntemin geçici problemlere de uygulanabileceğini gösteriyoruz.

Kaynakça

  • Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., Kudlur, M., Levenberg, J., Monga, R., Moore, S., Murray, D. G., Steiner, B., Tucker, P., Vasudevan, V., Warden, P., Wicke, M., Yu, Y., and Zheng, X. (2016). TensorFlow: A system for large-scale machine learning. In 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), pages 265–283.
  • Bairi, A., Zarco-Pernia, E., and De Maria, J.-M. G. (2014). A review on natural convection in enclosures for engineering applications. the particular case of the parallelogrammic diode cavity. Applied Thermal Engineering, 63(1):304–322.
  • Baydin, A. G., Pearlmutter, B. A., Radul, A. A., and Siskind, J. M. (2017). Automatic differentiation in machine learning: a survey. The Journal of Machine Learning Research, 18(1):5595–5637.
  • Cai, S., Mao, Z., Wang, Z., Yin, M., and Karniadakis, G. E. (2022). Physics-informed neural networks (PINNs) for fluid mechanics: a review. Acta Mechanica Sinica.
  • Cai, S., Wang, Z., Wang, S., Perdikaris, P., and Karniadakis, G. (2021). Physics-informed neural networks (PINNs) for heat transfer problems. Journal of Heat Transfer, 143.
  • De Vahl Davis, G. (1983). Natural convection of air in a square cavity: A bench mark numerical solution. International Journal for Numerical Methods in Fluids, 3(3).
  • Esmaeilzadeh, S., Azizzadenesheli, K., Kashinath, K., Mustafa, M., Tchelepi, H. A., Marcus, P., Prabhat, M., Anandkumar, A., et al. (2020). Meshfreeflownet: A physics-constrained deep continuous space-time super-resolution framework. In SC20: International Conference for High Performance Computing, Networking, Storage and Analysis, pages 1–15. IEEE. Fathony, R., Sahu, A. K., Willmott, D., and Kolter, J. Z. (2021). Multiplicative filter networks. In International Conference on Learning Representations.
  • Glorot, X. and Bengio, Y. (2010). Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pages 249–256. JMLR Workshop and Conference Proceedings. ISSN: 1938-7228.
  • Habchi, S. and Acharya, S. (1986). Laminar mixed convection in a partially blocked, vertical channel. International Journal of Heat and Mass Transfer, 29(11):1711–1722.
  • Hennigh, O., Narasimhan, S., Nabian, M. A., Subramaniam, A., Tangsali, K., Fang, Z., Rietmann, M., Byeon, W., and Choudhry, S. (2021). Nvidia SimNet™: An ai-accelerated multi-physics simulation framework. In International Conference on Computational Science, pages 447–461. Springer.
  • Hossain, M. Z., Cantwell, C. D., and Sherwin, S. J. (2021). A spectral/hp element method for thermal convection. International Journal for Numerical Methods in Fluids, 93(7):2380–2395.
  • Jacot, A., Gabriel, F., and Hongler, C. (2018). Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems, 31.
  • Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426:109951.
  • Karakus, A. (2022). An accelerated nodal discontinuous Galerkin method for thermal convection on unstructured meshes: Formulation and Validation. Journal of Thermal Science and Technology 42(1), 91-100.
  • Karakus, A., Chalmers, N., Swirydowicz, K., and Warburton, T. (2019). A GPU accelerated discontinuous Galerkin incompressible flow solver. Journal of Computational Physics, 390:380–404.
  • Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., and Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics
  • Kingma, D. P. and Ba, J. (2017). Adam: A method for stochastic optimization. arXiv:1412.6980 [cs].
  • Lagaris, I., Likas, A., and Fotiadis, D. (1998). Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9(5):987–1000.
  • Lee, H. and Kang, I. S. (1990). Neural algorithm for solving differential equations. Journal of Computational Physics, 91(1):110–131.
  • Lu, L., Meng, X., Mao, Z., and Karniadakis, G. E. (2021). DeepXDE: A deep learning library for solving differential equations. SIAM Review, 63(1):208–228.
  • Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., and Chintala, S. (2019). PyTorch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc.
  • Rahaman, N., Baratin, A., Arpit, D., Draxler, F., Lin, M., Hamprecht, F., Bengio, Y., and Courville, A. (2019). On the spectral bias of neural networks. In International Conference on Machine Learning, pages 5301–5310. PMLR.
  • Raissi, M., Perdikaris, P., and Karniadakis, G. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707.
  • Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2017a). Inferring solutions of differential equations using noisy multi-fidelity data. Journal of Computational Physics, 335:736–746.
  • Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2017b). Machine learning of linear differential equations using Gaussian processes. Journal of Computational Physics, 348:683–693.
  • Rao, C., Sun, H., and Liu, Y. (2020). Physics-informed deep learning for incompressible laminar flows. Theoretical and Applied Mechanics Letters, 10(3):207–212.
  • Sirignano, J. and Spiliopoulos, K. (2018). DGM: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 375:1339–1364.
  • Srivastava, R. K., Greff, K., and Schmidhuber, J. (2015). Training very deep networks. In Advances in Neural Information Processing Systems, volume 28. Curran Associates, Inc.
  • Stokos, K., Vrahliotis, S., Pappou, T., and Tsangaris, S. (2015). Development and validation of an incompressible Navier-Stokes solver including convective heat transfer. International Journal of Numerical Methods for Heat & Fluid Flow, 25(4):861–886.
  • Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., and Ng, R. (2020). Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems, 33:7537–7547.
  • Tang, L. Q. and Tsang, T. T. (1993). A least-squares finite element method for time-dependent incompressible flows with thermal convection. International Journal for Numerical Methods in Fluids, 17(4):271–289.
  • Wang, S., Teng, Y., and Perdikaris, P. (2021a). Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 43(5):A3055–A3081.
  • Wang, S., Wang, H., and Perdikaris, P. (2021b). On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 384:113938.
  • Willard, J., Jia, X., Xu, S., Steinbach, M., and Kumar, V. (2020). Integrating physics-based modeling with machine learning: A survey. arXiv preprint arXiv:2003.04919, 1(1):1–34.
  • Wu, H.-W. and Perng, S.-W. (1999). Effect of an oblique plate on the heat transfer enhancement of mixed convection over heated blocks in a horizontal channel. International Journal of Heat and Mass Transfer, 42(7):1217–1235.
  • Zubov, K., McCarthy, Z., Ma, Y., Calisto, F., Pagliarino, V., Azeglio, S., Bottero, L., Lujan, E., Sulzer, V., Bharambe, A., Vinchhi, N., Balakrishnan, K., Upadhyay, D., and Rackauckas, C. (2021). NeuralPDE: Automating physics-informed neural networks (PINNs) with error approximations. arXiv:2107.09443.

PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS

Yıl 2022, Cilt: 42 Sayı: 2, 221 - 232, 31.10.2022
https://doi.org/10.47480/isibted.1194992

Öz

Physics-informed neural networks (PINNs) have drawn attention in recent years in engineering problems due to their effectiveness and ability to tackle problems without generating complex meshes. PINNs use automatic differentiation to evaluate differential operators in conservation laws and hence do not need a discretization scheme. Using this ability, PINNs satisfy governing laws of physics in the loss function without any training data. In this work, we solve various incompressible thermal convection problems, and compare the results with numerical or analytical results. To evaluate the accuracy of the model we solve a channel problem with an analytical solution. The model is highly dependent on the weights of individual loss terms. Increasing the weight of boundary condition loss improves the accuracy if the flow inside the domain is not complicated. To assess the performance of different type of networks and ability to capture the Neumann boundary conditions, we solve a thermal convection problem in a closed enclosure in which the flow occurs due to the temperature gradients on the boundaries. The simple fully connected network performs well in thermal convection problems, and we do not need a Fourier mapping in the network since there is no multiscale behavior. Lastly, we consider steady and unsteady partially blocked channel problems resembling industrial applications to power electronics and show that the method can be applied to transient problems as well.

Kaynakça

  • Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., Kudlur, M., Levenberg, J., Monga, R., Moore, S., Murray, D. G., Steiner, B., Tucker, P., Vasudevan, V., Warden, P., Wicke, M., Yu, Y., and Zheng, X. (2016). TensorFlow: A system for large-scale machine learning. In 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), pages 265–283.
  • Bairi, A., Zarco-Pernia, E., and De Maria, J.-M. G. (2014). A review on natural convection in enclosures for engineering applications. the particular case of the parallelogrammic diode cavity. Applied Thermal Engineering, 63(1):304–322.
  • Baydin, A. G., Pearlmutter, B. A., Radul, A. A., and Siskind, J. M. (2017). Automatic differentiation in machine learning: a survey. The Journal of Machine Learning Research, 18(1):5595–5637.
  • Cai, S., Mao, Z., Wang, Z., Yin, M., and Karniadakis, G. E. (2022). Physics-informed neural networks (PINNs) for fluid mechanics: a review. Acta Mechanica Sinica.
  • Cai, S., Wang, Z., Wang, S., Perdikaris, P., and Karniadakis, G. (2021). Physics-informed neural networks (PINNs) for heat transfer problems. Journal of Heat Transfer, 143.
  • De Vahl Davis, G. (1983). Natural convection of air in a square cavity: A bench mark numerical solution. International Journal for Numerical Methods in Fluids, 3(3).
  • Esmaeilzadeh, S., Azizzadenesheli, K., Kashinath, K., Mustafa, M., Tchelepi, H. A., Marcus, P., Prabhat, M., Anandkumar, A., et al. (2020). Meshfreeflownet: A physics-constrained deep continuous space-time super-resolution framework. In SC20: International Conference for High Performance Computing, Networking, Storage and Analysis, pages 1–15. IEEE. Fathony, R., Sahu, A. K., Willmott, D., and Kolter, J. Z. (2021). Multiplicative filter networks. In International Conference on Learning Representations.
  • Glorot, X. and Bengio, Y. (2010). Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pages 249–256. JMLR Workshop and Conference Proceedings. ISSN: 1938-7228.
  • Habchi, S. and Acharya, S. (1986). Laminar mixed convection in a partially blocked, vertical channel. International Journal of Heat and Mass Transfer, 29(11):1711–1722.
  • Hennigh, O., Narasimhan, S., Nabian, M. A., Subramaniam, A., Tangsali, K., Fang, Z., Rietmann, M., Byeon, W., and Choudhry, S. (2021). Nvidia SimNet™: An ai-accelerated multi-physics simulation framework. In International Conference on Computational Science, pages 447–461. Springer.
  • Hossain, M. Z., Cantwell, C. D., and Sherwin, S. J. (2021). A spectral/hp element method for thermal convection. International Journal for Numerical Methods in Fluids, 93(7):2380–2395.
