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Conditional physics informed neural networks. (English) Zbl 1485.65086

Summary: We introduce conditional PINNs (physics informed neural networks) for estimating the solution of classes of eigenvalue problems. The concept of PINNs is expanded to learn not only the solution of one particular differential equation but the solutions to a class of problems. We demonstrate this idea by estimating the coercive field of permanent magnets which depends on the width and strength of local defects. When the neural network incorporates the physics of magnetization reversal, training can be achieved in an unsupervised way. There is no need to generate labeled training data. The presented test cases have been rigorously studied in the past. Thus, a detailed and easy comparison with analytical solutions is made. We show that a single deep neural network can learn the solution of partial differential equations for an entire class of problems. The method is demonstrated for the computation of the nucleation field related to defects in magnetic materials, which is an important problem in classical micromagnetics. We show that a single neural network can predict the nucleation field depending on the properties of the defect such as the defect width and its local intrinsic magnetic properties.

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
68T07 Artificial neural networks and deep learning
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References:

[1] Khan, A.; Ghorbanian, V.; Lowther, D., Deep learning for magnetic field estimation, IEEE Trans Magn, 55, 6, 1-4 (2019)
[2] Kim, B.; Azevedo, V. C.; Thuerey, N.; Kim, T.; Gross, M.; Solenthaler, B., Deep fluids: A generative network for parameterized fluid simulations, Comput Graph Forum, 38, 2, 59-70 (2019)
[3] Kovacs, A.; Fischbacher, J.; Oezelt, H.; Gusenbauer, M.; Exl, L.; Bruckner, F.; Suess, D.; Schrefl, T., Learning magnetization dynamics, J Magn Magn Mater, 491, Article 165548 pp. (2019)
[4] Raissi, M.; Perdikaris, P.; Karniadakis, G. E., 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 (2019) · Zbl 1415.68175
[5] Koryagin, A.; Khudorozkov, R.; Tsimfer, S., Pydens: A python framework for solving differential equations with neural networks (2019), arXiv preprint arXiv:1909.11544
[6] Kharazmi, E.; Zhang, Z.; Karniadakis, G. E., Variational physics-informed neural networks for solving partial differential equations (2019), arXiv preprint arXiv:1912.00873
[7] E, W.; Yu, B., The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems, Commun Math Stat, 6, 1, 1-12 (2018) · Zbl 1392.35306
[8] Hennigh, O.; Narasimhan, S.; Nabian, M. A.; Subramaniam, A.; Tangsali, K.; Rietmann, M.; Ferrandis, J.d. A.; Byeon, W.; Fang, Z.; Choudhry, S., NVIDIA SimNet^TM: An AI-Accelerated multi-physics simulation framework (2020), arXiv preprint arXiv:2012.07938
[9] Brown, W. F., Micromagnetics (1963), Interscience Publishers · Zbl 0129.23401
[10] Brown Jr, W. F., Criterion for uniform micromagnetization, Phys Rev, 105, 5, 1479 (1957)
[11] Kondorsky, E., On the stability of certain magnetic modes in fine ferromagnetic particles, IEEE Trans Magn, 15, 5, 1209-1214 (1979)
[12] Fredkin, D.; Koehler, T., Numerical micromagnetics by the finite element method, IEEE Trans Magn, 23, 5, 3385-3387 (1987)
[13] Schrefl, T.; Fischer, R.; Fidler, J.; Kronmüller, H., Two-and three-dimensional calculation of remanence enhancement of rare-earth based composite magnets, J Appl Phys, 76, 10, 7053-7058 (1994)
[14] Kantorovich, L. V.; Krylov, V. I., Approximate methods of higher analysis, Interscience (1958) · Zbl 0083.35301
[15] Komzsik, L., Applied calculus of variations for engineers (2019), CRC Press
[16] He, H.; Pathak, J., An unsupervised learning approach to solving heat equations on chip based on auto encoder and image gradient (2020), arXiv preprint arXiv:2007.09684
[17] Haghighat, E.; Juanes, R., Sciann: A keras/TensorFlow wrapper for scientific computations and physics-informed deep learning using artificial neural networks, Comput Methods Appl Mech Engrg, 373, Article 113552 pp. (2021) · Zbl 1506.65251
[18] Lu, L.; Jin, P.; Karniadakis, G. E., Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators (2019), arXiv preprint arXiv:1910.03193
[19] Chen, T.; Chen, H., 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 (1995)
