Abstract
Physics-informed neural networks learn not by example, but by checking physical patterns in a limited set of sampling points. Fully connected deep neural networks, implemented in deep learning libraries, for example, TensorFlow, are usually used as physics-informed neural networks for solving partial differential equations. An important role in the popularity of these libraries is played by the automatic differentiation implemented in them and modern learning algorithms. It is proposed to use radial basis functions networks as physics-informed neural networks, which are distinguished by a simple structure and the ability to adjust not only linear, but also nonlinear parameters. The authors have developed a fast algorithm for the Levenberg-Marquardt method for learning radial basis functions networks. Extensions of the TensorFlow library have been developed to implement the Levenberg-Marquardt algorithm and radial basis functions networks. The model problems solution has shown the advantages of using the radial basis functions networks implemented in TensorFlow as physics-informed neural networks.
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References
Lagaris, E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9(5), 987–1000 (1998)
Yadav, N., Yadav, A., Kumar, M.: An Introduction to Neural Network Methods for Differential Equations. Springer, Dordrecht (2015). https://doi.org/10.1007/978-94-017-9816-7
Cybenko, G.: Approximation by superposition of a sigmoidal function. Math. Control Signals Syst. 2, 303–314 (1989). https://doi.org/10.1007/BF02551274
Hanin, B.: Universal function approximation by deep neural nets with bounded width and ReLU activations (2017). https://arxiv.org/abs/1708.02691v2
Bavdin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res. 18(1), 1–43 (2018)
Raschka, S., Mirjalili, V.: Python Machine Learning: Machine Learning and Deep Learning with Python, Scikit-learn, and TensorFlow 2. Packt Publishing, Birmingham (2019)
Raissia, M., Perdikarisb, P., Karniadakisa, 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)
Tarkhov, D., Vasilyev, A.: Semi-Empirical Neural Network Modeling and Digital Twins Development. Academic Press, Cambridge (2019)
Park, J., Sandberg, I.W.: Universal approximation using radial-basis-function networks. Neural Comput. 3(2), 246–257 (1991)
Park, J., Sandberg, I.W.: Approximation and radial-basis-function networks. Neural Comput. 5(2), 305–316 (1993)
Gorbachenko, V.I., Zhukov, M.V.: Solving boundary value problems of mathematical physics using radial basis function networks. Comput. Math. Math. Phys. 57(1), 145–155 (2017). https://doi.org/10.1134/S0965542517010079
Gorbachenko, V.I., Alqezweeni, M.M.: Learning radial basis functions networks in solving boundary value problems. In: 2019 International Russian Automation Conference — RusAtoCon, Sochi, Russia, 8–14 September, pp. 1–6 (2019)
Gorbachenko, V.I., Alqezweeni, M.M.: Modeling of objects with distributed parameters on neural networks. Models Syst. Netw. Econ. Technol. Nat. Soc. 4(32), 50–64 (2019). (in Russian)
Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. https://arxiv.org/abs/1412.6980
Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2(2), 164–168 (1944)
Marquardt, D.: An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11, 431–441 (1963)
Transtrum, M.K., Sethna, J.P.: Improvements to the Levenberg-Marquardt algorithm for nonlinear least-squares minimization. https://arxiv.org/abs/1201.5885
Safarpoor, M., Takhtabnoos, F., Shirzadi, A.: A localized RBF-MLPG method and its application to elliptic PDEs. Eng. Comput. 36(1), 171–183 (2019). https://doi.org/10.1007/s00366-018-00692-y
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Alqezweeni, M.M., Glumskov, R.A., Gorbachenko, V.I., Stenkin, D.A. (2022). Solving Partial Differential Equations on Radial Basis Functions Networks and on Fully Connected Deep Neural Networks. In: Sharma, H., Vyas, V.K., Pandey, R.K., Prasad, M. (eds) Proceedings of the International Conference on Intelligent Vision and Computing (ICIVC 2021). ICIVC 2021. Proceedings in Adaptation, Learning and Optimization, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-030-97196-0_20
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