  • Jacot, A., Gabriel, F., and Hongler, C. (2018). Neural tangent kernel: Convergence and generalization in neural networks. Advances in neural information processing systems, 31.
  • Jin, X., Cai, S., Li, H., and Karniadakis, G. E. (2021). NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426:109951.
  • Karakus, A. (2022). An accelerated nodal discontinuous Galerkin method for thermal convection on unstructured meshes: Formulation and Validation. Journal of Thermal Science and Technology 42(1), 91-100.
  • Karakus, A., Chalmers, N., Swirydowicz, K., and Warburton, T. (2019). A GPU accelerated discontinuous Galerkin incompressible flow solver. Journal of Computational Physics, 390:380–404.
  • Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., and Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics
  • Kingma, D. P. and Ba, J. (2017). Adam: A method for stochastic optimization. arXiv:1412.6980 [cs].
  • Lagaris, I., Likas, A., and Fotiadis, D. (1998). Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9(5):987–1000.
  • Lee, H. and Kang, I. S. (1990). Neural algorithm for solving differential equations. Journal of Computational Physics, 91(1):110–131.
  • Lu, L., Meng, X., Mao, Z., and Karniadakis, G. E. (2021). DeepXDE: A deep learning library for solving differential equations. SIAM Review, 63(1):208–228.
  • Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., and Chintala, S. (2019). PyTorch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc.
  • Rahaman, N., Baratin, A., Arpit, D., Draxler, F., Lin, M., Hamprecht, F., Bengio, Y., and Courville, A. (2019). On the spectral bias of neural networks. In International Conference on Machine Learning, pages 5301–5310. PMLR.
  • Raissi, M., Perdikaris, P., and Karniadakis, G. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707.
  • Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2017a). Inferring solutions of differential equations using noisy multi-fidelity data. Journal of Computational Physics, 335:736–746.
  • Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2017b). Machine learning of linear differential equations using Gaussian processes. Journal of Computational Physics, 348:683–693.
  • Rao, C., Sun, H., and Liu, Y. (2020). Physics-informed deep learning for incompressible laminar flows. Theoretical and Applied Mechanics Letters, 10(3):207–212.
  • Sirignano, J. and Spiliopoulos, K. (2018). DGM: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 375:1339–1364.
  • Srivastava, R. K., Greff, K., and Schmidhuber, J. (2015). Training very deep networks. In Advances in Neural Information Processing Systems, volume 28. Curran Associates, Inc.
  • Stokos, K., Vrahliotis, S., Pappou, T., and Tsangaris, S. (2015). Development and validation of an incompressible Navier-Stokes solver including convective heat transfer. International Journal of Numerical Methods for Heat & Fluid Flow, 25(4):861–886.
  • Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., and Ng, R. (2020). Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems, 33:7537–7547.
  • Tang, L. Q. and Tsang, T. T. (1993). A least-squares finite element method for time-dependent incompressible flows with thermal convection. International Journal for Numerical Methods in Fluids, 17(4):271–289.
  • Wang, S., Teng, Y., and Perdikaris, P. (2021a). Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 43(5):A3055–A3081.
  • Wang, S., Wang, H., and Perdikaris, P. (2021b). On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 384:113938.
  • Willard, J., Jia, X., Xu, S., Steinbach, M., and Kumar, V. (2020). Integrating physics-based modeling with machine learning: A survey. arXiv preprint arXiv:2003.04919, 1(1):1–34.
  • Wu, H.-W. and Perng, S.-W. (1999). Effect of an oblique plate on the heat transfer enhancement of mixed convection over heated blocks in a horizontal channel. International Journal of Heat and Mass Transfer, 42(7):1217–1235.
  • Zubov, K., McCarthy, Z., Ma, Y., Calisto, F., Pagliarino, V., Azeglio, S., Bottero, L., Lujan, E., Sulzer, V., Bharambe, A., Vinchhi, N., Balakrishnan, K., Upadhyay, D., and Rackauckas, C. (2021). NeuralPDE: Automating physics-informed neural networks (PINNs) with error approximations. arXiv:2107.09443.
Toplam 36 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Makine Mühendisliği
Bölüm Araştırma Makalesi
Yazarlar