[20] Wang, S.; Wang, H.; Perdikaris, P., Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets (2021), arXiv preprint arXiv:2103.10974
[21] Henrot, A., Extremum problems for eigenvalues of elliptic operators (2006), Springer Science & Business Media · Zbl 1109.35081
[22] Gould, S. H., Variational methods for eigenvalue problems: An introduction to the methods of rayleigh, Ritz, Weinstein, and Aronszajn (2012), Courier Corporation · Zbl 0077.09603
[23] Skomski, R., Exact nucleation modes in arrays of magnetic particles, J. Appl Phys, 91, 10, 7053-7055 (2002)
[24] Caflisch, R. E., Monte carlo and quasi-monte carlo methods, Acta Numer, 1998, 1-49 (1998) · Zbl 0949.65003
[25] Head, T.; MechCoder, G. L., Scikit-optimize/scikit-optimize: v0. 5.2. 2018 (2018)
[26] Lu, L.; Meng, X.; Mao, Z.; Karniadakis, G. E., Deepxde: A deep learning library for solving differential equations, SIAM Rev, 63, 1, 208-228 (2021) · Zbl 1459.65002
[27] Kronmüller, H., Theory of nucleation fields in inhomogeneous ferromagnets, Phys Status Solidi (B), 144, 1, 385-396 (1987)
[28] Chollet, F., Deep learning with python (2018), Manning New York
[29] Nocedal, J.; Wright, S., Numerical optimization (2006), Springer Science & Business Media · Zbl 1104.65059
[30] Goodfellow, I.; Bengio, Y.; Courville, A., Deep learning (2016), MIT Press · Zbl 1373.68009
[31] Aharoni, A.; Shtrikman, S., Magnetization curve of the infinite cylinder, Phys Rev, 109, 5, 1522 (1958) · Zbl 0082.23405
[32] Skomski, R., Nucleation in inhomogeneous permanent magnets, Phys Status Solidi (B), 174, 2, K77-K80 (1992)
[33] Nieber, S.; Kronmüller, H., Nucleation fields in periodic multilayers, Phys Status Solidi (B), 153, 1, 367-375 (1989)
[34] Skomski, R.; Coey, J., Giant energy product in nanostructured two-phase magnets, Phys Rev B, 48, 21, 15812 (1993)
[35] Kronmüller, H.; Fähnle, M., Micromagnetism and the microstructure of ferromagnetic solids (2003), Cambridge University Press
[36] Aharoni, A., Introduction to the theory of ferromagnetism (2000), Clarendon Press
[37] Bance, S.; Oezelt, H.; Schrefl, T.; Ciuta, G.; Dempsey, N. M.; Givord, D.; Winklhofer, M.; Hrkac, G.; Zimanyi, G.; Gutfleisch, O., Influence of defect thickness on the angular dependence of coercivity in rare-earth permanent magnets, Appl Phys Lett, 104, 18, Article 182408 pp. (2014)
[38] Hirosawa, S., Development of industrial nanocomposite permanent magnets: a review, Trans Magnet Soc Japan, 4, 4-1, 101-112 (2004)
[39] Dong, S.-H., Factorization method in quantum mechanics (2007), Springer Science & Business Media · Zbl 1130.81001
[40] Landau, L.; Lifshitz, E., Quantum mechanics: Non-relativistic theory (1977), Pergamon · Zbl 0178.57901
[41] Skomski, R.; Coey, J., Permanent magnetism (1999), Institute of Physics Pub
[42] Skomski, R.; Coey, J., Exchange coupling and energy product in random two-phase aligned magnets, IEEE Trans Magn, 30, 2, 607-609 (1994)
[43] Goto, E.; Hayashi, N.; Miyashita, T.; Nakagawa, K., Magnetization and switching characteristics of composite thin magnetic films, J Appl Phys, 36, 9, 2951-2958 (1965)
[44] Schiff, L., Quantum mechanics (1955), McGraw-Hill · Zbl 0068.40202
[45] Kingma, D. P.; Ba, J., Adam: A method for stochastic optimization (2014), arXiv preprint arXiv:1412.6980
[46] Virtanen, P.; Gommers, R.; Oliphant, T. E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; van der Walt, S. J.; Brett, M.; Wilson, J.; Millman, K. J.; Mayorov, N.; Nelson, A. R.; Jones, E.; Kern, R.; Larson, E.; Carey, C. J.; Polat, I.; Feng, Y.; Moore, E. W.; VanderPlas, J.; Laxalde, D.; Perktold, J.; Cimrman, R.; Henriksen, I.; Quintero, E.; Harris, C. R.; Archibald, A. M.; Ribeiro, A. H.; Pedregosa, F.; van Mulbregt, P.; SciPy 1.0 Contributors, SciPy 1.0: Fundamental algorithms for scientific computing in python, Nature Methods, 17, 261-272 (2020)
[47] Ito, M.; Yano, M.; Sakuma, N.; Kishimoto, H.; Manabe, A.; Shoji, T.; Kato, A.; Dempsey, N.; Givord, D.; Zimanyi, G., Coercivity enhancement in Ce-Fe-B based magnets by core-shell grain structuring, Aip Advances, 6, 5, Article 056029 pp. (2016)
[48] Skomski, R.; Manchanda, P.; Takeuchi, I.; Cui, J., Geometry dependence of magnetization reversal in nanocomposite alloys, JOM, 66, 7, 1144-1150 (2014)
[49] Balachandran, P. V., Machine learning guided design of functional materials with targeted properties, Comput Mater Sci, 164, 82-90 (2019)
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