Atakan Aygun Bu kişi benim

Ali Karakus Bu kişi benim

Yayımlanma Tarihi 31 Ekim 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 42 Sayı: 2

Kaynak Göster

APA Aygun, A., & Karakus, A. (2022). PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS. Isı Bilimi Ve Tekniği Dergisi, 42(2), 221-232. https://doi.org/10.47480/isibted.1194992
AMA Aygun A, Karakus A. PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS. Isı Bilimi ve Tekniği Dergisi. Ekim 2022;42(2):221-232. doi:10.47480/isibted.1194992
Chicago Aygun, Atakan, ve Ali Karakus. “PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS”. Isı Bilimi Ve Tekniği Dergisi 42, sy. 2 (Ekim 2022): 221-32. https://doi.org/10.47480/isibted.1194992.
EndNote Aygun A, Karakus A (01 Ekim 2022) PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS. Isı Bilimi ve Tekniği Dergisi 42 2 221–232.
IEEE A. Aygun ve A. Karakus, “PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS”, Isı Bilimi ve Tekniği Dergisi, c. 42, sy. 2, ss. 221–232, 2022, doi: 10.47480/isibted.1194992.
ISNAD Aygun, Atakan - Karakus, Ali. “PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS”. Isı Bilimi ve Tekniği Dergisi 42/2 (Ekim 2022), 221-232. https://doi.org/10.47480/isibted.1194992.
JAMA Aygun A, Karakus A. PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS. Isı Bilimi ve Tekniği Dergisi. 2022;42:221–232.
MLA Aygun, Atakan ve Ali Karakus. “PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS”. Isı Bilimi Ve Tekniği Dergisi, c. 42, sy. 2, 2022, ss. 221-32, doi:10.47480/isibted.1194992.
Vancouver Aygun A, Karakus A. PHYSICS INFORMED NEURAL NETWORKS FOR TWO DIMENSIONAL INCOMPRESSIBLE THERMAL CONVECTION PROBLEMS. Isı Bilimi ve Tekniği Dergisi. 2022;42(2):221-32.